1939:"is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate assumptions about the existence of objects outside the "data" and to replace them by constructions on the basis of these data. The "data" are to understand in a relative sense here; i.e. in our case as logic without the assumption of the existence of classes and concepts]. The result has been in this case essentially negative; i.e. the classes and concepts introduced in this way do not have all the properties required from their use in mathematics. . . . All this is only a verification of the view defended above that logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away" (p. 132)
966:"71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. . . logically; the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal."(1903:69)
1210:. The reason is that, in Russell's detailed analysis, if a unit class becomes an entity in its own right, then it too can be an element in its own proposition; this causes the proposition to become "impredicative" and result in a "vicious circle". Rather, he states: "We saw in Chapter II that a cardinal number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes,
5238:
1926:
the first part that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by
Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics sees to rely on its inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false than that classical mathematics is false" (all quotes Gödel 1944:127).
887:. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words, unit, individual, and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. is in some sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term; and to deny that such and such a thing is a term must always be false" (Russell 1903:43)
333:(1928) used the word: "Without comment he used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244, explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik'; Behmann complained about its use in Carnap's manuscript so Carnap proposed the word 'Logizismus', but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately "the spread was mainly due to Carnap, from 1930 onwards." (G-G 2000:502).
685:
ultimate, and not reducible to adjectives of their terms or of the whole which these compose. . . . The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. . . . Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour." (Preface 1903:viii)
1947:"One should take a more conservative course, such as would consist in trying to make the meaning of terms "class" and "concept" clearer, and to set up a consistent theory of classes and concepts as objectively existing entities. This is the course which the actual development of mathematical logic has been taking and which Russell himself has been forced to enter upon in the more constructive parts of his work. Major among the attempts in this direction . . . are the simple
1135:, i.e. one-one correspondence of the elements, and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes; thus each is a class of classes" (Russell 1919:14).
2373:(cf. (26). Definition)). He observes that by establishing these conditions "we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relation to one another . . . by the order-setting transformation φ. . . . With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind." (p. 68)
991:, for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely. In the third place, we wish to define "number" in such a way that infinite numbers may be possible; thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them." (1919:13)
453:
was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography. Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed" (Frege 1879 in van
Heijenoort 1967:5).
1117:, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . .." (1919:184)
1955:, both of which have been successful at least to this extent, that they permit the derivation of modern mathematics and at the same time avoid all known paradoxes . . . ¶ It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others . . .." (p. 140)
625:, accusing both of "subjectivism": "It is part of the task of science to liberate us from arbitrariness, sentiment and habit and to protect us from the subjectivism that already made itself felt in Kronecker's views and, it seems to me, finds its culmination in intuitionism". Hilbert then states that "mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker . . ." (p. 479).
1357:: "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)
3119:
1932:: Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Russell's problem derives from his "constructivistic (or nominalistic") standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction" (p. 128).
1125:(1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.
2975:
921:, and this fact does not (except in exceptional cases) in any way involve the mind of the person who has the belief" (1912:130). If by investigation of the utterance and correspondence with "fact", Russell discovers that Emily is a rabbit, then his utterance is considered "false"; if Emily is a female human (a female "featherless biped" as Russell likes to call humans, following
676:. Note that he asserts that the belief: "Emily is a rabbit" is non-existent, and yet the truth of this non-existent proposition is independent of any knowing mind; if Emily really is a rabbit, the fact of this truth exists whether or not Russell or any other mind is alive or dead, and the relation of Emily to rabbit-hood is "ultimate":
948:". But "if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential." (1909 p. 66)
1895:" The axiom of reducibility is dropped, and it is stated explicitly that all primitive predicates belong to the lowest type and that the only purpose of variables (and evidently also of constants) of higher orders and types is to make it possible to assert more complicated truth-functions of atomic propositions" (Gödel 1944 in
1395:. . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked" (in Mancosu 1998:243).
1183:: Russell defined a new characteristic "hereditary" (cf Frege's 'ancestral'), a property of certain classes with the ability to "inherit" a characteristic from another class (which may be a class of classes) i.e. "A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number
2587:
sweatshirt and mine are different objects generalized by the word "sweatshirt", you have a relation to yours and I have a relation to mine. And
Russell "treated relations on par with other universals" (p. xii). But Gödel is saying that Russell's "no-class" theory denies the numbers the status of "universals".
1050:. "Here instead we presuppose the totality of all properties of cardinal numbers, as existing in logic, prior to the definition of the natural number sequence" (Kleene 1952:44). The problem will appear, even in the finite example presented here, when Russell deals with the unit class (cf. Russell 1903:517).
1770:
are to be classified by their "order", where functions of individuals are of order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defines the "type" of a function's arguments (the function's "inputs") to be their "range of significance", i.e. what are those inputs
1466:
With respect to the philosophy that might underlie these foundations, Gödel considered
Russell's "no-class theory" as embodying a "nominalistic kind of constructivism . . . which might better be called fictionalism" (cf. footnote 1 in Gödel 1944:119) – to be faulty. See more in "Gödel's criticism and
1431:
Mancosu states that
Brouwer concluded that: "the classical laws or principles of logic are part of perceived regularity ; they are derived from the post factum record of mathematical constructions . . . Theoretical logic . . . an empirical science and an application of mathematics" (Brouwer quoted
648:
knowledge, while empiricism would contribute the role of experiential knowledge (induction from experience). Russell would credit Kant with the idea of "a priori" knowledge, but he offers an objection to Kant he deems "fatal": "The facts must always conform to logic and arithmetic. To say that logic
465:
In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions of intuitions of space and time, that I consider it an immediate result from the laws of thought . . . numbers are free creations of the human mind .
452:
consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed I had to bend every effort to keep the chain of inferences free of gaps . . . I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I
387:
In a logicist derivation of the natural numbers and their properties, no "intuition" of number should "sneak in" either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the real numbers, from some chosen "laws of thought" alone, without
1809:
By "predicative", Russell meant that the function must be of an order higher than the "type" of its variable(s). Thus a function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes (type 1) and individuals (type 0), as these are lower types.
1718:
It is as if a rancher were to round up all his livestock (sheep, cows and horses) into three fictitious corrals (one for the sheep, one for the cows, and one for the horses) that are located in his fictitious ranch. What actually exist are the sheep, the cows and the horses (the extensions), but not
1510:
Suppose a librarian wants to index her collection into a single book (call it Ι for "index"). Her index will list all the books and their locations in the library. As it turns out, there are only three books, and these have titles Ά, β, and Γ. To form her index I, she goes out and buys a book of 200
1097:
do not belong to the propositional function aka "class" created by the function. The class itself is not to be regarded as a unitary object in its own right, it exists only as a kind of useful fiction: "We have avoided the decision as to whether a class of things has in any sense an existence as one
908:
of a proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is. It is a characteristic of the terms of a proposition that anyone of them may be replaced by any other entity without our ceasing to have a proposition. Thus
866:
on the rational numbers, rational numbers in turn being derived from the naturals. While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophical difficulties appear in a logicist derivation of the natural numbers, these problems should
756:
The logicism of Frege and
Dedekind is similar to that of Russell, but with differences in the particulars (see Criticisms, below). Overall, the logicist derivations of the natural numbers are different from derivations from, for example, Zermelo's axioms for set theory ('Z'). Whereas, in derivations
713:
How did
Russell arrive in this situation? Gödel observes that Russell is a surprising "realist" with a twist: he cites Russell's 1919:169 "Logic is concerned with the real world just as truly as zoology" (Gödel 1944:120). But he observes that "when he started on a concrete problem, the objects to be
1632:
Russell avoided this problem by declaring a class to be more or a "fiction". By this he meant that a class could designate only those elements that satisfied its propositional function and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a
1152:
when it has at least one member . . . the class which has no members is called the "null class" . . . "α is the null-class" is equivalent to "α does not exist". The question naturally arises whether the null class itself 'exists'? Difficulties related to this question occur in
Russell's 1903 work.
705:"In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x. proves that something is amiss, but what this is I have hitherto failed to discover. (Preface to Russell 1903:vi)"
478:
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language. ¶ That is why it is of the utmost importance to examine attentively the very words we use. My goal
2531:
input x, x is an element of set y if and only if x satisfies the given function Φ.) Note that (i) input x is unrestricted as to the "type" of thing that it can be (it can be a thing, or a class), and (ii) function Φ is unrestricted as well. Pick the following tricky function Φ(x) = ¬(x ε x). (This
1925:
Gödel, in his 1944 work, identifies the place where he considers
Russell's logicism to fail and offers suggestions to rectify the problems. He submits the "vicious circle principle" to re-examination, splitting it into three parts "definable only in terms of", "involving" and "presupposing". It is
1769:
can be treated in his logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicative definitions. This supposition requires the notions of function-"orders" and argument-"types". First, functions (and their classes-as-extensions, i.e. "matrices")
1398:
Hilbert 1931:266-7, like
Bernays, considers there is "something extra-logical" in mathematics: "Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian
321:
Apparently the first (and only) usage by Russell appeared in his 1919: "Russell referred several time to Frege, introducing him as one 'who first succeeded in "logicising" mathematics' (p. 7). Apart from the misrepresentation (which Russell partly rectified by explaining his own view of the
717:
In an observation pertinent to Russell's brand of logicism, Perry remarks that Russell went through three phases of realism: extreme, moderate and constructive (Perry 1997:xxv). In 1903 he was in his extreme phase; by 1905 he would be in his moderate phase. In a few years he would "dispense with
1686:
Gödel 1944:131 observes that "Russell adduces two reasons against the extensional view of classes, namely the existence of (1) the null class, which cannot very well be a collection, and (2) the unit classes, which would have to be identical with their single elements." He suggests that Russell
1412:
that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is
1390:
Bernays 1930–1931 observes that this notion "two things" already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things; it means, simply, "a thing and one more thing. . . . With respect to this simple
709:
Gödel in his 1944 would disagree with the young Russell of 1903 (" allow mathematics to be true") but would probably agree with Russell's statement quoted above ("something is amiss"); Russell's theory had failed to arrive at a satisfactory foundation of mathematics: the result was "essentially
255:
as part of logic. On the face of it, this damages the logicist programme also, albeit only for those already doubtful concerning 'infinitary methods'. Nonetheless, positions deriving from both logicism and from Hilbertian finitism have continued to be propounded since the publication of Gödel's
2262:
The exact quote from Russell 1919 is the following: "It is time now to turn to the considerations which make it necessary to advance beyond the standpoint of Peano, who represents the last perfection of the "arithmetisation" of mathematics, to that of Frege, who first succeeded in "logicising"
1908:
Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy must be considered as unsolved at the present time". Gödel
769:
number axiom exists in the various logicist axiom systems allowing the derivation of the natural numbers. Note that the axioms needed to derive the definition of a number may differ between axiom systems for set theory in any case. For instance, in ZF and ZFC, the axiom of pairing, and hence
909:
we shall say that "Socrates is human" is a proposition having only one term; of the remaining component of the proposition, one is the verb, the other is a predicate.. . . Predicates, then, are concepts, other than verbs, which occur in propositions having only one term or subject." (1903:45)
857:
The importance to the logicist programme of the construction of the natural numbers derives from Russell's contention "That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected" (1919:4). One derivation of the
436:
1952 states it this way: "Leibniz (1666) first conceived of logic as a science containing the ideas and principles underlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematical notions in terms of logical ones, and Peano (1889, 1894–1908) in
2586:
Perry observes that Plato and Russell are "enthusiastic" about "universals", then in the next sentence writes: " 'Nominalists' think that all that particulars really have in common are the words we apply to them" (Perry in his 1997 Introduction to Russell 1912:xi). Perry adds that while your
684:
nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are
437:
expressing mathematical theorems in a logical symbolism" (p. 43); in the previous paragraph he includes Russell and Whitehead as exemplars of the "logicistic school", the other two "foundational" schools being the intuitionistic and the "formalistic or axiomatic school" (p. 43).
1609:
As described above, Both Frege's and Russell's constructions of the natural numbers begin with the formation of equinumerous classes of classes ("bundles"), followed by an assignment of a unique "numeral" to each bundle, and then by the placing of the bundles into an order via a relation
879:
For Russell, collections (classes) are aggregates of "things" specified by proper names, that come about as the result of propositions (assertions of fact about a thing or things). Russell analysed this general notion. He begins with "terms" in sentences, which he analysed as follows:
1109:"When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than
466:. . only through the purely logical process of building up the science of numbers . . . are we prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind" (Dedekind 1887 Dover republication 1963 :31).
2272:
For example, von Neumann 1925 would cite Kronecker as follows: "The denumerable infinite . . . is nothing more the general notion of the positive integer, on which mathematics rests and of which even Kronecker and Brouwer admit that it was "created by God"" (von Neumann 1925
1293:
character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the
420:, through Frege and Peano to Russell: "Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic", whereas Peano "was more interested in its applications within mathematics". But "It was only
1387:: Kleene notes that "the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration" (Kleene 1952:46)
870:
One attempt to construct the natural numbers is summarized by Bernays 1930–1931. But rather than use Bernays' précis, which is incomplete in some details, an attempt at a paraphrase of Russell's construction, incorporating some finite illustrations, is set out below:
2541:
Russell's letter to Frege announcing the "discovery", and Frege's letter back to Russell in sad response, together with commentary, can be found in van Heijenoort 1967:124-128. Zermelo in his 1908 claimed priority to the discovery; cf. footnote 9 on page 191 in van
1677:
This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchy of "types" of classes.
448:"I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with the sole support of those laws of thought that transcend all particulars. My initial step was to attempt to reduce the concept of ordering in a sequence to that of
2386:
Russell refers to such assumptions (there are others) as "primitive propositions" ("pp" as opposed to "axioms" (there are some of those, too). But the reader is never certain whether these pp are axioms/axiom-schemas or construction-devices (like substitution or
2532:
says: Φ(x) is satisfied when x is NOT an element of x)). Because y (a class) is also "unrestricted" we can plug "y" in as input: ∃y. This says that "there exists a class y that is an element of itself only if it is NOT and element of itself. That is the paradox.
932:"The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate" (1903 p. 55). Classes can be specified by extension (listing their members) or by intension, i.e. by a "propositional function" such as "
1714:"Thus the final conclusion is, that the correct theory of classes is even more extensional than that of Chapter VI; that the class as many is the only object always defined by a propositional function, and that this is adequate for formal purposes" (1903:518).
322:
role of arithmetic in mathematics), the passage is notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used the word again, so that 'logicism' did not emerge until the later 1920s" (G-G 2002:434).
287:
assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system.
2445:"If the predicates are partitioned into classes with respect to equinumerosity in such a way that all predicates of a class are equinumerous to one another and predicates of different classes are not equinumerous, then each such class represents the
898:; the former are the terms indicated by proper names, the latter those indicated by all other words . . . Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs" (1903:44).
561:
every object of our thought"; we humans use symbols to discuss these "things" of our minds; "A thing is completely determined by all that can be affirmed or thought concerning it" (p. 44). In a subsequent paragraph Dedekind discusses what a "system
808:
this way to the natural numbers, and these numbers end up all of the same "type" – as classes of classes – whereas in some set theoretical constructions – for instance the von Neumann and the Zermelo numerals – each number has its predecessor as a
353:
begins with a very reduced set of marks (non-arithmetic symbols), a few "logical" axioms that embody the "laws of thought", and rules of inference that dictate how the marks are to be assembled and manipulated – for instance substitution and
2527:, Kluwer Academic Publishers, Dordrecht, The Netherlands, ISBN. They give a demonstration of how to create the paradox (pages 1–2), as follows: Define an aggregate/class/set y this way: ∃y∀x. (This says: There exists a class y such that for
1511:
blank pages and labels it "I". Now she has four books: I, Ά, β, and Γ. Her task is not difficult. When completed, the contents of her index I are 4 pages, each with a unique title and unique location (each entry abbreviated as Title.Location
1427:
modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable." (Hilbert 1931 in Mancoso 1998:267). The incompleteness theorems arguably constitute a similar obstacle for Hilbertian finitism.
644:. To begin with, "Russell was a realist about two key issues: universals and material objects" (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell the observer. Rationalism would contribute the notion of
912:
Suppose one were to point to an object and say: "This object in front of me named 'Emily' is a woman." This is a proposition, an assertion of the speaker's belief, which is to be tested against the "facts" of the outer world: "Minds do not
653:
knowledge that we possess is "about things, and not merely about thoughts" (1912:89). And in this Russell's epistemology seems different from that of Dedekind's belief that "numbers are free creations of the human mind" (Dedekind 1887:31)
388:
any tacit assumptions of "before" and "after" or "less" and "more" or to the point: "successor" and "predecessor". Gödel 1944 summarized Russell's logicistic "constructions", when compared to "constructions" in the foundational systems of
1073:
does not belong to the function, but the two together make a whole (ib. p. 6 " (Russell 1903:505).) For example, a particular "unity" could be given a name; suppose a family Fα has the children with the names Annie, Barbie and Charles:
1884:, and real numbers generally, can no longer be adequately dealt with. . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom" (
1447:"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it so greatly lacking in formal precision in the foundations (contained in *1–*21 of
1702:"The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not a single term, but essentially many terms; this is the case with all propositions asserting numbers other than 0 and 1" (1903:516).
2330:"It must be admitted . . . that logical principles are known to us, and cannot be themselves proved by experience, since all proof presupposes them. In this, therefore . . . the rationalists were in the right" (Russell 1912:74).
2352:
He drives the point home (pages 67-68) where he defines four conditions that determine what we call "the numbers" (cf. (71)). Definition, page 67: the successor set N' is a part of the collection N, there is a starting-point
461:. He believed that in the "foundations of the simplest science; viz., that part of logic which deals with the theory of numbers" had not been properly argued – "nothing capable of proof ought to be accepted without proof":
146:
This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of
995:
To illustrate, consider the following finite example: Suppose there are 12 families on a street. Some have children, some do not. To discuss the names of the children in these households requires 12 propositions asserting
1672:
as the numerical conjunction is not a single term . . . Thus a class of classes will be many many's; its constituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents."
1039:" on the finite street of a finite number of families, Russell apparently intended the following to extend to all propositional functions extending over an infinite domain so as to allow the creation of all the numbers.
499:
deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles" (Preface
209:), from which no inconsistencies have as yet been derived. Thus, elements of the logicist programmes have proved viable, but in the process theories of classes, sets and mappings, and higher-order logics other than with
1564:
in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In response to these difficulties, Russell advocated a strong prohibition, his "vicious circle principle":
1903:
Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). He deduces that "one obtains integers of different orders" (p. 134-135); the proof in Russell 1927
1858:
of 1927, though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function "all the way down" to its elementary propositions, linked together with logical operators:
801:": placing the elements of collections in one-to-one correspondence) and "ordering" (using "the successor of" relation to order the collections of the equinumerous classes)". The logicistic derivation equates the
376:). Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization"—the notions "
1559:
By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that
701:. In two Appendices added at the last minute he devoted 28 pages to both a detailed analysis of Frege's theory contrasted against his own, and a fix for the paradox. But he was not optimistic about the outcome:
988:
In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed, for example, that there are an infinite collection of trios in the
1407:
is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain
2572:
1913/1962 edition:56, the original uses x with a circumflex). Here φẑ indicates the function with variable ẑ, i.e. φ(x) where x is argument "z"; φz indicates the value of the function given argument "z";
1451:) that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism" (cf. footnote 1 in Gödel 1944
1403:
mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The
1967:
describes a range of views considered by their proponents to be successors of the original logicist program. More narrowly, neo-logicism may be seen as the attempt to salvage some or all elements of
1157:
he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a "doctrine of types"; see more about the unit class, the problem of
1595:
Within the algebra of, say, rational numbers the equation is satisfied when α = 0.5. But within, for instance, a Boolean algebra, where only "truth values" 0 and 1 are permitted, then the equality
538:
the customary "laws of thought" concerning simple propositional statements (usually of belief); these laws would be sufficient in themselves if augmented with theory of classes and relations (e.g.
1413:
immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction." (Hilbert 1931 in Mancosu 1998: 266, 267).
1141:: Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection.
1376:. It is easy to see, and not difficult to prove, that the relation "less than", so defined, is asymmetrical, transitive, and connected, and has the numbers for its field ." (1919:35)
1569:"No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Gödel 1944 appearing in
621:, to the extent that he accepts the integer with its essential properties as a dogma and does not look back" and equated his extreme constructivist stance with that of Brouwer's
883:
For Russell, "terms" are either "things" or "concepts": "Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a
396:("the Hilbert School") as follows: "Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell's
1004:" applied to this collection of households on the particular street of families with names F1, F2, . . . F12. Each of the 12 propositions regards whether or not the "argument"
1847:
types of variables are relevant; thus the lowest type occurring in a given context may be called that of individuals" (p. 161). But the axiom of reducibility proposes that
251:. Gödel's result suggests that in order to maintain a logicist position, while still retaining as much as possible of classical mathematics, one must accept some axiom of
1606:
that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but also from the function's own output.
1144:
The resulting entity may be called "the null class" or "the empty class". Russell symbolized the null/empty class with Λ. So what exactly is the Russellian null class? In
714:
analyzed (e.g. the classes or propositions) soon for the most part turned into "logical fictions" . . . only that we have no direct perception of them." (Gödel 1944:120)
17:
1199:– the "children", the inheritors of the "successor" – of 0 with respect to the relation "the immediate predecessor of (which is the converse of "successor") (1919:23).
602:, Vol. 99, pp. 334-336) has endeavored to impose certain limitations upon the free formation of concepts in mathematics which I do not believe to be justified" (p. 45).
1880:
The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but
1706:
In the following notice the wording "the class as many"—a class is an aggregate of those terms (things) that satisfy the propositional function, but a class is not a
1261:
the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class
505:"A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the
741:
exist as real objects, and sentences containing these terms are meaningful only as they can be interpreted as ... a manner of speaking about other things" (p. 125).
1843:
th order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions. "For in practice only the
1625:
But, since the class with numeral 1 is a single object or unit in its own right, it too must be included in the class of unit classes. This inclusion results in an
1177:
class of classes as the appropriate class to fill this role, this being the class of classes having no members. This null class of classes may be labelled "0"
1093:. Russell's solution was to define the notion of a class to be only those elements that satisfy the proposition, his argument being that, indeed, the arguments
797:, begins its construction of the numbers from primitive propositions such as "class", "propositional function", and in particular, relations of "similarity" ("
1730:
problem did not go away; rather, it arrived in a new form: "It will now be necessary to distinguish (1) terms, (2) classes, (3) classes of classes, and so on
426:
that full use was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms. In addition, the young
2854:
Mario Livio, 2011 "Why Math Works: Is math invented or discovered? A leading astrophysicist suggests that the answer to the millennia-old question is both",
669:
and what makes a belief true is a fact, "and this fact does not (except in exceptional cases) involve the mind of the person who has the belief" (1912:130).
2564:
function, i.e. there is a predicative function which is true when φz is true and false when φz is false. In symbols, the axiom is: ⊦ :(∃ψ) : φz. ≡
2227:
2779:. Contains two essays—I. "Continuity and Irrational Numbers" with original Preface, II. "The Nature and Meaning of Numbers" with two Prefaces (1887, 1893).
2263:
mathematics, i.e. in reducing to logic the arithmetical notions which his predecessors had shown to be sufficient for mathematics." (Russell 1919/2005:17).
3617:
845:], where we presupposed an intuitive conception of the natural number sequence, and elicited from it the principle that, whenever a particular property
140:
118:. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.
710:
negative; i.e. the classes and concepts introduced this way do not have all the properties required for the use of mathematics" (Gödel 1944:132).
718:
physical or material objects as basic bits of the furniture of the world. He would attempt to construct them out of sense-data" in his next book
192:, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their
4292:
2690:
2193:
2117:
1057:
or should be. For Dedekind and Frege, a class is a distinct entity in its own right, a 'unity' that can be identified with all those entities
680:"On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the
201:
Today, the bulk of extant mathematics is believed to be derivable logically from a small number of extralogical axioms, such as the axioms of
5274:
1810:
Type 3 can only entertain types 2, 1 or 0, and so forth. But these types can be mixed (for example, for this sentence to be (sort of) true: "
1459:
In particular he pointed out that "The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their
4375:
3516:
2799:
The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and The Foundations of Mathematics from Cantor Through Russell to Gödel
2624:
2391:), or what, exactly. Gödel 1944:120 comments on this absence of formal syntax and the absence of a clearly specified substitution process.
2140:
1167:: For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary.
5464:
1987:
397:
341:
The overt intent of logicism is to derive all of mathematics from symbolic logic (Frege, Dedekind, Peano, Russell.) As contrasted with
1863:"All propositions, of whatever order, are derived from a matrix composed of elementary propositions combined by means of the stroke" (
3015:
1877:– adopted for the 2nd edition of PM – a single two argument logical function from which all other logical functions may be defined.)
1298:
of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)
1641:"The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus
3215:
4689:
1131:: These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by
901:"The former kind will often be called predicates or class-concepts; the latter are always or almost always relations." (1903:44)
1577:
To illustrate what a pernicious instance of impredicativity might be, consider the consequence of inputting argument α into the
1423:
Hilbert dismissed logicism as a "false path": "Some tried to define the numbers purely logically; others simply took the usual
606:
Indeed he awaits Kronecker's "publishing his reasons for the necessity or merely the expediency of these limitations" (p. 45).
1994:) with some 'safer' axiom so as to prevent the derivation of the known paradoxes. The most cited candidate to replace BLV is
4847:
2932:
2907:
2203:
393:
3635:
1602:
Some of the difficulties in the logicist programme may derive from the α = NOT-α paradox Russell discovered in Frege's 1879
243:
whereby 'infinitary' theories – such as that of PM – were to be proved consistent from finitary theories, with the aim that
4702:
4025:
1782:), yield a meaningful output ω. Note that this means that a "type" can be of mixed order, as the following example shows:
224:
2653:
2166:
1814:
won the 1947 World Series" could accept the individual (type 0) "Joe DiMaggio" and/or the names of his other teammates,
748:
661:
when applied to logical principles, cf. 1912:74) is intricate. He would strongly, unambiguously express support for the
5545:
4707:
4697:
4434:
4287:
3640:
2858:(ISSN 0036-8733), Volume 305, Number 2, August 2011, Scientific American division of Nature America, Inc, New York, NY.
2577:
indicates "equivalence for all z"; ψ!z indicates a predicative function, i.e. one with no variables except individuals.
1806:
an aggregate "Yankees" as its input. Thus the composite-sentence has a (mixed) type of 2, mixed as to order (1 and 2).
276:
3631:
1285:: The process of creating a successor requires the relation " . . . is the successor of . . .", which may be denoted "
5267:
4843:
2892:
2877:
2848:
2821:
2806:
2791:
2776:
1891:
Gödel 1944 agrees that Russell's logicist project was stymied; he seems to disagree that even the integers survived:
1065:. (This symbolism appears in Russell, attributed there to Frege: "The essence of a function is what is left when the
986:"In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration.
4185:
3422:
4940:
4684:
3509:
318:, and was used by Russell and others from then on, in versions appropriate for various languages." (G-G 2000:501).
5540:
4245:
3938:
2246:
202:
3679:
5550:
5201:
4903:
4666:
4661:
4486:
3907:
3591:
3483:
3088:
3043:
586:(1); it is completely determined when with respect to every thing it is determined whether it is an element of
259:
One argument that programmes derived from logicism remain valid might be that the incompleteness theorems are '
110:
plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to
2088:
Another quasi-neo-logicist approach has been suggested by M. Randall Holmes. In this kind of amendment to the
159:
1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an
5196:
4979:
4896:
4609:
4540:
4417:
3659:
3473:
3008:
509:
of Dynamics. . . . I was led to a re-examination of the principles of Geometry, thence to the philosophy of
172:
122:
2771:, English translation published by Open Court Publishing Company 1901, Dover publication 1963, Mineola, NY,
5260:
5121:
4947:
4633:
4267:
3866:
3281:
3169:
444:: He started with a consideration of arithmetic: did it derive from "logic" or from "facts of experience"?
315:
2560:"The axiom of reducibility is the assumption that, given any function φẑ, there is a formally equivalent,
1555:
all objects that are dependent upon the notion defined, that is, that can in any way be determined by it".
665:"universals" (cf. 1912:91-118) and he would conclude that truth and falsity are "out there"; minds create
231:
for the natural numbers may be derived – such as Russell's systems in PM – can decide all the well-formed
4999:
4994:
4604:
4343:
4272:
3601:
3502:
3322:
775:
1221:
For his definition of successor, Russell will use for his "unit" a single entity or "term" as follows:
1208:
Observe in particular that Russell does not use the unit class of classes "1" to construct the successor
849:
of natural numbers is given such that (1) and (2), then any given natural number must have the property
4928:
4518:
3912:
3880:
3571:
3296:
3286:
3189:
649:
and arithmetic are contributed by us does not account for this" (1912:87); Russell concludes that the
240:
2505:
Zermelo 1908 in van Heijenoort 1967:190. See the discussion of this very quotation in Mancosu 1998:68.
1917:) must be described by finite combinations of symbols (all quotes and content derived from page 135).
5218:
5167:
5064:
4562:
4523:
4000:
3645:
3337:
3327:
3078:
1301:
Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of
1267:
Russell's definition requires a new "term" which is "added into" the collections inside the bundles.
82:
Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the
31:
3674:
2686:
5059:
4989:
4528:
4380:
4363:
4086:
3566:
3306:
3301:
3291:
3001:
510:
2946:, Cambridge at the University Press, Cambridge UK, no ISBN. Second edition, abridged to *56, with
2709:, "Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's
1622:. But Frege, unlike Russell, allowed the class of unit classes to be identified as a unit itself:
1202:
Note Russell has used a few words here without definition, in particular "number series", "number
5484:
4891:
4868:
4829:
4715:
4656:
4302:
4222:
4066:
4010:
3623:
3437:
3347:
3342:
3332:
3184:
1758:
will become a meaningless proposition; and in this way the contradiction is avoided" (1903:517).
1499:
1478:. Set theory, meanwhile had moved on with its reduction of relation to the ordered pair of sets.
1098:
object. A decision of this question in either way is indifferent to our logic" (First edition of
214:
210:
2836:
with commentary by van Heijenoort, letters to Frege from Russell and from Russell to Frege, etc.
1968:
1547:". He seems to have considered that only predicative definitions can be allowed in mathematics:
5304:
5181:
4908:
4886:
4853:
4746:
4592:
4577:
4550:
4501:
4385:
4320:
4145:
4111:
4106:
3980:
3811:
3788:
3412:
3255:
3194:
3179:
3174:
3138:
2621:
2136:
2092:, BLV remains intact, save for a restriction to stratifiable formulae in the manner of Quine's
2082:
2060:
2030:
1589:-α, with truth values 1 and 0. When input α = 0, output ω = 1; when input α = 1, output ω = 0.
1578:
842:
786:
the Frege numerals may be derived in an analogous way to their derivation in the Grundgesetze.
381:
115:
59:
47:
837:"The viewpoint here is very different from that of 's maxim that 'God made the integers' plus
5519:
5111:
4964:
4756:
4474:
4210:
4116:
3975:
3960:
3841:
3816:
3478:
3276:
3250:
3235:
3220:
3098:
2068:
1823:
813:. Kleene observes the following. (Kleene's assumptions (1) and (2) state that 0 has property
629:
617:, all else is the work of man" had his foes, among them Hilbert. Hilbert called Kronecker a "
422:
307:
194:
189:
2161:
1839:
function of the appropriate order. A careful reading of the first edition indicates that an
1105:
Russell continues to hold this opinion in his 1919; observe the words "symbolic fictions":
982:
collection of entities. Russell makes this clear in the second (italicized) sentence below.
749:
An example of a logicist construction of the natural numbers: Russell's construction in the
275:. The former can be proven using finistic methods, while the latter – in general – cannot.
5084:
5046:
4923:
4727:
4567:
4491:
4469:
4297:
4255:
4154:
4121:
3985:
3773:
3684:
3468:
3240:
3093:
1952:
1761:
This is Russell's "doctrine of types". To guarantee that impredicative expressions such as
1592:
To make the function "impredicative", identify the input with the output, yielding α = 1−α
1086:
690:
377:
156:
2942:
Alfred North Whitehead and Bertrand Russell, 1927 2nd edition, (first edition 1910–1913),
1416:
In brief, according to Hilbert and Bernays, the notion of "sequence" or "successor" is an
922:
267:'. However, that argument appears not to acknowledge the distinction between theorems of
8:
5456:
5283:
5213:
5104:
5089:
5069:
5026:
4913:
4863:
4789:
4734:
4671:
4464:
4459:
4407:
4175:
4164:
3836:
3736:
3664:
3655:
3651:
3586:
3581:
3442:
3245:
3210:
3073:
2751:
2449:, which applies to the predicates that belong to it" (Bernays 1930-1 in Mancosu 1998:240.
1995:
1873:
1734:; we shall have to hold that no member of one set is a member of any other set, and that
1326:
1302:
951:"The characteristic of a class concept, as distinguished from terms in general, is that "
779:
672:
Where did Russell derive these epistemic notions? He tells us in the Preface to his 1903
433:
296:
152:
38:
is a programme comprising one or more of the theses that – for some coherent meaning of '
1085:
This notion of collection or class as object, when used without restriction, results in
590:
or not.*" (p. 45, italics added). The * indicates a footnote where he states that:
5347:
5242:
5011:
4974:
4959:
4952:
4935:
4739:
4721:
4587:
4513:
4496:
4449:
4262:
4171:
4005:
3990:
3950:
3902:
3887:
3875:
3831:
3806:
3576:
3525:
3387:
3260:
3153:
3148:
3048:
3038:
2988:
2238:
2076:
1976:
1691:
classes (such as the class-of-classes that define the numbers 2, 3, etc) are fictions.
1490:(1914). Perhaps because of "residual annoyance, Russell did not react at all". By 1914
1354:
890:"Among terms, it is possible to distinguish two kinds, which I shall call respectively
272:
260:
4195:
3447:
697:, derived from Frege's Basic Law V and he was determined not to repeat it in his 1903
5379:
5364:
5291:
5237:
5177:
4984:
4794:
4784:
4676:
4557:
4392:
4368:
4149:
4133:
4038:
4015:
3892:
3861:
3826:
3721:
3556:
3143:
3053:
2980:
2928:
2903:
2888:
2873:
2844:
2817:
2802:
2787:
2772:
2523:
One source for more detail is Fairouz Kamareddine, Twan Laan and Rob Nderpelt, 2004,
2479:
Russell deemed Wiener "the infant phenomenon . . . more infant than phenomenon"; see
2199:
2105:
1881:
1687:
should have regarded these as fictitious, but not derive the further conclusion that
838:
771:
595:
268:
111:
2601:
2242:
1540:
1505:
867:
be sufficient to stop the program until these are resolved (see Criticisms, below).
247:
could be reassured that their use should provably not result in the derivation of a
135:
commitments of then-extant accounts of the natural numbers, and his conviction that
5509:
5401:
5191:
5186:
5079:
5036:
4858:
4819:
4814:
4799:
4625:
4582:
4479:
4277:
4227:
3801:
3763:
3427:
3417:
3382:
3103:
1991:
1626:
1479:
1470:
A complicated theory of relations continued to strangle Russell's explanatory 1919
758:
496:
457:
Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his
330:
232:
213:
have come to be regarded as extralogical in nature, in part under the influence of
185:
164:
67:
55:
2870:
From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s
2230:
Frege's Influence on Wittgenstein: Reversing Metaphysics via the Context Principle
1909:
concluded that it wouldn't matter anyway because propositional functions of order
430:
was enriched by a new instrument, the abstract theory of relations" (p. 120-121).
5494:
5343:
5172:
5162:
5116:
5099:
5054:
5016:
4918:
4838:
4645:
4572:
4545:
4533:
4439:
4353:
4327:
4282:
4250:
4051:
3853:
3796:
3746:
3711:
3669:
3463:
3367:
3024:
2730:
2706:
2694:
2644:
2628:
2597:
2157:
2144:
2093:
2052:
2045:
1707:
1561:
1491:
1158:
1090:
1047:
1043:
802:
783:
733:" derived from Russell's "more radical idea, the no-class theory" (p. 125):
342:
91:
1031:
Whereas the preceding example is finite over the finite propositional function "
5296:
5157:
5136:
5094:
5074:
4969:
4824:
4422:
4412:
4402:
4397:
4331:
4205:
4081:
3970:
3965:
3943:
3544:
3407:
3063:
2887:(with Introduction by John Perry 1997), Oxford University Press, New York, NY,
2648:
2100:
then 'goes through'. The resulting system has the same consistency strength as
2026:
2006:
1726:
classes are useful fictions he solved the problem of the "unit" class, but the
1487:
566:
is: it is an aggregate, a manifold, a totality of associated elements (things)
479:
has been to undertake this examination" (Peano 1889 in van Heijenoort 1967:85).
350:
311:
177:
107:
71:
5534:
5131:
4809:
4316:
4101:
4091:
4061:
4046:
3716:
3432:
3392:
3230:
2101:
1582:
1544:
1424:
1046:
definition that he will have to resolve, or risk deriving something like the
346:
326:
248:
236:
160:
63:
3397:
917:
truth or falsehood. They create beliefs . . . what makes a belief true is a
220:
5489:
5474:
5031:
4878:
4779:
4771:
4651:
4599:
4508:
4444:
4427:
4358:
4217:
4076:
3778:
3561:
3402:
3377:
3372:
3225:
2072:
863:
762:
730:
726:
622:
513:
and infinity, and then, with a view to discovering the meaning of the word
389:
356:
292:
228:
128:
51:
2733:, "Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics",
5469:
5359:
5329:
5141:
5021:
4200:
4190:
4137:
3821:
3741:
3726:
3606:
3551:
3068:
3058:
2079:
1983:
1971:
programme through the use of a modified version of Frege's system in the
1948:
1774:(individuals? classes? classes-of-classes? etc.) that, when plugged into
1486:
Russell ignored this reduction that had been achieved by his own student
637:
87:
43:
1943:
He concludes his essay with the following suggestions and observations:
1818:
it could accept the class (type 1) of individual players "The Yankees".
534:
seem less well-defined than that of Russell, but both seem accepting as
295:
and later Dummett – was a significant contributor to the development of
163:
in Frege's system set out in the Grundgesetze der Arithmetik. Note that
5514:
5504:
5411:
5386:
5339:
4071:
3926:
3897:
3703:
3133:
1495:
641:
506:
148:
95:
5252:
2667:
Bob Hale and Crispin Wright (2002), "Benacerraf's dilemma revisited",
1694:
But Russell did not do this. After a detailed analysis in Appendix A:
1681:
757:
from Z, one definition of "number" uses an axiom of that system – the
5446:
5426:
5334:
5309:
5223:
5126:
4179:
4096:
4056:
4020:
3956:
3768:
3758:
3731:
3494:
3108:
2042:
2010:
1277:
Step 8: For every class of equinumerous classes, create its successor
798:
662:
284:
121:
The philosophical impetus behind Frege's logicist programme from the
1008:
applies to a child in a particular household. The children's names (
557:
Dedekind's argument begins with "1. In what follows I understand by
291:
Logicism – especially through the influence of Frege on Russell and
5479:
5431:
5416:
5354:
5314:
5006:
4454:
4159:
3753:
2814:
From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931
2181:"On the philosophical relevance of Gödel's incompleteness theorems"
2180:
1586:
1551:"a definition is 'predicative' and logically admissible only if it
1069:
is taken away, i.e in the above instance, 2( ) + ( ). The argument
1024:), where the function is "name of a child in the family with name F
633:
527:
252:
244:
181:
132:
103:
2993:
2051:
Other major proponents of neo-logicism include Bernard Linsky and
1746:
should be of a set of a degree lower by one than the set to which
1399:
epistemology retains its significance: to ascertain the intuitive
1372:
possesses every hereditary property possessed by the successor of
1214:
must be defined so as not to assume that we know what is meant by
5406:
5396:
5391:
5374:
4804:
3596:
2784:
Foundations and Fundamental Concepts of Mathematics Third Edition
2525:
A Modern Perspective on Type Theory, From its Origins Until Today
2198:(Volume 153 ed.). Amsterdam: Elsevier, inc. pp. 59–90.
1506:
The unit class, impredicativity, and the vicious circle principle
614:
427:
413:
264:
99:
5441:
5436:
5421:
5319:
1802:
an individual "Joe DiMaggio" as its input, the other takes for
810:
770:
ultimately the notion of an ordered pair is derivable from the
280:
2816:, 3rd printing 1976, Harvard University Press, Cambridge, MA,
4348:
3694:
3539:
1391:
definition, the Number concept turns out to be an elementary
1289:", between the various "numerals". "We must now consider the
926:
531:
483:
Russell 1903 describes his intent in the Preface to his 1903
83:
39:
310:
states that the French word 'Logistique' was "introduced by
5369:
5324:
1195:". (1903:21). He asserts that "the natural numbers are the
657:
But his epistemology about the innate (he prefers the word
440:
Frege 1879 describes his intent in the Preface to his 1879
136:
2492:
See van Heijenoort's commentary and Norbert Wiener's 1914
1581:
f with output ω = 1−α. This may be seen as the equivalent
491:"THE present work has two main objects. One of these, the
139:'s use of truths about the natural numbers as examples of
2436:
has a domain consisting of the 12 families on the street.
1798:
won the 1947 World Series". The first sentence takes for
1786:"Joe DiMaggio and the Yankees won the 1947 World Series".
1309:(" . . . is the successor of . . . ") between two terms
610:
206:
3118:
2402:
The Philosophy of Mathematics and Hilbert's Proof Theory
2357:" , this "1" is not contained in any successor, for any
1385:
The presumption of an 'extralogical' notion of iteration
1206:", and "successor". He will define these in due course.
969:
470:
Peano 1889 states his intent in his Preface to his 1889
46:
is an extension of logic, some or all of mathematics is
1835:
order can be reduced to (or replaced by) an equivalent
1660:
s." This must not be taken as a relation of two terms,
1935:
Indeed, Russell's "no class" theory, Gödel concludes:
521:
2841:
Introduction To Metamathematics 1991 10th impression,
2674:(1):101–129, esp. "6. Objections and Qualifications".
1637:
but is not in Russell's view "worthy of thing-hood":
1113:. And if we can find any way of dealing with them as
412:
Gödel 1944 summarized the historical background from
2970:
2865:, Cambridge: at the University Press, Cambridge, UK.
2832:
with commentary by Willard V. Quine, Zermelo's 1908
2514:
This same definition appears also in Kleene 1952:42.
2343:
except by the help of experience" (Russell 1912:74).
2021:. This kind of neo-logicism is often referred to as
1930:
Russell's no-class theory is the root of the problem
1920:
1225:"It remains to define "successor". Given any number
2843:, North-Holland Publishing Company, Amsterdam, NY,
2461:
Cf. sections 487ff (pages 513ff in the Appendix A).
2195:
Studies In Logic And The Foundations Of Mathematics
1790:
This sentence can be decomposed into two clauses: "
1682:
A solution to impredicativity: a hierarchy of types
974:In the Prinicipia, the natural numbers derive from
959:" is a propositional function when, and only when,
725:
These constructions in what Gödel 1944 would call "
384:", "class" (collection, aggregate) and "relation".
2830:Mathematical logic as based on the theory of types
2828:with commentary by van Heijenoort, Russell's 1908
2361:in the collection there exists a transformation φ(
2312:Perry in his 1997 Introduction to Russell 1912:ix)
1585:expression to the 'symbolic-logic' expression ω =
1494:would provide another, equivalent definition, and
1443:(either edition), Gödel in 1944 was disappointed:
2834:A new proof of the possibility of a well-ordering
1271:Step 7: Construct the successor of the null class
1129:Step 2: Collect "similar" classes into 'bundles'
127:onwards was in part his dissatisfaction with the
5532:
1439:aspects of Russellian logicism as it appears in
1161:and Russell's "vicious circle principle" below.
729:constructivism ... which might better be called
302:
279:shows that Gödel numbering can be used to prove
1998:, the contextual definition of '#' given by '#
1696:The Logical and Arithmetical Doctrines of Frege
1257:members. Thus we have the following definition:
689:In 1902 Russell discovered a "vicious circle" (
584:as an object of our thought is likewise a thing
336:
50:to logic, or some or all of mathematics may be
2989:"Logicism" at the Encyclopaedia of Mathematics
2432:s domain is "all childnames", and family name
2118:Aristotelian realist philosophy of mathematics
176:in 1903 using the paradox and developments of
5268:
3510:
3009:
2682:
2680:
1719:the fictitious "concepts" corrals and ranch.
1173:Following Frege, Russell picked the empty or
1053:The question arises what precisely a "class"
1042:Kleene considers that Russell has set out an
2801:, Princeton University Press, Princeton NJ,
2096:and related systems. Essentially all of the
1851:a reduction "all the way down" is possible.
929:), then his utterance is considered "true".
94:. This and related ideas convinced him that
1629:of increasing type and increasing content.
1539:This sort of definition of I was deemed by
1153:After he discovered the paradox in Frege's
782:(but not the Zermelo numerals), whereas in
745:See more in the Criticism sections, below.
517:, to Symbolic Logic" (Preface 1903:vi-vii).
18:Scottish School (philosophy of mathematics)
5275:
5261:
3702:
3517:
3503:
3016:
3002:
2952:Propositions Containing Apparent Variables
2910:. This is a non-mathematical companion to
2677:
2494:A simplification of the logic of relations
2288:On the foundations of logic and arithmetic
1988:axiom schema of unrestricted comprehension
609:Kronecker, famous for his assertion that "
227:show that no formal system from which the
188:in geometry and set theory as well as the
27:Programme in the philosophy of mathematics
2944:Principia Mathematica to *56,1962 Edition
2927:, Barnes & Noble, Inc, New York, NY,
2902:, Barnes & Noble, Inc, New York, NY,
2872:, Oxford University Press, New York, NY,
1420:notion that lies outside symbolic logic.
1165:Step 4: Assign a "numeral" to each bundle
1061:that satisfy some propositional function
778:and is required in the definition of the
2950:pages Xiii-xlvi, and new Appendix A (*8
2786:, Dover Publications, Inc, Mineola, NY,
1181:Step 6: Define the notion of "successor"
978:propositions that can be asserted about
737:"according to which classes or concepts
62:championed this programme, initiated by
5282:
2925:Introduction to Mathematical Philosophy
2900:Introduction to Mathematical Philosophy
2551:van Heijenoort 1967:3 and pages 124-128
2025:. Proponents of neo-Fregeanism include
1482:observes that in the second edition of
1472:Introduction to Mathematical Philosophy
1364:is said to be less than another number
636:, with portions borrowed from European
245:those uneasy about 'infinitary methods'
14:
5533:
3524:
2639:
2637:
2191:
1148:Russell says that "A class is said to
329:(1929), but apparently independently,
5256:
3498:
2997:
2622:PHIL 30067: Logicism and Neo-Logicism
2457:
2455:
970:The definition of the natural numbers
632:served him as an antidote to British
235:of that system. This result damaged
2863:The Principles of Mathematics Vol. I
2225:
349:) that employs arithmetic concepts,
316:International Congress of Philosophy
167:also suffers from this difficulty.
3023:
2654:Stanford Encyclopedia of Philosophy
2634:
2481:Russell's confrontation with Wiener
2167:Stanford Encyclopedia of Philosophy
1325:. Second, he defines the notion of
1241:be a term which is not a member of
862:numbers derives from the theory of
761:– that leads to the definition of "
720:Our knowledge of the External World
522:Epistemology, ontology and logicism
24:
2948:Introduction to the Second Edition
2452:
1353:. Third, he defines the notion of
1000:is the name of a child in family F
578:"; he asserts that "such a system
184:. Since he treated the subject of
25:
5562:
2966:
2643:
2156:
2071:. Modal neo-logicism derives the
1921:Gödel's criticism and suggestions
459:The Nature and Meaning of Numbers
170:On the other hand, Russell wrote
5236:
3117:
2973:
1982:For instance, one might replace
1975:(which may be seen as a kind of
1698:in his 1903, Russell concludes:
1305:i.e. given the relation such as
874:
554:linked by the generalization R.
2839:Stephen C. Kleene, 1971, 1952,
2769:Essays on the Theory of Numbers
2761:
2744:
2724:
2700:
2661:
2615:
2590:
2580:
2554:
2545:
2535:
2517:
2508:
2499:
2486:
2483:in Grattan-Guinness 2000:419ff.
2473:
2464:
2439:
2407:
2394:
2376:
2346:
2333:
2324:
2315:
2306:
2275:An axiomatization of set theory
1959:
1474:and his 1927 second edition of
1245:. Then the class consisting of
963:is a class-concept." (1903:56)
277:Tarski's undefinability theorem
3484:Tractatus Logico-Philosophicus
3089:Problem of multiple generality
2767:Richard Dedekind, 1858, 1878,
2715:Journal of Philosophical Logic
2669:European Journal of Philosophy
2606:The Bulletin of Symbolic Logic
2301:The Foundations of Mathematics
2293:
2280:
2266:
2256:
2219:
2185:
2174:
2150:
2130:
1794:won the 1947 World Series" + "
1410:extra-logical concrete objects
299:during the twentieth century.
66:and subsequently developed by
13:
1:
5197:History of mathematical logic
3474:The Principles of Mathematics
2496:in van Heijenoort 1967:224ff.
2404:1930:1931 in Mancosu, p. 242.
2299:Pages 474–5 in Hilbert 1927,
2123:
1360:He concludes: ". . . number
1139:Step 3: Define the null class
1121:And in the second edition of
303:Origin of the name 'logicism'
173:The Principles of Mathematics
5122:Primitive recursive function
3170:Commutativity of conjunction
2956:Theory of Apparent Variables
2303:in: van Heijenoort 1967:475.
2277:in van Heijenoort 1967:413).
2033:, sometimes also called the
1379:
1016:in a propositional function
337:Intent, or goal, of logicism
7:
2923:to Bertrand Russell, 1919,
2812:Jean van Heijenoort, 1967,
2797:I. Grattan-Guinness, 2000,
2290:in van Heijenoort 1967:130.
2111:
1035:of the children in family F
1012:) can be thought of as the
695:Grundgesetze der Arithmetik
418:Characteristica universalis
203:Zermelo–Fraenkel set theory
77:
10:
5567:
4186:Schröder–Bernstein theorem
3913:Monadic predicate calculus
3572:Foundations of mathematics
3190:Monotonicity of entailment
2960:Truth-Functions and Others
2885:The Problems of Philosophy
2649:"Logicism and Neologicism"
793:, like its forerunner the
407:
241:foundations of mathematics
5546:Philosophy of mathematics
5455:
5402:Parsimony (Occam's razor)
5290:
5232:
5219:Philosophy of mathematics
5168:Automated theorem proving
5150:
5045:
4877:
4770:
4622:
4339:
4315:
4293:Von Neumann–Bernays–Gödel
4238:
4132:
4036:
3934:
3925:
3852:
3787:
3693:
3615:
3532:
3456:
3360:
3315:
3269:
3203:
3162:
3126:
3115:
3079:Idempotency of entailment
3031:
2752:"Repairing Frege’s Logic"
2339:"Nothing can be known to
1283:Step 9: Order the numbers
1159:impredicative definitions
1091:impredicative definitions
699:Principles of Mathematics
674:Principles of Mathematics
485:Principles of Mathematics
124:Grundlagen der Arithmetik
32:philosophy of mathematics
2898:Bertrand Russell, 1919,
2883:Bertrand Russell, 1912,
2861:Bertrand Russell, 1903,
2824:. Includes Frege's 1879
2067:, who espouse a form of
2057:Stanford–Edmonton School
2041:, who espouse a form of
2039:abstractionist Platonism
1867:1927 Appendix A, p. 385)
1722:When Russell proclaimed
472:Principles of Arithmetic
141:synthetic a priori truth
4869:Self-verifying theories
4690:Tarski's axiomatization
3641:Tarski's undefinability
3636:incompleteness theorems
3438:Willard Van Orman Quine
2192:Gabbay, Dov M. (2009).
2162:"Principia Mathematica"
2055:, sometimes called the
1827:is the hypothesis that
1571:Collected Works Vol. II
1089:; see more below about
325:About the same time as
314:and others at the 1904
225:incompleteness theorems
5541:Abstract object theory
5243:Mathematics portal
4854:Proof of impossibility
4502:propositional variable
3812:Propositional calculus
3413:Charles Sanders Peirce
3256:Hypothetical syllogism
2741:(1–2) (2000), 219–265.
2602:"What is Neologicism?"
2108:'s Axiom of Counting.
2061:abstract structuralism
1854:By the 2nd edition of
1498:in 1921 would provide
904:"I shall speak of the
843:mathematical induction
546:) between individuals
526:The epistemologies of
106:were reducible to the
90:using certain sets of
60:Alfred North Whitehead
5551:Theories of deduction
5457:Theories of deduction
5112:Kolmogorov complexity
5065:Computably enumerable
4965:Model complete theory
4757:Principia Mathematica
3817:Propositional formula
3646:Banach–Tarski paradox
3479:Principia Mathematica
3251:Disjunctive syllogism
3236:modus ponendo tollens
2912:Principia Mathematica
2868:Paolo Mancosu, 1998,
2069:axiomatic metaphysics
1824:axiom of reducibility
1463:" (Russell 1944:120)
1441:Principia Mathematica
1233:be a class which has
1191:+1, the successor of
1187:, it also belongs to
1100:Principia Mathematica
722:" (Perry 1997:xxvi).
423:Principia Mathematica
308:Ivor Grattan-Guinness
195:Principia Mathematica
190:calculus of relations
5060:Church–Turing thesis
5047:Computability theory
4256:continuum hypothesis
3774:Square of opposition
3632:Gödel's completeness
3469:Function and Concept
3241:Constructive dilemma
3216:Material implication
2721:(6) (1999): 619–660.
2321:Cf. Russell 1912:74.
2226:Reck, Erich (1997),
1953:axiomatic set theory
1614:that is asymmetric:
1500:the one in use today
1467:suggestions" below.
1435:With respect to the
1432:by Mancosu 1998:9).
853:." (Kleene 1952:44).
780:von Neumann numerals
776:Axiom of Replacement
283:constructs, but not
263:just like any other
5284:Philosophical logic
5214:Mathematical object
5105:P versus NP problem
5070:Computable function
4864:Reverse mathematics
4790:Logical consequence
4667:primitive recursive
4662:elementary function
4435:Free/bound variable
4288:Tarski–Grothendieck
3807:Logical connectives
3737:Logical equivalence
3587:Logical consequence
3443:Ludwig Wittgenstein
3246:Destructive dilemma
3074:Well-formed formula
2856:Scientific American
2782:Howard Eves, 1990,
2750:M. Randall Holmes,
2713:in Object Theory",
2596:Bernard Linsky and
2413:To be precise both
2382:In his 1903 and in
1329:for three numerals
1253:added on will have
364:materially implies
297:analytic philosophy
86:characterizing the
5348:Unity of opposites
5012:Transfer principle
4975:Semantics of logic
4960:Categorical theory
4936:Non-standard model
4450:Logical connective
3577:Information theory
3526:Mathematical logic
3388:Augustus De Morgan
2693:2006-12-24 at the
2627:2011-07-17 at the
2612:(1) (2006): 60–99.
2369:(distinguishable)
2143:2008-02-20 at the
2065:modal neo-logicism
1986:(analogous to the
1977:second-order logic
1393:structural concept
1171:Step 5: Define "0"
925:'s anecdote about
273:higher-order logic
217:'s later thought.
205:(or its extension
155:1896, Zermelo and
5528:
5527:
5380:List of fallacies
5365:Explanatory power
5292:Critical thinking
5250:
5249:
5182:Abstract category
4985:Theories of truth
4795:Rule of inference
4785:Natural deduction
4766:
4765:
4311:
4310:
4016:Cartesian product
3921:
3920:
3827:Many-valued logic
3802:Boolean functions
3685:Russell's paradox
3660:diagonal argument
3557:First-order logic
3492:
3491:
3356:
3355:
2981:Philosophy portal
2958:, and Appendix C
2933:978-1-4114-2942-0
2908:978-1-4114-2942-0
2754:, August 5, 2018.
2205:978-0-444-52012-8
1871:(The "stroke" is
1633:"unit". It is an
1583:'algebraic-logic'
1237:members, and let
1115:symbolic fictions
1111:symbolic fictions
1087:Russell's paradox
923:Diogenes Laërtius
772:Axiom of Infinity
691:Russell's paradox
400:" (Gödel 1944 in
372:, one may derive
269:first-order logic
261:proved with logic
239:'s programme for
186:primitive notions
16:(Redirected from
5558:
5510:Platonic realism
5277:
5270:
5263:
5254:
5253:
5241:
5240:
5192:History of logic
5187:Category of sets
5080:Decision problem
4859:Ordinal analysis
4800:Sequent calculus
4698:Boolean algebras
4638:
4637:
4612:
4583:logical/constant
4337:
4336:
4323:
4246:Zermelo–Fraenkel
3997:Set operations:
3932:
3931:
3869:
3700:
3699:
3680:Löwenheim–Skolem
3567:Formal semantics
3519:
3512:
3505:
3496:
3495:
3428:Henry M. Sheffer
3418:Bertrand Russell
3383:Richard Dedekind
3267:
3266:
3211:De Morgan's laws
3185:Noncontradiction
3127:Classical logics
3121:
3018:
3011:
3004:
2995:
2994:
2983:
2978:
2977:
2976:
2954:) to replace *9
2919:Amit Hagar 2005
2755:
2748:
2742:
2728:
2722:
2704:
2698:
2687:st-andrews.ac.uk
2684:
2675:
2665:
2659:
2658:
2645:Zalta, Edward N.
2641:
2632:
2619:
2613:
2594:
2588:
2584:
2578:
2558:
2552:
2549:
2543:
2539:
2533:
2521:
2515:
2512:
2506:
2503:
2497:
2490:
2484:
2477:
2471:
2468:
2462:
2459:
2450:
2443:
2437:
2431:
2421:and family name
2411:
2405:
2398:
2392:
2380:
2374:
2350:
2344:
2337:
2331:
2328:
2322:
2319:
2313:
2310:
2304:
2297:
2291:
2284:
2278:
2270:
2264:
2260:
2254:
2253:
2251:
2245:, archived from
2236:
2223:
2217:
2216:
2214:
2212:
2189:
2183:
2178:
2172:
2171:
2158:Zalta, Edward N.
2154:
2148:
2134:
1996:Hume's principle
1992:naive set theory
1874:Sheffer's stroke
1659:
1627:infinite regress
1480:Grattan-Guinness
1425:number-theoretic
821:+1 has property
803:cardinal numbers
759:axiom of pairing
683:
600:Crelle's Journal
497:pure mathematics
271:and theorems of
211:Henkin semantics
165:naive set theory
92:rational numbers
68:Richard Dedekind
56:Bertrand Russell
21:
5566:
5565:
5561:
5560:
5559:
5557:
5556:
5555:
5531:
5530:
5529:
5524:
5495:Logical atomism
5451:
5344:Socratic method
5295:
5286:
5281:
5251:
5246:
5235:
5228:
5173:Category theory
5163:Algebraic logic
5146:
5117:Lambda calculus
5055:Church encoding
5041:
5017:Truth predicate
4873:
4839:Complete theory
4762:
4631:
4627:
4623:
4618:
4610:
4330: and
4326:
4321:
4307:
4283:New Foundations
4251:axiom of choice
4234:
4196:Gödel numbering
4136: and
4128:
4032:
3917:
3867:
3848:
3797:Boolean algebra
3783:
3747:Equiconsistency
3712:Classical logic
3689:
3670:Halting problem
3658: and
3634: and
3622: and
3621:
3616:Theorems (
3611:
3528:
3523:
3493:
3488:
3464:Begriffsschrift
3452:
3448:Jan Łukasiewicz
3368:Bernard Bolzano
3352:
3323:Double negation
3311:
3282:Double negation
3265:
3199:
3175:Excluded middle
3158:
3122:
3113:
3027:
3025:Classical logic
3022:
2979:
2974:
2972:
2969:
2826:Begriffsschrift
2764:
2759:
2758:
2749:
2745:
2731:Edward N. Zalta
2729:
2725:
2707:Edward N. Zalta
2705:
2701:
2695:Wayback Machine
2685:
2678:
2666:
2662:
2642:
2635:
2629:Wayback Machine
2620:
2616:
2598:Edward N. Zalta
2595:
2591:
2585:
2581:
2576:
2567:
2559:
2555:
2550:
2546:
2540:
2536:
2522:
2518:
2513:
2509:
2504:
2500:
2491:
2487:
2478:
2474:
2470:1909 Appendix A
2469:
2465:
2460:
2453:
2444:
2440:
2429:
2425:are variables.
2412:
2408:
2399:
2395:
2381:
2377:
2356:
2351:
2347:
2338:
2334:
2329:
2325:
2320:
2316:
2311:
2307:
2298:
2294:
2285:
2281:
2271:
2267:
2261:
2257:
2249:
2234:
2224:
2220:
2210:
2208:
2206:
2190:
2186:
2179:
2175:
2155:
2151:
2145:Wayback Machine
2135:
2131:
2126:
2114:
2053:Edward N. Zalta
2046:foundationalism
2035:Scottish School
1962:
1949:theory of types
1923:
1897:Collected Works
1708:thing-in-itself
1684:
1657:
1604:Begriffsschrift
1562:impredicativity
1534:
1530:
1526:
1522:
1514:
1508:
1453:Collected Works
1382:
1081:
1048:Russell paradox
972:
877:
754:
682:non-existential
681:
524:
442:Begriffsschrift
410:
402:Collected Works
343:algebraic logic
339:
305:
143:was incorrect.
129:epistemological
108:natural numbers
80:
28:
23:
22:
15:
12:
11:
5:
5564:
5554:
5553:
5548:
5543:
5526:
5525:
5523:
5522:
5517:
5512:
5507:
5502:
5497:
5492:
5487:
5482:
5477:
5472:
5467:
5465:Constructivism
5461:
5459:
5453:
5452:
5450:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5384:
5383:
5382:
5372:
5367:
5362:
5357:
5352:
5351:
5350:
5332:
5327:
5322:
5317:
5312:
5307:
5301:
5299:
5297:informal logic
5288:
5287:
5280:
5279:
5272:
5265:
5257:
5248:
5247:
5233:
5230:
5229:
5227:
5226:
5221:
5216:
5211:
5206:
5205:
5204:
5194:
5189:
5184:
5175:
5170:
5165:
5160:
5158:Abstract logic
5154:
5152:
5148:
5147:
5145:
5144:
5139:
5137:Turing machine
5134:
5129:
5124:
5119:
5114:
5109:
5108:
5107:
5102:
5097:
5092:
5087:
5077:
5075:Computable set
5072:
5067:
5062:
5057:
5051:
5049:
5043:
5042:
5040:
5039:
5034:
5029:
5024:
5019:
5014:
5009:
5004:
5003:
5002:
4997:
4992:
4982:
4977:
4972:
4970:Satisfiability
4967:
4962:
4957:
4956:
4955:
4945:
4944:
4943:
4933:
4932:
4931:
4926:
4921:
4916:
4911:
4901:
4900:
4899:
4894:
4887:Interpretation
4883:
4881:
4875:
4874:
4872:
4871:
4866:
4861:
4856:
4851:
4841:
4836:
4835:
4834:
4833:
4832:
4822:
4817:
4807:
4802:
4797:
4792:
4787:
4782:
4776:
4774:
4768:
4767:
4764:
4763:
4761:
4760:
4752:
4751:
4750:
4749:
4744:
4743:
4742:
4737:
4732:
4712:
4711:
4710:
4708:minimal axioms
4705:
4694:
4693:
4692:
4681:
4680:
4679:
4674:
4669:
4664:
4659:
4654:
4641:
4639:
4620:
4619:
4617:
4616:
4615:
4614:
4602:
4597:
4596:
4595:
4590:
4585:
4580:
4570:
4565:
4560:
4555:
4554:
4553:
4548:
4538:
4537:
4536:
4531:
4526:
4521:
4511:
4506:
4505:
4504:
4499:
4494:
4484:
4483:
4482:
4477:
4472:
4467:
4462:
4457:
4447:
4442:
4437:
4432:
4431:
4430:
4425:
4420:
4415:
4405:
4400:
4398:Formation rule
4395:
4390:
4389:
4388:
4383:
4373:
4372:
4371:
4361:
4356:
4351:
4346:
4340:
4334:
4317:Formal systems
4313:
4312:
4309:
4308:
4306:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4259:
4258:
4253:
4242:
4240:
4236:
4235:
4233:
4232:
4231:
4230:
4220:
4215:
4214:
4213:
4206:Large cardinal
4203:
4198:
4193:
4188:
4183:
4169:
4168:
4167:
4162:
4157:
4142:
4140:
4130:
4129:
4127:
4126:
4125:
4124:
4119:
4114:
4104:
4099:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4054:
4049:
4043:
4041:
4034:
4033:
4031:
4030:
4029:
4028:
4023:
4018:
4013:
4008:
4003:
3995:
3994:
3993:
3988:
3978:
3973:
3971:Extensionality
3968:
3966:Ordinal number
3963:
3953:
3948:
3947:
3946:
3935:
3929:
3923:
3922:
3919:
3918:
3916:
3915:
3910:
3905:
3900:
3895:
3890:
3885:
3884:
3883:
3873:
3872:
3871:
3858:
3856:
3850:
3849:
3847:
3846:
3845:
3844:
3839:
3834:
3824:
3819:
3814:
3809:
3804:
3799:
3793:
3791:
3785:
3784:
3782:
3781:
3776:
3771:
3766:
3761:
3756:
3751:
3750:
3749:
3739:
3734:
3729:
3724:
3719:
3714:
3708:
3706:
3697:
3691:
3690:
3688:
3687:
3682:
3677:
3672:
3667:
3662:
3650:Cantor's
3648:
3643:
3638:
3628:
3626:
3613:
3612:
3610:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3548:
3547:
3536:
3534:
3530:
3529:
3522:
3521:
3514:
3507:
3499:
3490:
3489:
3487:
3486:
3481:
3476:
3471:
3466:
3460:
3458:
3454:
3453:
3451:
3450:
3445:
3440:
3435:
3430:
3425:
3423:Ernst Schröder
3420:
3415:
3410:
3408:Giuseppe Peano
3405:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3364:
3362:
3358:
3357:
3354:
3353:
3351:
3350:
3345:
3340:
3335:
3330:
3325:
3319:
3317:
3313:
3312:
3310:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3273:
3271:
3264:
3263:
3258:
3253:
3248:
3243:
3238:
3233:
3228:
3223:
3218:
3213:
3207:
3205:
3201:
3200:
3198:
3197:
3192:
3187:
3182:
3177:
3172:
3166:
3164:
3160:
3159:
3157:
3156:
3151:
3146:
3141:
3136:
3130:
3128:
3124:
3123:
3116:
3114:
3112:
3111:
3106:
3101:
3096:
3091:
3086:
3081:
3076:
3071:
3066:
3064:Truth function
3061:
3056:
3051:
3046:
3041:
3035:
3033:
3029:
3028:
3021:
3020:
3013:
3006:
2998:
2992:
2991:
2985:
2984:
2968:
2967:External links
2965:
2964:
2963:
2940:
2939:
2938:
2937:
2936:
2896:
2881:
2866:
2859:
2852:
2837:
2810:
2795:
2780:
2763:
2760:
2757:
2756:
2743:
2723:
2699:
2676:
2660:
2633:
2614:
2589:
2579:
2574:
2565:
2553:
2544:
2534:
2516:
2507:
2498:
2485:
2472:
2463:
2451:
2438:
2406:
2393:
2375:
2354:
2345:
2332:
2323:
2314:
2305:
2292:
2279:
2265:
2255:
2218:
2204:
2184:
2173:
2149:
2128:
2127:
2125:
2122:
2121:
2120:
2113:
2110:
2027:Crispin Wright
2023:neo-Fregeanism
2007:if and only if
1961:
1958:
1957:
1956:
1941:
1940:
1922:
1919:
1901:
1900:
1869:
1868:
1788:
1787:
1750:belongs. Thus
1742:requires that
1716:
1715:
1704:
1703:
1683:
1680:
1675:
1674:
1653:is one of the
1599:be satisfied.
1575:
1574:
1557:
1556:
1537:
1536:
1532:
1528:
1524:
1520:
1512:
1507:
1504:
1488:Norbert Wiener
1457:
1456:
1381:
1378:
1265:
1264:
1258:
1119:
1118:
1083:
1082:
1079:
993:
992:
971:
968:
876:
873:
855:
854:
841:of number and
839:Peano's axioms
799:equinumerosity
753:
747:
743:
742:
707:
706:
687:
686:
604:
603:
598:not long ago (
523:
520:
519:
518:
502:
501:
481:
480:
468:
467:
455:
454:
409:
406:
398:constructivism
351:symbolic logic
338:
335:
304:
301:
178:Giuseppe Peano
79:
76:
72:Giuseppe Peano
26:
9:
6:
4:
3:
2:
5563:
5552:
5549:
5547:
5544:
5542:
5539:
5538:
5536:
5521:
5518:
5516:
5513:
5511:
5508:
5506:
5503:
5501:
5498:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5462:
5460:
5458:
5454:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5381:
5378:
5377:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5349:
5345:
5341:
5338:
5337:
5336:
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5302:
5300:
5298:
5293:
5289:
5285:
5278:
5273:
5271:
5266:
5264:
5259:
5258:
5255:
5245:
5244:
5239:
5231:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5203:
5200:
5199:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5159:
5156:
5155:
5153:
5149:
5143:
5140:
5138:
5135:
5133:
5132:Recursive set
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5082:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5052:
5050:
5048:
5044:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5001:
4998:
4996:
4993:
4991:
4988:
4987:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4954:
4951:
4950:
4949:
4946:
4942:
4941:of arithmetic
4939:
4938:
4937:
4934:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4906:
4905:
4902:
4898:
4895:
4893:
4890:
4889:
4888:
4885:
4884:
4882:
4880:
4876:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4849:
4848:from ZFC
4845:
4842:
4840:
4837:
4831:
4828:
4827:
4826:
4823:
4821:
4818:
4816:
4813:
4812:
4811:
4808:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4777:
4775:
4773:
4769:
4759:
4758:
4754:
4753:
4748:
4747:non-Euclidean
4745:
4741:
4738:
4736:
4733:
4731:
4730:
4726:
4725:
4723:
4720:
4719:
4717:
4713:
4709:
4706:
4704:
4701:
4700:
4699:
4695:
4691:
4688:
4687:
4686:
4682:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4649:
4647:
4643:
4642:
4640:
4635:
4629:
4624:Example
4621:
4613:
4608:
4607:
4606:
4603:
4601:
4598:
4594:
4591:
4589:
4586:
4584:
4581:
4579:
4576:
4575:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4552:
4549:
4547:
4544:
4543:
4542:
4539:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4516:
4515:
4512:
4510:
4507:
4503:
4500:
4498:
4495:
4493:
4490:
4489:
4488:
4485:
4481:
4478:
4476:
4473:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4452:
4451:
4448:
4446:
4443:
4441:
4438:
4436:
4433:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4410:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4387:
4384:
4382:
4381:by definition
4379:
4378:
4377:
4374:
4370:
4367:
4366:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4341:
4338:
4335:
4333:
4329:
4324:
4318:
4314:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4268:Kripke–Platek
4266:
4264:
4261:
4257:
4254:
4252:
4249:
4248:
4247:
4244:
4243:
4241:
4237:
4229:
4226:
4225:
4224:
4221:
4219:
4216:
4212:
4209:
4208:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4181:
4177:
4173:
4170:
4166:
4163:
4161:
4158:
4156:
4153:
4152:
4151:
4147:
4144:
4143:
4141:
4139:
4135:
4131:
4123:
4120:
4118:
4115:
4113:
4112:constructible
4110:
4109:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4044:
4042:
4040:
4035:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3996:
3992:
3989:
3987:
3984:
3983:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3962:
3958:
3954:
3952:
3949:
3945:
3942:
3941:
3940:
3937:
3936:
3933:
3930:
3928:
3924:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3889:
3886:
3882:
3879:
3878:
3877:
3874:
3870:
3865:
3864:
3863:
3860:
3859:
3857:
3855:
3851:
3843:
3840:
3838:
3835:
3833:
3830:
3829:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3794:
3792:
3790:
3789:Propositional
3786:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3748:
3745:
3744:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3720:
3718:
3717:Logical truth
3715:
3713:
3710:
3709:
3707:
3705:
3701:
3698:
3696:
3692:
3686:
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3657:
3653:
3649:
3647:
3644:
3642:
3639:
3637:
3633:
3630:
3629:
3627:
3625:
3619:
3614:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3546:
3543:
3542:
3541:
3538:
3537:
3535:
3531:
3527:
3520:
3515:
3513:
3508:
3506:
3501:
3500:
3497:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3461:
3459:
3455:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3433:Alfred Tarski
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3393:Gottlob Frege
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3365:
3363:
3359:
3349:
3346:
3344:
3341:
3339:
3338:Biconditional
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3320:
3318:
3314:
3308:
3305:
3303:
3300:
3298:
3297:Biconditional
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3274:
3272:
3268:
3262:
3259:
3257:
3254:
3252:
3249:
3247:
3244:
3242:
3239:
3237:
3234:
3232:
3231:modus tollens
3229:
3227:
3224:
3222:
3221:Transposition
3219:
3217:
3214:
3212:
3209:
3208:
3206:
3202:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3161:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3139:Propositional
3137:
3135:
3132:
3131:
3129:
3125:
3120:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3094:Associativity
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3036:
3034:
3030:
3026:
3019:
3014:
3012:
3007:
3005:
3000:
2999:
2996:
2990:
2987:
2986:
2982:
2971:
2961:
2957:
2953:
2949:
2945:
2941:
2934:
2930:
2926:
2922:
2918:
2917:
2916:
2915:
2913:
2909:
2905:
2901:
2897:
2894:
2893:0-19-511552-X
2890:
2886:
2882:
2879:
2878:0-19-509632-0
2875:
2871:
2867:
2864:
2860:
2857:
2853:
2850:
2849:0-7204-2103-9
2846:
2842:
2838:
2835:
2831:
2827:
2823:
2822:0-674-32449-8
2819:
2815:
2811:
2808:
2807:0-691-05858-X
2804:
2800:
2796:
2793:
2792:0-486-69609-X
2789:
2785:
2781:
2778:
2777:0-486-21010-3
2774:
2770:
2766:
2765:
2753:
2747:
2740:
2736:
2732:
2727:
2720:
2716:
2712:
2708:
2703:
2696:
2692:
2688:
2683:
2681:
2673:
2670:
2664:
2656:
2655:
2650:
2646:
2640:
2638:
2630:
2626:
2623:
2618:
2611:
2607:
2603:
2599:
2593:
2583:
2571:
2563:
2557:
2548:
2538:
2530:
2526:
2520:
2511:
2502:
2495:
2489:
2482:
2476:
2467:
2458:
2456:
2448:
2442:
2435:
2428:
2424:
2420:
2416:
2410:
2403:
2397:
2390:
2385:
2379:
2372:
2368:
2364:
2360:
2349:
2342:
2336:
2327:
2318:
2309:
2302:
2296:
2289:
2286:Hilbert 1904
2283:
2276:
2269:
2259:
2252:on 2018-08-24
2248:
2244:
2240:
2233:
2232:
2229:
2222:
2207:
2201:
2197:
2196:
2188:
2182:
2177:
2169:
2168:
2163:
2159:
2153:
2146:
2142:
2138:
2133:
2129:
2119:
2116:
2115:
2109:
2107:
2103:
2099:
2095:
2091:
2086:
2084:
2083:object theory
2081:
2078:
2074:
2070:
2066:
2062:
2058:
2054:
2049:
2047:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2005:
2001:
1997:
1993:
1989:
1985:
1980:
1978:
1974:
1970:
1966:
1954:
1950:
1946:
1945:
1944:
1938:
1937:
1936:
1933:
1931:
1927:
1918:
1916:
1912:
1907:
1898:
1894:
1893:
1892:
1889:
1887:
1883:
1878:
1876:
1875:
1866:
1862:
1861:
1860:
1857:
1852:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1825:
1819:
1817:
1813:
1807:
1805:
1801:
1797:
1793:
1785:
1784:
1783:
1781:
1777:
1773:
1768:
1764:
1759:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1720:
1713:
1712:
1711:
1709:
1701:
1700:
1699:
1697:
1692:
1690:
1679:
1671:
1667:
1663:
1656:
1652:
1648:
1644:
1640:
1639:
1638:
1636:
1630:
1628:
1623:
1621:
1617:
1613:
1607:
1605:
1600:
1598:
1593:
1590:
1588:
1584:
1580:
1572:
1568:
1567:
1566:
1563:
1554:
1550:
1549:
1548:
1546:
1545:impredicative
1542:
1518:
1517:
1516:
1503:
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1468:
1464:
1462:
1454:
1450:
1446:
1445:
1444:
1442:
1438:
1433:
1429:
1426:
1421:
1419:
1414:
1411:
1406:
1402:
1396:
1394:
1388:
1386:
1377:
1375:
1371:
1367:
1363:
1358:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1299:
1297:
1292:
1288:
1284:
1280:
1278:
1274:
1272:
1268:
1262:
1259:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1223:
1222:
1219:
1217:
1213:
1209:
1205:
1200:
1198:
1194:
1190:
1186:
1182:
1178:
1176:
1172:
1168:
1166:
1162:
1160:
1156:
1151:
1147:
1142:
1140:
1136:
1134:
1130:
1126:
1124:
1116:
1112:
1108:
1107:
1106:
1103:
1101:
1096:
1092:
1088:
1077:
1076:
1075:
1072:
1068:
1064:
1060:
1056:
1051:
1049:
1045:
1044:impredicative
1040:
1038:
1034:
1029:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
990:
985:
984:
983:
981:
977:
967:
964:
962:
958:
954:
949:
947:
943:
939:
935:
930:
928:
924:
920:
916:
910:
907:
902:
899:
897:
893:
888:
886:
881:
875:Preliminaries
872:
868:
865:
864:Dedekind cuts
861:
852:
848:
844:
840:
836:
835:
834:
832:
829:has property
828:
824:
820:
816:
812:
807:
804:
800:
796:
792:
787:
785:
781:
777:
773:
768:
764:
760:
752:
746:
740:
736:
735:
734:
732:
728:
723:
721:
715:
711:
704:
703:
702:
700:
696:
693:) in Frege's
692:
679:
678:
677:
675:
670:
668:
664:
660:
655:
652:
647:
643:
639:
635:
631:
626:
624:
620:
616:
612:
607:
601:
597:
593:
592:
591:
589:
585:
581:
577:
573:
569:
565:
560:
555:
553:
549:
545:
541:
537:
533:
529:
516:
512:
508:
504:
503:
498:
494:
490:
489:
488:
486:
477:
476:
475:
473:
464:
463:
462:
460:
451:
447:
446:
445:
443:
438:
435:
431:
429:
425:
424:
419:
415:
405:
403:
399:
395:
391:
385:
383:
379:
375:
371:
367:
363:
359:
358:
352:
348:
347:Boolean logic
344:
334:
332:
328:
327:Rudolf Carnap
323:
319:
317:
313:
309:
300:
298:
294:
289:
286:
282:
278:
274:
270:
266:
262:
257:
254:
250:
249:contradiction
246:
242:
238:
237:David Hilbert
234:
230:
226:
222:
218:
216:
212:
208:
204:
199:
197:
196:
191:
187:
183:
180:'s school of
179:
175:
174:
168:
166:
162:
161:inconsistency
158:
154:
150:
144:
142:
138:
134:
130:
126:
125:
119:
117:
113:
109:
105:
101:
97:
93:
89:
85:
75:
73:
69:
65:
64:Gottlob Frege
61:
57:
53:
49:
45:
41:
37:
33:
19:
5499:
5490:Intuitionism
5475:Fictionalism
5234:
5208:
5032:Ultraproduct
4879:Model theory
4844:Independence
4780:Formal proof
4772:Proof theory
4755:
4728:
4685:real numbers
4657:second-order
4568:Substitution
4445:Metalanguage
4386:conservative
4359:Axiom schema
4303:Constructive
4273:Morse–Kelley
4239:Set theories
4218:Aleph number
4211:inaccessible
4117:Grothendieck
4001:intersection
3888:Higher-order
3876:Second-order
3822:Truth tables
3779:Venn diagram
3562:Formal proof
3403:Hugh MacColl
3378:Georg Cantor
3373:George Boole
3270:Introduction
3226:modus ponens
3154:Higher-order
3149:Second-order
3099:Distribution
3083:
3059:Truth tables
2959:
2955:
2951:
2947:
2943:
2924:
2921:Introduction
2920:
2911:
2899:
2884:
2869:
2862:
2855:
2840:
2833:
2829:
2825:
2813:
2798:
2783:
2768:
2762:Bibliography
2746:
2738:
2734:
2726:
2718:
2714:
2711:Grundgesetze
2710:
2702:
2671:
2668:
2663:
2652:
2617:
2609:
2605:
2592:
2582:
2569:
2561:
2556:
2547:
2537:
2528:
2524:
2519:
2510:
2501:
2493:
2488:
2480:
2475:
2466:
2446:
2441:
2433:
2426:
2422:
2418:
2414:
2409:
2401:
2396:
2389:modus ponens
2388:
2383:
2378:
2370:
2366:
2362:
2358:
2348:
2340:
2335:
2326:
2317:
2308:
2300:
2295:
2287:
2282:
2274:
2268:
2258:
2247:the original
2231:
2228:
2221:
2209:. Retrieved
2194:
2187:
2176:
2165:
2152:
2132:
2098:Grundgesetze
2097:
2090:Grundgesetze
2089:
2087:
2077:second-order
2073:Peano axioms
2064:
2056:
2050:
2038:
2034:
2022:
2018:
2014:
2003:
1999:
1981:
1973:Grundgesetze
1972:
1965:Neo-logicism
1964:
1963:
1960:Neo-logicism
1942:
1934:
1929:
1928:
1924:
1914:
1910:
1905:
1902:
1896:
1890:
1885:
1879:
1872:
1870:
1864:
1855:
1853:
1848:
1844:
1840:
1836:
1832:
1831:function of
1828:
1822:
1820:
1815:
1811:
1808:
1803:
1799:
1795:
1791:
1789:
1779:
1775:
1771:
1766:
1762:
1760:
1755:
1751:
1747:
1743:
1739:
1735:
1732:ad infinitum
1731:
1727:
1723:
1721:
1717:
1705:
1695:
1693:
1688:
1685:
1676:
1669:
1665:
1661:
1654:
1650:
1646:
1642:
1634:
1631:
1624:
1619:
1615:
1611:
1608:
1603:
1601:
1596:
1594:
1591:
1576:
1570:
1558:
1552:
1538:
1509:
1483:
1475:
1471:
1469:
1465:
1460:
1458:
1452:
1448:
1440:
1436:
1434:
1430:
1422:
1417:
1415:
1409:
1404:
1400:
1397:
1392:
1389:
1384:
1383:
1373:
1369:
1365:
1361:
1359:
1350:
1346:
1342:
1338:
1334:
1330:
1327:transitivity
1322:
1318:
1314:
1310:
1306:
1300:
1295:
1290:
1286:
1282:
1281:
1276:
1275:
1270:
1269:
1266:
1263:." (1919:23)
1260:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1220:
1218:(1919:181).
1215:
1212:unit classes
1211:
1207:
1203:
1201:
1196:
1192:
1188:
1184:
1180:
1179:
1174:
1170:
1169:
1164:
1163:
1155:Grundgesetze
1154:
1149:
1145:
1143:
1138:
1137:
1132:
1128:
1127:
1122:
1120:
1114:
1110:
1104:
1099:
1094:
1084:
1070:
1066:
1062:
1058:
1054:
1052:
1041:
1036:
1032:
1030:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
994:
987:
979:
975:
973:
965:
960:
956:
952:
950:
945:
941:
937:
933:
931:
918:
914:
911:
905:
903:
900:
895:
891:
889:
884:
882:
878:
869:
859:
856:
850:
846:
830:
826:
822:
818:
814:
805:
795:Grundgesetze
794:
790:
788:
766:
763:ordered pair
755:
750:
744:
738:
731:fictionalism
727:nominalistic
724:
719:
716:
712:
708:
698:
694:
688:
673:
671:
666:
658:
656:
650:
645:
640:and British
627:
623:intuitionism
618:
608:
605:
599:
587:
583:
579:
575:
571:
567:
563:
558:
556:
551:
547:
543:
539:
535:
525:
514:
492:
484:
482:
471:
469:
458:
456:
449:
441:
439:
432:
421:
417:
411:
401:
390:Intuitionism
386:
373:
369:
365:
361:
360:(i.e. from
357:modus ponens
355:
340:
324:
320:
306:
293:Wittgenstein
290:
258:
229:Peano axioms
219:
200:
193:
171:
169:
145:
123:
120:
88:real numbers
81:
35:
29:
5470:Dialetheism
5360:Explanation
5330:Credibility
5142:Type theory
5090:undecidable
5022:Truth value
4909:equivalence
4588:non-logical
4201:Enumeration
4191:Isomorphism
4138:cardinality
4122:Von Neumann
4087:Ultrafilter
4052:Uncountable
3986:equivalence
3903:Quantifiers
3893:Fixed-point
3862:First-order
3742:Consistency
3727:Proposition
3704:Traditional
3675:Lindström's
3665:Compactness
3607:Type theory
3552:Cardinality
3348:Disjunction
3343:Conjunction
3328:Existential
3316:Elimination
3307:Disjunction
3302:Conjunction
3287:Existential
3144:First-order
3069:Truth value
3039:Quantifiers
2562:predicative
2542:Heijenoort.
2417:= variable
2211:1 September
2009:there is a
1984:Basic Law V
1888:1927:xiv).
1882:irrationals
1837:predicative
1673:(1903:516).
1649:will mean "
1078:{ a, b, c }
806:constructed
638:rationalism
404:1990:119).
281:syntactical
133:ontological
44:mathematics
5535:Categories
5515:Pragmatism
5505:Nominalism
5412:Propaganda
5387:Hypothesis
5340:Antithesis
4953:elementary
4646:arithmetic
4514:Quantifier
4492:functional
4364:Expression
4082:Transitive
4026:identities
4011:complement
3944:hereditary
3927:Set theory
3398:Kurt Gödel
3261:Absorption
3163:Principles
3049:Connective
2735:Erkenntnis
2124:References
1951:. . . and
1668:, because
1635:assemblage
1573:1990:125).
1496:Kuratowski
1455:1990:120).
1102:1927:24).
1033:childnames
642:empiricism
628:Russell's
511:continuity
507:philosophy
221:Kurt Gödel
149:set theory
96:arithmetic
54:in logic.
5485:Formalism
5447:Vagueness
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3602:Theory
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