663:
7391:
7700:
them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction
Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834).
1500:
6882:
1036:
6510:, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.
1495:{\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}
214:
3766:
6868:
5103:
3376:
7400:, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered
7603:
An illustration of 'interpretation' is
Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of
5165:
The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative uses an order relation symbol instead of the successor
4332:
Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or
7699:
So
Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of
4445:
of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom. The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of
4896:
6623:. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.
6498:
shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism. This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.
3761:{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}
4391:
that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true
4360:. Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε
1025:
4372:, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.
6973:
The non-contiguous set satisfies axiom 1 as it has a 0 element, 2–5 as it doesn't affect equality relations, 6 & 8 as all pieces have a successor, bar the zero element and axiom 7 as no two dominos topple, or are toppled by, the same
3258:
1833:
4255:
6102:
5098:{\displaystyle \forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}}
2363:
4383:, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be
8064:
5795:
5420:
6184:
5584:
7556:
7120:
6251:
702:(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0.
6011:
318:(∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the
3031:
5928:
7680:
4593:
4842:
4157:
3381:
3140:
1743:
1041:
920:
5655:
2477:
172:
The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about
5481:
5260:
7258:
2542:
1949:
915:
7170:
6853:
4719:
6773:
6529:. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible
5851:
7308:
4449:
First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the
7028:
5712:
4767:
4515:
5325:
2654:
6298:
6720:
4333:
stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using
4641:
3353:
7404:
in wihich each set of positive elements has a least member. was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.".
7202:
2147:
2054:
1591:
375:
6368:
4375:
The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as
4305:; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900,
7340:
6672:
5135:
3819:), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
314:
was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for
4417:
1649:
3135:
6346:
1738:
1535:
4453:, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the
2396:
2205:
2176:
2086:
2013:
1981:
1865:
155:
proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book
4431:. The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is
6326:
4419:
theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a
Hilbert-style proof of "0=1").
2106:
1669:
1611:
4152:
7898:
5153:. The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formula
8671:]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Fratres Bocca. pp. 83–97.
6018:
176:; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic". The next three axioms are
2823:, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are
3042:, i.e., "What are the numbers and what are they good for?") that any two models of the Peano axioms (including the second-order induction axiom) are
7523:
414:. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.
7951:
7372:
4446:
natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).
8912:
2210:
555:
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "
165:
8410:
5723:
5336:
6109:
5488:
3280:
7033:
6191:
2722:
The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤":
8409:
Meseguer, José; Goguen, Joseph A. (Dec 1986). "Initiality, induction, and computability". In
Maurice Nivat and John C. Reynolds (ed.).
7880:
6525:, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is
6409:
but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as
4439:
over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order
844:
induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section
4329:, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself, if Peano arithmetic is consistent.
189:
5939:
188:. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing the
8201:(1983). "Mathematics, the Empirical Facts, and Logical Necessity". In Hempel, Carl G.; Putnam, Hilary; Essler, Wilhelm K. (eds.).
9037:
180:
statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a
6424:
for deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence,
5862:
2398:
is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and
328:, published in 1879. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of
8896:
8872:
8846:
8822:
8633:
8609:
8450:
8423:
8397:
8373:
8249:
8218:
8186:
7974:
7607:
4521:
6394:
4773:
278:
6559:, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
6460:(more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of the
8933:
250:
5595:
2408:
9042:
6513:
It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as
8049:
6417:
5427:
5176:
1020:{\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}
662:
8573:
8541:
8345:
8283:
7207:
878:
and addition and multiplication are often added as axioms. The respective functions and relations are constructed in
297:
184:
statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to
6429:
2486:
1877:
333:
257:
6915:
6518:
6473:
3268:
17:
7125:
6805:
4847:
In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a
4647:
4460:
The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of
398:
Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in
8304:
6728:
6507:
3294:
6495:
5806:
3370:
that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:
1983:
is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:
8716:
4376:
4342:
235:
7263:
264:
8964:
8601:
8365:
6986:
5670:
4725:
4470:
4354:
4326:
231:
5271:
9032:
7872:
6920:
2619:
6262:
9047:
9027:
8959:
7520:
6945:
6681:
4314:
4284:
246:
135:, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the
8813:
Buss, Samuel R. (1998). "Chapter II: First-Order Proof Theory of
Arithmetic". In Buss, Samuel R. (ed.).
4599:
1732:
is a function mapping two natural numbers to another one. Given addition, it is defined recursively as:
8178:
6895:
6494:
implies that the existence of nonstandard elements cannot be excluded in first-order logic. The upward
6437:
4848:
682:
to the chain of light pieces ("no junk") as only light dominoes will fall when the nearest is toppled.
8992:
5170:(axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):
3311:
8683:
7175:
5931:
2111:
2018:
1540:
355:
8954:
6351:
4384:
148:
8337:
8130:
7966:
7507:
6940:
6522:
4388:
3808:
2727:
780:
224:
185:
7695:
7604:
natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form
7537:
7313:
6648:
5167:
5111:
8478:
8261:
7599:
6905:
6545:
be the order type of the rationals, the order type of any countable nonstandard model of PA is
6461:
3836:
3253:{\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}
1672:
675:
560:
411:
408:
400:
173:
144:
140:
8364:. Derives the basic number systems from the Peano axioms. English/German vocabulary included.
8170:
4395:
1828:{\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}
674:
satisfied by the set of all dominoes — whether light or dark — taken together. The 9th axiom (
8525:
8010:
6925:
6441:
5587:
4457:
of Peano arithmetic, and axioms are included that relate the three operations to each other.
4346:
4318:
4070:
3363:
1698:
1616:
584:
547:
315:
8755:"Self-verifying axiom systems, the incompleteness theorem and related reflection principles"
8239:
6253:, i.e. zero and one are distinct and there is no element between them. In other words, 0 is
4368:
describing a suitable order on the integers, or more abstractly as consisting of the finite
709:
sufficiently many times to zero requires an additional axiom, which is sometimes called the
8789:
8514:
6964:
the nearest light piece corresponding to 0, and a neighbor piece corresponding to successor
6331:
5328:
5263:
4297:
and others agreed that these axioms implicitly defined what we mean by a "natural number".
3850:
1508:
687:
666:
The chain of light dominoes on the right, starting with the nearest, can represent the set
106:
2372:
2181:
2152:
2062:
1989:
1957:
1841:
869:
271:
8:
8858:
8704:
8621:
7884:
6935:
6526:
6491:
6410:
5798:
4461:
2719:; because there is no natural number between 0 and 1, it is a discrete ordered semiring.
2560:
relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:
1709:
514:
8662:
8466:
3271:. (This is not the case with any first-order reformulation of the Peano axioms, below.)
9011:
8906:
8793:
8777:
8736:
8728:
8594:
8565:
8558:
8502:
8224:
8139:
8108:
8076:
8050:"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"
8031:
7992:
7945:
7937:
7532:
7501:
6900:
6887:
6487:
6311:
6186:, i.e. given any two distinct elements, the larger is the smaller plus another element.
5854:
4450:
4436:
4432:
4427:
All of the Peano axioms except the ninth axiom (the induction axiom) are statements in
3816:
3804:
3776:
3264:
2091:
1654:
1596:
883:
837:
711:
636:
556:
469:
434:
378:
311:
181:
136:
31:
8973:
8892:
8868:
8842:
8834:
8818:
8629:
8605:
8569:
8537:
8530:
8446:
8419:
8393:
8385:
8369:
8341:
8289:
8279:
8245:
8228:
8214:
8182:
8150:
8080:
8035:
7970:
7928:
6930:
6881:
6873:
6402:
6254:
5662:
4454:
4428:
4369:
4250:{\displaystyle {\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}
3800:
898:
841:
346:
177:
132:
102:
8740:
8321:
4298:
8797:
8769:
8720:
8646:
8494:
8436:
8316:
8206:
8198:
8134:
8126:
8068:
8023:
7894:
6448:, and thus definable by existentially quantified formulas (with free variables) of
6104:, i.e. the ordering is preserved under multiplication by the same positive element.
6097:{\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}
5715:
5658:
4294:
3298:
3022:
2399:
1868:
152:
69:
41:
8976:
7030:" can be proven from the other axioms (in first-order logic) as follows. Firstly,
6308:. It is also incomplete and one of its important properties is that any structure
8862:
8785:
8510:
8440:
8265:
8154:
8065:
On
Formally Undecidable Propositions of Principia Mathematica and Related Systems
8005:
7960:
7527:
7401:
6457:
6445:
4338:
3840:
3828:
3035:
1701:
341:
320:
94:
8684:"Introduction to Peano Arithmetic (Gödel Incompleteness and Nonstandard Models)"
8210:
4379:. A small number of philosophers and mathematicians, some of whom also advocate
8658:
8553:
8462:
7819:
7557:
File:Inductive proofs of properties of add, mult from recursive definitions.pdf
6479:
4365:
4350:
3796:
3290:
1729:
861:
160:
113:
90:
86:
8839:
Introduction to
Mathematical Logic (Discrete Mathematics and Its Applications)
8653:. On p. 100, he restates and defends his axioms of 1888. pp. 98–103.
8390:
Introduction to
Mathematical Logic (Discrete Mathematics and Its Applications)
9021:
8884:
8642:
Contains translations of the following two papers, with valuable commentary:
8589:
8357:
8293:
4380:
4306:
2958:
2839:
894:
325:
98:
8944:
8754:
8482:
8104:
8045:
4322:
8109:"Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes"
8084:
6910:
4441:
3103:
2981:
2716:
690:
of the intuitive notion of natural numbers: the number 1 can be defined as
329:
193:
705:
The intuitive notion that each natural number can be obtained by applying
381:
for the axioms consist of a constant symbol 0 and a unary function symbol
8750:
8166:
7367:
4334:
4301:
was more cautious, saying they only defined natural numbers if they were
4288:
3799:
with several weak systems of set theory. One such system is ZFC with the
3043:
2557:
2358:{\displaystyle S(0)\cdot S(a)=S(0)+S(0)\cdot a=S(0)+a=a+S(0)=S(a+0)=S(a)}
1683:
865:
97:
investigations, including research into fundamental questions of whether
7310:
by substitution, contradicting irreflexivity. Therefore it must be that
840:. It is now common to replace this second-order principle with a weaker
9007:
8781:
8732:
8506:
8113:
8072:
8027:
6530:
6503:
3286:
2824:
879:
128:
117:
8981:
7962:
From Kant to
Hilbert: A Source Book in the Foundations of Mathematics
6421:
5790:{\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}
3812:
3301:, starts from a definition of 0 as the empty set, ∅, and an operator
2828:
1705:
906:
856:
If we use the second-order induction axiom, it is possible to define
8773:
8724:
8498:
6401:(if consistent) is incomplete. Consequently, there are sentences of
6013:, i.e. the ordering is preserved under addition of the same element.
5415:{\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}
213:
8939:
Includes a discussion of Poincaré's critique of the Peano's axioms.
8626:
From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931
7761:
7577:
6179:{\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}
5579:{\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}
4310:
2545:
857:
657:
is false. That is, there is no natural number whose successor is 0.
8928:
7115:{\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}
6246:{\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}
200:
is sometimes used for specifically naming this restricted system.
6432:. Undecidability arises already for the existential sentences of
1717:
388:
The first axiom states that the constant 0 is a natural number:
6514:
4852:
2831:
2616:
This relation is stable under addition and multiplication: for
1686:
851:
770:
The induction axiom is sometimes stated in the following form:
8528:. In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).
7460:
7458:
93:. These axioms have been used nearly unchanged in a number of
8008:(1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie".
7260:
by addition of the same element and commutativity, and hence
6006:{\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}
4364:
can be encoded in terms of finite objects (for example, as a
82:
53:
9003:
8943:
Podnieks, Karlis (2015-01-25). "3. First Order Arithmetic".
7836:
8971:
8471:(V ed.). Turin, Bocca frères, Ch. Clausen. p. 27.
8418:. Cambridge: Cambridge University Press. pp. 459–541.
7682:
of which the series of the natural numbers is one instance.
7455:
7431:
4069:) has an initial object; this initial object is known as a
845:
546:
is also a natural number. That is, the natural numbers are
59:
47:
7443:
6328:
satisfying this theory has an initial segment (ordered by
4309:
posed the problem of proving their consistency using only
3803:
replaced by its negation. Another such system consists of
836:
In Peano's original formulation, the induction axiom is a
8581:
8121:(3–4). Reprinted in English translation in 1990. Gödel's
4422:
8993:"What are numbers, and what is their meaning?: Dedekind"
7824:
7809:
7785:
6416:
Closely related to the above incompleteness result (via
6385:
is obtained by adding the first-order induction schema.
5923:{\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}
886:, and can be shown to be unique using the Peano axioms.
583:) is a natural number. That is, the natural numbers are
8864:
The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs
8669:
The principles of arithmetic, presented by a new method
7675:{\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n},\ldots }
4588:{\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}
339:
The Peano axioms define the arithmetical properties of
1439:
1390:
1353:
1279:
1239:
1202:
1152:
1118:
1081:
203:
157:
The principles of arithmetic presented by a new method
7994:
Bertrand Russell's construction of the external world
7848:
7797:
7610:
7316:
7266:
7210:
7178:
7128:
7036:
6989:
6808:
6731:
6684:
6651:
6354:
6334:
6314:
6265:
6257:
by 1, which suggests that these numbers are discrete.
6194:
6112:
6021:
5942:
5865:
5809:
5726:
5673:
5598:
5491:
5430:
5339:
5274:
5179:
5114:
4899:
4837:{\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}
4776:
4728:
4650:
4602:
4524:
4473:
4398:
4155:
3379:
3314:
3138:
2622:
2489:
2411:
2375:
2213:
2184:
2155:
2114:
2094:
2065:
2021:
1992:
1960:
1880:
1844:
1741:
1657:
1619:
1599:
1543:
1511:
1039:
918:
358:
56:
8434:
8203:
Methodology, Epistemology, and Philosophy of Science
7773:
7767:
7379:
6863:
6517:
in 1933 provided an explicit construction of such a
6388:
3822:
3297:. The standard construction of the naturals, due to
3267:. This means that the second-order Peano axioms are
89:
presented by the 19th-century Italian mathematician
44:
7749:
7703:
7482:
7122:by distributivity and additive identity. Secondly,
6722:is a formula in the language of arithmetic so that
2926:Thus, by the strong induction principle, for every
238:. Unsourced material may be challenged and removed.
50:
8593:
8557:
8529:
7991:
7674:
7419:
7334:
7302:
7252:
7196:
7164:
7114:
7022:
6847:
6767:
6714:
6666:
6362:
6340:
6320:
6292:
6245:
6178:
6096:
6005:
5922:
5845:
5789:
5706:
5650:{\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}
5649:
5578:
5475:
5414:
5319:
5254:
5129:
5097:
4836:
4761:
4713:
4635:
4587:
4509:
4411:
4249:
4061:is said to satisfy the Dedekind–Peano axioms if US
3760:
3347:
3252:
2648:
2536:
2472:{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}
2471:
2390:
2357:
2199:
2170:
2141:
2100:
2080:
2048:
2007:
1975:
1943:
1859:
1827:
1663:
1643:
1605:
1585:
1529:
1494:
1019:
369:
310:When Peano formulated his axioms, the language of
8707:(June 1957). "The Axiomatization of Arithmetic".
7715:
7470:
7407:
6502:When interpreted as a proof within a first-order
5090:
5047:
5040:
4967:
4924:
4917:
2918:, for otherwise it would be the least element of
9019:
9012:Creative Commons Attribution/Share-Alike License
8156:Lehrbuch der Arithmetik für höhere Lehranstalten
5476:{\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}
5255:{\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}
1505:To prove commutativity of addition, first prove
876:with first-order induction, this is not possible
8946:What is Mathematics: Gödel's Theorem and Around
8596:A Formalization of Set Theory without Variables
8067:for details on English translations.: 173–198.
8018:. Reprinted in English translation in his 1969
7253:{\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}
6483:
6304:The theory defined by these axioms is known as
4260:This is precisely the recursive definition of 0
8620:
8532:Studies in the Logic of Charles Sanders Peirce
8408:
7543:
7464:
7449:
7437:
3827:The Peano axioms can also be understood using
2537:{\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}
1944:{\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}
8664:Arithmetices principia, nova methodo exposita
8600:. AMS Colloquium Publications. Vol. 41.
8309:Bulletin of the American Mathematical Society
6405:(FOL) that are true in the standard model of
5160:
846:§ Peano arithmetic as first-order theory
166:Arithmetices principia, nova methodo exposita
120:provided by Peano axioms is commonly called
8588:
8536:. Indiana University Press. pp. 43–52.
7950:: CS1 maint: multiple names: authors list (
7583:
7373:Random House Webster's Unabridged Dictionary
7165:{\displaystyle x\cdot 0=0\lor x\cdot 0>0}
6848:{\displaystyle M\vDash \phi (c,{\bar {a}}).}
4714:{\displaystyle \forall x,y\ (x+S(y)=S(x+y))}
3751:
3733:
3727:
3724:
3712:
3697:
3691:
3688:
3676:
3673:
3667:
3655:
3646:
3634:
3596:
3584:
3578:
3575:
3569:
3560:
3554:
3551:
3545:
3542:
3536:
3530:
3521:
3515:
3477:
3471:
3465:
3459:
3453:
3447:
3342:
3336:
852:Defining arithmetic operations and relations
670:of natural numbers. However, axioms 1–8 are
8315:(10). Translated by Winton, Maby: 437–479.
6768:{\displaystyle M\vDash \phi (b,{\bar {a}})}
4293:When the Peano axioms were first proposed,
3362:is defined as the intersection of all sets
3281:Set-theoretic definition of natural numbers
2842:—one can reason as follows. Let a nonempty
131:was not well appreciated until the work of
8911:: CS1 maint: location missing publisher (
8681:
8526:"3. Peirce's Axiomatization of Arithmetic"
7881:Courant Institute of Mathematical Sciences
6537:be the order type of the natural numbers,
6533:of a countable nonstandard model. Letting
6486:, there are other models as well (called "
6374:elements, while other elements are called
5846:{\displaystyle \forall x\ (\neg (x<x))}
5665:for multiplication (actually superfluous).
8833:
8384:
8320:
8149:
8138:
7791:
7546:, sections 2.3 (p. 464) and 4.1 (p. 471).
7385:
6356:
4882:in the language of Peano arithmetic, the
4112:is any other object, then the unique map
2819:This form of the induction axiom, called
2642:
2494:
360:
298:Learn how and when to remove this message
9002:This article incorporates material from
8942:
8857:
8645:
8197:
7905:What are and what should the numbers be?
7893:
7779:
7303:{\displaystyle x\cdot 0+0>x\cdot 0+0}
661:
8883:
8749:
8523:
8302:
8237:
8004:
7755:
7744:
7709:
7572:
7425:
7023:{\displaystyle \forall x\ (x\cdot 0=0)}
6541:be the order type of the integers, and
5707:{\displaystyle \forall x\ (x\cdot 1=x)}
4762:{\displaystyle \forall x\ (x\cdot 0=0)}
4510:{\displaystyle \forall x\ (0\neq S(x))}
3274:
3102:of the Peano axioms, there is a unique
151:of natural-number arithmetic. In 1888,
14:
9020:
8990:
8891:(Second ed.). Mineola, New York.
8841:(6th ed.). Chapman and Hall/CRC.
8552:
8477:
8356:
8273:
8241:The Logical Foundations of Mathematics
8175:Henri Poincaré: A scientific biography
7830:
7815:
7568:
7413:
6795:that is greater than every element of
6370:. Elements in that segment are called
5483:, i.e., multiplication is commutative.
5422:, i.e., multiplication is associative.
5320:{\displaystyle \forall x,y\ (x+y=y+x)}
4423:Peano arithmetic as first-order theory
8972:
8657:
8461:
8103:
8044:
7989:
7958:
7926:
7871:
7733:
7721:
7595:
7488:
7476:
6637:be a nonstandard model of PA and let
6467:
6420:for FOL) it follows that there is no
3285:The Peano axioms can be derived from
2649:{\displaystyle a,b,c\in \mathbb {N} }
901:two natural numbers (two elements of
68:
8812:
8703:
8331:
8165:
8159:. Verlag von Theod. Chr. Fr. Enslin.
7877:Computability. Notes by Barry Jacobs
7854:
7842:
7803:
7691:
7397:
6619:is a cut that is a proper subset of
6293:{\displaystyle \forall x\ (x\geq 0)}
874:directly using the axioms. However,
236:adding citations to reliable sources
207:
8934:Internet Encyclopedia of Philosophy
8442:Mathematical Methods in Linguistics
7900:Was sind und was sollen die Zahlen?
6715:{\displaystyle \phi (x,{\bar {a}})}
6300:, i.e. zero is the minimum element.
3040:Was sind und was sollen die Zahlen?
827:) is true for every natural number
204:Historical second-order formulation
24:
8806:
8276:Introduction to Mathematical Logic
8140:10.1111/j.1746-8361.1958.tb01464.x
7818:, VI.4.3, presenting a theorem of
7768:Partee, Ter Meulen & Wall 2012
6990:
6266:
6207:
6143:
6113:
6022:
5943:
5866:
5822:
5810:
5727:
5674:
5599:
5492:
5431:
5340:
5275:
5180:
5055:
4959:
4900:
4777:
4729:
4651:
4636:{\displaystyle \forall x\ (x+0=x)}
4603:
4525:
4474:
4464:, is sufficient for this purpose:
4400:
3462:
3450:
3441:
3432:
3394:
3046:. In particular, given two models
2860:Because 0 is the least element of
2369:Therefore, by the induction axiom
2365:, using commutativity of addition.
407:The next four axioms describe the
25:
9059:
8926:
8920:
8307:[Mathematical Problems].
8256:Derives the Peano axioms (called
7927:Beman, Wooster, Woodruff (1901).
6389:Undecidability and incompleteness
3823:Interpretation in category theory
3293:and axioms of set theory such as
3027:The Nature and Meaning of Numbers
3000:is a (necessarily infinite) set,
2588:if and only if there exists some
1723:
1613:. Using both results, then prove
6916:Non-standard model of arithmetic
6880:
6866:
6474:Non-standard model of arithmetic
6436:, due to the negative answer to
3348:{\displaystyle s(a)=a\cup \{a\}}
2984:of the Peano axioms is a triple
905:) to another one. It is defined
212:
40:
8487:American Journal of Mathematics
8322:10.1090/s0002-9904-1902-00923-3
7930:Essays on the Theory of Numbers
7738:
7727:
7685:
7589:
7562:
7549:
7512:
7494:
6977:
6967:
6958:
6440:, whose proof implies that all
6395:Gödel's incompleteness theorems
5930:, i.e., the ordering satisfies
5853:, i.e., the '<' operator is
5797:, i.e., the '<' operator is
2551:
1712:. The smallest group embedding
223:needs additional citations for
9010:, which is licensed under the
8997:Commentary on Dedekind's work.
8817:. New York: Elsevier Science.
8717:Association for Symbolic Logic
8682:Van Oosten, Jaap (June 1999).
8580:Derives the Peano axioms from
8412:Algebraic Methods in Semantics
7990:Fritz, Charles A. Jr. (1952).
7361:
7079:
7067:
7017:
6999:
6839:
6833:
6818:
6762:
6756:
6741:
6709:
6703:
6688:
6658:
6287:
6275:
6240:
6228:
6216:
6173:
6170:
6152:
6140:
6128:
6091:
6067:
6043:
6000:
5976:
5964:
5917:
5881:
5840:
5837:
5825:
5819:
5784:
5772:
5748:
5701:
5683:
5644:
5608:
5573:
5570:
5558:
5552:
5540:
5534:
5522:
5513:
5470:
5446:
5409:
5406:
5394:
5376:
5364:
5361:
5314:
5290:
5249:
5246:
5234:
5216:
5204:
5201:
5166:operation and the language of
5121:
5085:
5079:
5064:
5052:
5035:
5029:
5017:
5011:
5005:
4999:
4996:
4990:
4975:
4953:
4947:
4932:
4909:
4831:
4810:
4804:
4792:
4756:
4738:
4708:
4705:
4693:
4684:
4678:
4666:
4630:
4612:
4582:
4570:
4567:
4561:
4552:
4546:
4540:
4504:
4501:
4495:
4483:
4278:
4237:
4228:
4208:
4199:
4169:
4163:
3649:
3631:
3622:
3616:
3524:
3512:
3503:
3497:
3435:
3429:
3420:
3414:
3324:
3318:
3243:
3240:
3234:
3228:
3208:
3205:
3199:
3186:
3159:
3146:
2531:
2528:
2522:
2490:
2466:
2454:
2448:
2436:
2430:
2418:
2385:
2379:
2352:
2346:
2337:
2325:
2316:
2310:
2289:
2283:
2268:
2262:
2253:
2247:
2238:
2232:
2223:
2217:
2194:
2188:
2165:
2159:
2124:
2118:
2075:
2069:
2031:
2025:
2002:
1996:
1970:
1964:
1920:
1908:
1896:
1890:
1854:
1848:
1815:
1803:
1787:
1781:
1580:
1568:
1553:
1547:
1475:
1472:
1466:
1460:
1433:
1430:
1427:
1421:
1415:
1409:
1384:
1372:
1347:
1341:
1306:
1300:
1273:
1270:
1264:
1258:
1233:
1221:
1196:
1190:
1143:
1137:
1112:
1100:
1075:
1069:
998:
986:
973:
967:
765:contains every natural number.
127:The importance of formalizing
27:Axioms for the natural numbers
13:
1:
9038:Formal theories of arithmetic
8762:The Journal of Symbolic Logic
8709:The Journal of Symbolic Logic
8602:American Mathematical Society
8238:Hatcher, William S. (2014) .
8022:, M. E. Szabo, ed.: 132–213.
7998:. New York, Humanities Press.
7349:
7197:{\displaystyle x\cdot 0>0}
6615:is closed under successor. A
4327:second incompleteness theorem
3849:, and define the category of
2178:is also the left identity of
2142:{\displaystyle S(0)\cdot a=a}
2049:{\displaystyle S(0)\cdot 0=0}
1586:{\displaystyle S(a)+b=S(a+b)}
370:{\displaystyle \mathbb {N} .}
8628:. Harvard University Press.
7555:For formal proofs, see e.g.
7354:
6562:
6418:Gödel's completeness theorem
6363:{\displaystyle \mathbb {N} }
5168:discretely ordered semirings
4089:is this initial object, and
3792:satisfies the Peano axioms.
3021:satisfies the axioms above.
7:
8991:Burris, Stanley N. (2001).
8960:Encyclopedia of Mathematics
8278:. Hochschultext. Springer.
8211:10.1007/978-94-015-7676-5_8
6946:Typographical Number Theory
6859:
4884:first-order induction axiom
3358:The set of natural numbers
2964:being a nonempty subset of
2015:is the left identity of 0:
889:
686:Axioms 1, 6, 7, 8 define a
345:, usually represented as a
10:
9064:
8392:(4th ed.). Springer.
8334:Models of Peano arithmetic
8179:Princeton University Press
8151:Grassmann, Hermann Günther
8057:Monatshefte für Mathematik
7959:Ewald, William B. (1996).
7923:Two English translations:
7864:
7544:Meseguer & Goguen 1986
7335:{\displaystyle x\cdot 0=0}
6896:Foundations of mathematics
6674:is a tuple of elements of
6667:{\displaystyle {\bar {a}}}
6471:
6381:Finally, Peano arithmetic
5161:Equivalent axiomatizations
5130:{\displaystyle {\bar {y}}}
4851:and even decidable set of
4343:a proof of the consistency
4282:
3278:
804:) being true implies that
9043:Logic in computer science
8592:; Givant, Steven (1987).
4345:of Peano's axioms, using
3025:proved in his 1888 book,
2975:
1689:with identity element 0.
792:for every natural number
734:for every natural number
642:For every natural number
571:For every natural number
419:For every natural number
192:axiom with a first-order
8815:Handbook of Proof Theory
8483:"On the Logic of Number"
8305:"Mathematische Probleme"
7584:Tarski & Givant 1987
6951:
6921:Paris–Harrington theorem
6496:Löwenheim–Skolem theorem
4412:{\displaystyle \Pi _{1}}
4285:Hilbert's second problem
3775:together with 0 and the
2088:is the left identity of
594:For all natural numbers
532:is a natural number and
475:For all natural numbers
440:For all natural numbers
8439:; Wall, Robert (2012).
8362:Grundlagen Der Analysis
8338:Oxford University Press
8303:Hilbert, David (1902).
8131:Oxford University Press
7967:Oxford University Press
7508:Simon Fraser University
6941:Second-order arithmetic
6571:in a nonstandard model
6438:Hilbert's tenth problem
5586:, i.e., multiplication
5137:is an abbreviation for
4389:self-verifying theories
4387:. Curiously, there are
1838:It is easy to see that
1644:{\displaystyle a+b=b+a}
1593:, each by induction on
866:total (linear) ordering
773:
718:
569:
513:. That is, equality is
468:. That is, equality is
433:. That is, equality is
417:
391:
186:second-order arithmetic
8867:. Dover Publications.
8524:Shields, Paul (1997).
8468:Formulario Mathematico
8366:AMS Chelsea Publishing
8332:Kaye, Richard (1991).
8262:axiomatic set theories
7676:
7521:Mathematical Induction
7503:Mathematical Induction
7336:
7304:
7254:
7198:
7166:
7116:
7024:
6849:
6769:
6716:
6668:
6521:. On the other hand,
6482:satisfy the axioms of
6462:arithmetical hierarchy
6364:
6342:
6322:
6294:
6247:
6180:
6098:
6007:
5924:
5847:
5791:
5708:
5651:
5580:
5477:
5416:
5321:
5256:
5131:
5099:
4849:recursively enumerable
4838:
4763:
4715:
4637:
4589:
4511:
4413:
4251:
3762:
3349:
3254:
3039:
2856:has no least element.
2650:
2538:
2473:
2392:
2359:
2201:
2172:
2143:
2102:
2082:
2050:
2009:
1977:
1945:
1867:is the multiplicative
1861:
1829:
1665:
1645:
1607:
1587:
1531:
1496:
1021:
683:
401:Formulario mathematico
393:0 is a natural number.
371:
190:second-order induction
164:
145:Charles Sanders Peirce
8274:Hermes, Hans (1973).
8011:Mathematische Annalen
7677:
7337:
7305:
7255:
7199:
7167:
7117:
7025:
6926:Presburger arithmetic
6850:
6770:
6717:
6669:
6575:is a nonempty subset
6442:computably enumerable
6365:
6343:
6341:{\displaystyle \leq }
6323:
6295:
6248:
6181:
6099:
6008:
5925:
5848:
5792:
5709:
5661:for addition, and an
5652:
5581:
5478:
5417:
5322:
5257:
5132:
5100:
4839:
4764:
4716:
4638:
4590:
4512:
4414:
4347:transfinite induction
4319:twenty-three problems
4283:Further information:
4252:
4071:natural number object
3851:pointed unary systems
3763:
3350:
3289:constructions of the
3255:
2972:has a least element.
2651:
2539:
2474:
2393:
2360:
2202:
2173:
2144:
2103:
2083:
2051:
2010:
1978:
1946:
1862:
1830:
1666:
1646:
1608:
1588:
1532:
1530:{\displaystyle 0+b=b}
1497:
1022:
665:
372:
75:Dedekind–Peano axioms
73:), also known as the
8859:Smullyan, Raymond M.
8689:. Utrecht University
8651:Letter to Keferstein
8622:Van Heijenoort, Jean
8560:Axiomatic Set Theory
8205:. pp. 167–192.
7969:. pp. 787–832.
7608:
7314:
7264:
7208:
7176:
7126:
7034:
6987:
6806:
6729:
6682:
6649:
6587:is downward closed (
6523:Tennenbaum's theorem
6428:is an example of an
6352:
6332:
6312:
6263:
6192:
6110:
6019:
5940:
5863:
5807:
5724:
5671:
5596:
5489:
5428:
5337:
5327:, i.e., addition is
5272:
5262:, i.e., addition is
5177:
5112:
4897:
4774:
4726:
4648:
4600:
4522:
4471:
4396:
4153:
3795:Peano arithmetic is
3377:
3312:
3305:on sets defined as:
3275:Set-theoretic models
3136:
2884:, suppose for every
2852:be given and assume
2760:) is true for every
2707:Thus, the structure
2620:
2487:
2409:
2391:{\displaystyle S(0)}
2373:
2211:
2200:{\displaystyle S(a)}
2182:
2171:{\displaystyle S(0)}
2153:
2112:
2092:
2081:{\displaystyle S(0)}
2063:
2019:
2008:{\displaystyle S(0)}
1990:
1976:{\displaystyle S(0)}
1958:
1878:
1860:{\displaystyle S(0)}
1842:
1739:
1655:
1617:
1597:
1541:
1509:
1037:
916:
724:is a set such that:
688:unary representation
356:
232:improve this article
9033:Mathematical axioms
7885:New York University
7465:Van Heijenoort 1967
7450:Van Heijenoort 1967
7438:Van Heijenoort 1967
6936:Robinson arithmetic
6906:Goodstein's theorem
6641:be a proper cut of
6631: —
6492:compactness theorem
6488:non-standard models
6478:Although the usual
6411:Robinson arithmetic
5718:for multiplication.
5657:, i.e., zero is an
4855:. For each formula
4462:Robinson arithmetic
4451:successor operation
3817:axiom of adjunction
3811:, existence of the
3771:and so on. The set
897:is a function that
379:non-logical symbols
137:successor operation
9048:Mathematical logic
9028:1889 introductions
8974:Weisstein, Eric W.
8949:. pp. 93–121.
8835:Mendelson, Elliott
8566:Dover Publications
8479:Peirce, C. S.
8388:(December 1997) .
8386:Mendelson, Elliott
8073:10.1007/bf01700692
8028:10.1007/bf01565428
7938:Dover Publications
7672:
7533:Harvard University
7526:2 May 2013 at the
7518:Gerardo con Diaz,
7332:
7300:
7250:
7194:
7162:
7112:
7020:
6888:Mathematics portal
6845:
6765:
6712:
6664:
6629:
6468:Nonstandard models
6430:undecidable theory
6360:
6338:
6318:
6290:
6243:
6176:
6094:
6003:
5920:
5843:
5787:
5714:, i.e., one is an
5704:
5647:
5576:
5473:
5412:
5317:
5252:
5127:
5095:
4834:
4759:
4711:
4633:
4585:
4507:
4409:
4247:
4245:
3805:general set theory
3777:successor function
3758:
3756:
3345:
3250:
3248:
2864:, it must be that
2646:
2534:
2469:
2388:
2355:
2197:
2168:
2139:
2098:
2078:
2046:
2005:
1973:
1941:
1857:
1825:
1823:
1661:
1641:
1603:
1583:
1527:
1492:
1490:
1443:
1394:
1357:
1283:
1243:
1206:
1156:
1122:
1085:
1017:
1015:
884:second-order logic
838:second-order axiom
712:axiom of induction
684:
367:
312:mathematical logic
32:mathematical logic
8898:978-0-486-49073-1
8874:978-0-486-49705-1
8861:(December 2013).
8848:978-1-4822-3772-6
8824:978-0-444-89840-1
8647:Dedekind, Richard
8635:978-0-674-32449-7
8611:978-0-8218-1041-5
8452:978-94-009-2213-6
8437:Ter Meulen, Alice
8435:Partee, Barbara;
8425:978-0-521-26793-9
8399:978-0-412-80830-2
8375:978-0-8284-0141-8
8251:978-1-4831-8963-5
8220:978-90-481-8389-0
8199:Harsanyi, John C.
8188:978-0-691-15271-4
7976:978-0-19-853271-2
7895:Dedekind, Richard
7857:, pp. 70ff..
7806:, pp. 16–18.
6998:
6931:Skolem arithmetic
6874:Philosophy portal
6836:
6759:
6706:
6661:
6627:
6519:nonstandard model
6403:first-order logic
6321:{\displaystyle M}
6274:
6215:
6151:
6127:
6042:
5963:
5880:
5818:
5747:
5682:
5663:absorbing element
5607:
5512:
5445:
5360:
5289:
5200:
5124:
5082:
5032:
4993:
4950:
4912:
4791:
4737:
4665:
4611:
4539:
4482:
4429:first-order logic
3865:The objects of US
3801:axiom of infinity
2544:is a commutative
2101:{\displaystyle a}
1664:{\displaystyle b}
1606:{\displaystyle b}
1485:
1442:
1393:
1356:
1282:
1242:
1205:
1155:
1121:
1084:
1010:
951:
308:
307:
300:
282:
133:Hermann Grassmann
70:[peˈaːno]
16:(Redirected from
9055:
8996:
8987:
8986:
8977:"Peano's Axioms"
8968:
8950:
8938:
8929:"Henri Poincaré"
8916:
8910:
8902:
8878:
8852:
8828:
8801:
8759:
8744:
8698:
8696:
8694:
8688:
8672:
8654:
8639:
8615:
8599:
8579:
8563:
8547:
8535:
8518:
8472:
8456:
8429:
8417:
8403:
8379:
8351:
8326:
8324:
8297:
8255:
8232:
8192:
8160:
8144:
8142:
8127:Solomon Feferman
8098:
8096:
8095:
8089:
8083:. Archived from
8054:
8039:
8006:Gentzen, Gerhard
7999:
7997:
7980:
7955:
7949:
7941:
7935:
7920:
7918:
7916:
7910:
7888:
7858:
7852:
7846:
7840:
7834:
7828:
7822:
7813:
7807:
7801:
7795:
7789:
7783:
7777:
7771:
7765:
7759:
7753:
7747:
7742:
7736:
7731:
7725:
7719:
7713:
7707:
7701:
7689:
7683:
7681:
7679:
7678:
7673:
7665:
7664:
7646:
7645:
7633:
7632:
7620:
7619:
7593:
7587:
7581:
7575:
7566:
7560:
7553:
7547:
7541:
7535:
7516:
7510:
7498:
7492:
7486:
7480:
7474:
7468:
7462:
7453:
7447:
7441:
7435:
7429:
7423:
7417:
7411:
7405:
7395:
7389:
7383:
7377:
7365:
7343:
7341:
7339:
7338:
7333:
7309:
7307:
7306:
7301:
7259:
7257:
7256:
7251:
7203:
7201:
7200:
7195:
7172:by Axiom 15. If
7171:
7169:
7168:
7163:
7121:
7119:
7118:
7113:
7029:
7027:
7026:
7021:
6996:
6981:
6975:
6971:
6965:
6962:
6890:
6885:
6884:
6876:
6871:
6870:
6869:
6854:
6852:
6851:
6846:
6838:
6837:
6829:
6787:Then there is a
6774:
6772:
6771:
6766:
6761:
6760:
6752:
6721:
6719:
6718:
6713:
6708:
6707:
6699:
6673:
6671:
6670:
6665:
6663:
6662:
6654:
6632:
6558:
6446:diophantine sets
6397:, the theory of
6369:
6367:
6366:
6361:
6359:
6348:) isomorphic to
6347:
6345:
6344:
6339:
6327:
6325:
6324:
6319:
6299:
6297:
6296:
6291:
6272:
6252:
6250:
6249:
6244:
6213:
6185:
6183:
6182:
6177:
6149:
6125:
6103:
6101:
6100:
6095:
6040:
6012:
6010:
6009:
6004:
5961:
5929:
5927:
5926:
5921:
5878:
5852:
5850:
5849:
5844:
5816:
5796:
5794:
5793:
5788:
5745:
5713:
5711:
5710:
5705:
5680:
5656:
5654:
5653:
5648:
5605:
5585:
5583:
5582:
5577:
5510:
5482:
5480:
5479:
5474:
5443:
5421:
5419:
5418:
5413:
5358:
5326:
5324:
5323:
5318:
5287:
5261:
5259:
5258:
5253:
5198:
5136:
5134:
5133:
5128:
5126:
5125:
5117:
5104:
5102:
5101:
5096:
5094:
5093:
5084:
5083:
5075:
5051:
5050:
5044:
5043:
5034:
5033:
5025:
4995:
4994:
4986:
4971:
4970:
4952:
4951:
4943:
4928:
4927:
4921:
4920:
4914:
4913:
4905:
4890:is the sentence
4881:
4843:
4841:
4840:
4835:
4789:
4768:
4766:
4765:
4760:
4735:
4720:
4718:
4717:
4712:
4663:
4642:
4640:
4639:
4634:
4609:
4594:
4592:
4591:
4586:
4537:
4516:
4514:
4513:
4508:
4480:
4418:
4416:
4415:
4410:
4408:
4407:
4295:Bertrand Russell
4256:
4254:
4253:
4248:
4246:
4227:
4226:
4188:
4187:
4145:
4111:
4088:
4052:
4026:
4008:
3941:
3922:
3900:is an object of
3895:
3791:
3767:
3765:
3764:
3759:
3757:
3354:
3352:
3351:
3346:
3299:John von Neumann
3259:
3257:
3256:
3251:
3249:
3227:
3226:
3198:
3197:
3178:
3177:
3158:
3157:
3128:
3101:
3073:
3034:
3020:
3006:
2995:
2956:
2945:
2935:
2917:
2903:
2893:
2883:
2870:
2851:
2821:strong induction
2804:
2779:
2769:
2751:
2739:(0) is true, and
2717:ordered semiring
2714:
2713:, +, ·, 1, 0, ≤)
2665:
2655:
2653:
2652:
2647:
2645:
2611:
2597:
2587:
2577:
2543:
2541:
2540:
2535:
2497:
2478:
2476:
2475:
2470:
2400:distributes over
2397:
2395:
2394:
2389:
2364:
2362:
2361:
2356:
2206:
2204:
2203:
2198:
2177:
2175:
2174:
2169:
2148:
2146:
2145:
2140:
2107:
2105:
2104:
2099:
2087:
2085:
2084:
2079:
2055:
2053:
2052:
2047:
2014:
2012:
2011:
2006:
1982:
1980:
1979:
1974:
1950:
1948:
1947:
1942:
1866:
1864:
1863:
1858:
1834:
1832:
1831:
1826:
1824:
1696:
1681:
1670:
1668:
1667:
1662:
1651:by induction on
1650:
1648:
1647:
1642:
1612:
1610:
1609:
1604:
1592:
1590:
1589:
1584:
1536:
1534:
1533:
1528:
1501:
1499:
1498:
1493:
1491:
1488:
1486:
1483:
1444:
1440:
1399:
1395:
1391:
1362:
1358:
1354:
1312:
1284:
1280:
1248:
1244:
1240:
1211:
1207:
1203:
1161:
1157:
1153:
1127:
1123:
1119:
1090:
1086:
1082:
1026:
1024:
1023:
1018:
1016:
1012:
1011:
1008:
953:
952:
949:
877:
789:(0) is true, and
656:
630:
620:
541:
512:
467:
457:
432:
376:
374:
373:
368:
363:
303:
296:
292:
289:
283:
281:
240:
216:
208:
198:Peano arithmetic
153:Richard Dedekind
122:Peano arithmetic
95:metamathematical
79:Peano postulates
72:
66:
65:
62:
61:
58:
55:
52:
49:
46:
21:
18:Peano arithmetic
9063:
9062:
9058:
9057:
9056:
9054:
9053:
9052:
9018:
9017:
8953:
8923:
8904:
8903:
8899:
8875:
8849:
8825:
8809:
8807:Further reading
8804:
8774:10.2307/2695030
8757:
8751:Willard, Dan E.
8725:10.2307/2964176
8692:
8690:
8686:
8659:Peano, Giuseppe
8636:
8612:
8576:
8554:Suppes, Patrick
8544:
8499:10.2307/2369151
8463:Peano, Giuseppe
8453:
8426:
8415:
8400:
8376:
8348:
8286:
8266:category theory
8260:) from several
8252:
8221:
8189:
8181:. p. 133.
8123:Collected Works
8093:
8091:
8087:
8052:
8020:Collected works
7977:
7943:
7942:
7933:
7914:
7912:
7908:
7867:
7862:
7861:
7853:
7849:
7845:, Section 11.3.
7841:
7837:
7829:
7825:
7814:
7810:
7802:
7798:
7790:
7786:
7780:Harsanyi (1983)
7778:
7774:
7766:
7762:
7754:
7750:
7743:
7739:
7732:
7728:
7720:
7716:
7708:
7704:
7698:
7690:
7686:
7660:
7656:
7641:
7637:
7628:
7624:
7615:
7611:
7609:
7606:
7605:
7602:
7594:
7590:
7582:
7578:
7567:
7563:
7554:
7550:
7542:
7538:
7528:Wayback Machine
7517:
7513:
7499:
7495:
7487:
7483:
7475:
7471:
7463:
7456:
7448:
7444:
7436:
7432:
7424:
7420:
7412:
7408:
7402:integral domain
7396:
7392:
7384:
7380:
7366:
7362:
7357:
7352:
7347:
7346:
7315:
7312:
7311:
7265:
7262:
7261:
7209:
7206:
7205:
7177:
7174:
7173:
7127:
7124:
7123:
7035:
7032:
7031:
6988:
6985:
6984:
6982:
6978:
6972:
6968:
6963:
6959:
6954:
6901:Frege's theorem
6886:
6879:
6872:
6867:
6865:
6862:
6857:
6828:
6827:
6807:
6804:
6803:
6751:
6750:
6730:
6727:
6726:
6698:
6697:
6683:
6680:
6679:
6653:
6652:
6650:
6647:
6646:
6645:. Suppose that
6630:
6628:Overspill lemma
6565:
6546:
6480:natural numbers
6476:
6470:
6458:quantifier rank
6391:
6355:
6353:
6350:
6349:
6333:
6330:
6329:
6313:
6310:
6309:
6264:
6261:
6260:
6193:
6190:
6189:
6111:
6108:
6107:
6020:
6017:
6016:
5941:
5938:
5937:
5864:
5861:
5860:
5808:
5805:
5804:
5725:
5722:
5721:
5672:
5669:
5668:
5597:
5594:
5593:
5490:
5487:
5486:
5429:
5426:
5425:
5338:
5335:
5334:
5273:
5270:
5269:
5178:
5175:
5174:
5163:
5152:
5143:
5116:
5115:
5113:
5110:
5109:
5089:
5088:
5074:
5073:
5046:
5045:
5039:
5038:
5024:
5023:
4985:
4984:
4966:
4965:
4942:
4941:
4923:
4922:
4916:
4915:
4904:
4903:
4898:
4895:
4894:
4879:
4870:
4856:
4775:
4772:
4771:
4727:
4724:
4723:
4649:
4646:
4645:
4601:
4598:
4597:
4523:
4520:
4519:
4472:
4469:
4468:
4425:
4403:
4399:
4397:
4394:
4393:
4377:Gentzen's proof
4363:
4358:
4339:Gerhard Gentzen
4313:methods as the
4291:
4281:
4274:
4265:
4244:
4243:
4222:
4218:
4211:
4193:
4192:
4183:
4179:
4172:
4156:
4154:
4151:
4150:
4143:
4134:
4113:
4109:
4100:
4090:
4078:
4064:
4048:
4039:
4028:
4025:
4019:
4010:
3996:
3990:
3981:
3971:
3962:
3932:
3924:
3917:
3911:
3905:
3893:
3884:
3874:
3868:
3856:
3848:
3841:terminal object
3829:category theory
3825:
3779:
3755:
3754:
3606:
3600:
3599:
3487:
3481:
3480:
3404:
3398:
3397:
3387:
3380:
3378:
3375:
3374:
3313:
3310:
3309:
3291:natural numbers
3283:
3277:
3247:
3246:
3222:
3218:
3211:
3193:
3189:
3180:
3179:
3173:
3169:
3162:
3153:
3149:
3139:
3137:
3134:
3133:
3127:
3118:
3106:
3099:
3090:
3084:
3075:
3071:
3062:
3056:
3047:
3030:
3008:
3001:
2985:
2978:
2947:
2937:
2927:
2905:
2895:
2885:
2875:
2865:
2843:
2796:
2795:then for every
2771:
2761:
2743:
2708:
2657:
2641:
2621:
2618:
2617:
2599:
2589:
2579:
2565:
2554:
2493:
2488:
2485:
2484:
2410:
2407:
2406:
2374:
2371:
2370:
2212:
2209:
2208:
2183:
2180:
2179:
2154:
2151:
2150:
2113:
2110:
2109:
2093:
2090:
2089:
2064:
2061:
2060:
2020:
2017:
2016:
1991:
1988:
1987:
1959:
1956:
1955:
1879:
1876:
1875:
1843:
1840:
1839:
1822:
1821:
1790:
1769:
1768:
1755:
1742:
1740:
1737:
1736:
1726:
1690:
1675:
1656:
1653:
1652:
1618:
1615:
1614:
1598:
1595:
1594:
1542:
1539:
1538:
1510:
1507:
1506:
1489:
1487:
1482:
1479:
1478:
1438:
1436:
1397:
1396:
1389:
1387:
1360:
1359:
1352:
1350:
1325:
1313:
1310:
1309:
1278:
1276:
1246:
1245:
1238:
1236:
1209:
1208:
1201:
1199:
1174:
1162:
1159:
1158:
1151:
1149:
1125:
1124:
1117:
1115:
1088:
1087:
1080:
1078:
1053:
1040:
1038:
1035:
1034:
1014:
1013:
1007:
1006:
1004:
976:
955:
954:
948:
947:
945:
932:
919:
917:
914:
913:
892:
875:
854:
834:
768:
660:
647:
622:
603:
553:
550:under equality.
533:
504:
459:
449:
424:
396:
359:
357:
354:
353:
342:natural numbers
321:Begriffsschrift
304:
293:
287:
284:
241:
239:
229:
217:
206:
87:natural numbers
43:
39:
28:
23:
22:
15:
12:
11:
5:
9061:
9051:
9050:
9045:
9040:
9035:
9030:
8999:
8998:
8988:
8969:
8955:"Peano axioms"
8951:
8940:
8927:Murzi, Mauro.
8922:
8921:External links
8919:
8918:
8917:
8897:
8885:Takeuti, Gaisi
8880:
8879:
8873:
8854:
8853:
8847:
8837:(June 2015) .
8830:
8829:
8823:
8808:
8805:
8803:
8802:
8768:(2): 536–596.
8746:
8745:
8700:
8699:
8678:
8677:
8676:
8675:
8674:
8673:
8655:
8634:
8617:
8616:
8610:
8590:Tarski, Alfred
8585:
8584:
8574:
8549:
8548:
8542:
8520:
8519:
8474:
8473:
8458:
8457:
8451:
8431:
8430:
8424:
8405:
8404:
8398:
8381:
8380:
8374:
8358:Landau, Edmund
8353:
8352:
8346:
8328:
8327:
8299:
8298:
8284:
8270:
8269:
8250:
8234:
8233:
8219:
8194:
8193:
8187:
8171:"The Essayist"
8162:
8161:
8146:
8145:
8100:
8099:
8041:
8040:
8001:
8000:
7986:
7985:
7984:
7983:
7982:
7981:
7975:
7956:
7890:
7889:
7868:
7866:
7863:
7860:
7859:
7847:
7835:
7823:
7820:Thoralf Skolem
7808:
7796:
7794:, p. 155.
7792:Mendelson 1997
7784:
7772:
7770:, p. 215.
7760:
7748:
7737:
7726:
7714:
7702:
7684:
7671:
7668:
7663:
7659:
7655:
7652:
7649:
7644:
7640:
7636:
7631:
7627:
7623:
7618:
7614:
7588:
7586:, Section 7.6.
7576:
7561:
7548:
7536:
7511:
7493:
7481:
7469:
7454:
7442:
7430:
7418:
7406:
7390:
7386:Grassmann 1861
7378:
7359:
7358:
7356:
7353:
7351:
7348:
7345:
7344:
7331:
7328:
7325:
7322:
7319:
7299:
7296:
7293:
7290:
7287:
7284:
7281:
7278:
7275:
7272:
7269:
7249:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7193:
7190:
7187:
7184:
7181:
7161:
7158:
7155:
7152:
7149:
7146:
7143:
7140:
7137:
7134:
7131:
7111:
7108:
7105:
7102:
7099:
7096:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7063:
7060:
7057:
7054:
7051:
7048:
7045:
7042:
7039:
7019:
7016:
7013:
7010:
7007:
7004:
7001:
6995:
6992:
6976:
6966:
6956:
6955:
6953:
6950:
6949:
6948:
6943:
6938:
6933:
6928:
6923:
6918:
6913:
6908:
6903:
6898:
6892:
6891:
6877:
6861:
6858:
6856:
6855:
6844:
6841:
6835:
6832:
6826:
6823:
6820:
6817:
6814:
6811:
6785:
6784:
6764:
6758:
6755:
6749:
6746:
6743:
6740:
6737:
6734:
6711:
6705:
6702:
6696:
6693:
6690:
6687:
6660:
6657:
6625:
6564:
6561:
6472:Main article:
6469:
6466:
6452:. Formulas of
6390:
6387:
6358:
6337:
6317:
6302:
6301:
6289:
6286:
6283:
6280:
6277:
6271:
6268:
6258:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6212:
6209:
6206:
6203:
6200:
6197:
6187:
6175:
6172:
6169:
6166:
6163:
6160:
6157:
6154:
6148:
6145:
6142:
6139:
6136:
6133:
6130:
6124:
6121:
6118:
6115:
6105:
6093:
6090:
6087:
6084:
6081:
6078:
6075:
6072:
6069:
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6045:
6039:
6036:
6033:
6030:
6027:
6024:
6014:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5960:
5957:
5954:
5951:
5948:
5945:
5935:
5919:
5916:
5913:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5877:
5874:
5871:
5868:
5858:
5842:
5839:
5836:
5833:
5830:
5827:
5824:
5821:
5815:
5812:
5802:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5744:
5741:
5738:
5735:
5732:
5729:
5719:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5679:
5676:
5666:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5604:
5601:
5591:
5590:over addition.
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5509:
5506:
5503:
5500:
5497:
5494:
5484:
5472:
5469:
5466:
5463:
5460:
5457:
5454:
5451:
5448:
5442:
5439:
5436:
5433:
5423:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5357:
5354:
5351:
5348:
5345:
5342:
5332:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5286:
5283:
5280:
5277:
5267:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5203:
5197:
5194:
5191:
5188:
5185:
5182:
5162:
5159:
5148:
5141:
5123:
5120:
5106:
5105:
5092:
5087:
5081:
5078:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5049:
5042:
5037:
5031:
5028:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4992:
4989:
4983:
4980:
4977:
4974:
4969:
4964:
4961:
4958:
4955:
4949:
4946:
4940:
4937:
4934:
4931:
4926:
4919:
4911:
4908:
4902:
4875:
4868:
4845:
4844:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4788:
4785:
4782:
4779:
4769:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4734:
4731:
4721:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4662:
4659:
4656:
4653:
4643:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4608:
4605:
4595:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4536:
4533:
4530:
4527:
4517:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4479:
4476:
4424:
4421:
4406:
4402:
4366:Turing machine
4361:
4356:
4299:Henri Poincaré
4280:
4277:
4270:
4261:
4258:
4257:
4242:
4239:
4236:
4233:
4230:
4225:
4221:
4217:
4214:
4212:
4210:
4207:
4204:
4201:
4198:
4195:
4194:
4191:
4186:
4182:
4178:
4175:
4173:
4171:
4168:
4165:
4162:
4159:
4158:
4139:
4130:
4105:
4096:
4062:
4055:
4054:
4044:
4035:
4021:
4015:
3986:
3977:
3967:
3958:
3947:
3928:
3913:
3907:
3889:
3880:
3873:) are triples
3866:
3861:) as follows:
3854:
3844:
3824:
3821:
3809:extensionality
3797:equiconsistent
3769:
3768:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3607:
3605:
3602:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3488:
3486:
3483:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3405:
3403:
3400:
3399:
3396:
3393:
3390:
3388:
3386:
3383:
3382:
3356:
3355:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3279:Main article:
3276:
3273:
3261:
3260:
3245:
3242:
3239:
3236:
3233:
3230:
3225:
3221:
3217:
3214:
3212:
3210:
3207:
3204:
3201:
3196:
3192:
3188:
3185:
3182:
3181:
3176:
3172:
3168:
3165:
3163:
3161:
3156:
3152:
3148:
3145:
3142:
3141:
3123:
3114:
3095:
3086:
3080:
3067:
3058:
3052:
2977:
2974:
2924:
2923:
2872:
2817:
2816:
2815:
2814:
2793:
2740:
2705:
2704:
2686:
2644:
2640:
2637:
2634:
2631:
2628:
2625:
2614:
2613:
2553:
2550:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2496:
2492:
2481:
2480:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2387:
2384:
2381:
2378:
2367:
2366:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2196:
2193:
2190:
2187:
2167:
2164:
2161:
2158:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2097:
2077:
2074:
2071:
2068:
2057:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2004:
2001:
1998:
1995:
1972:
1969:
1966:
1963:
1952:
1951:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1869:right identity
1856:
1853:
1850:
1847:
1836:
1835:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1791:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1770:
1767:
1764:
1761:
1758:
1756:
1754:
1751:
1748:
1745:
1744:
1730:multiplication
1725:
1724:Multiplication
1722:
1660:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1602:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1526:
1523:
1520:
1517:
1514:
1503:
1502:
1481:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1437:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1400:
1398:
1388:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1363:
1361:
1351:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1326:
1324:
1321:
1318:
1315:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1277:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1249:
1247:
1237:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1212:
1210:
1200:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1175:
1173:
1170:
1167:
1164:
1163:
1160:
1150:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1128:
1126:
1116:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1091:
1089:
1079:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1054:
1052:
1049:
1046:
1043:
1042:
1028:
1027:
1005:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
977:
975:
972:
969:
966:
963:
960:
957:
956:
946:
944:
941:
938:
935:
933:
931:
928:
925:
922:
921:
891:
888:
862:multiplication
853:
850:
833:
832:
818:
817:
790:
772:
767:
766:
760:
759:
732:
717:
659:
658:
640:
592:
568:
552:
551:
518:
473:
438:
416:
404:include zero.
395:
394:
390:
366:
362:
316:set membership
306:
305:
247:"Peano axioms"
220:
218:
211:
205:
202:
149:axiomatization
114:axiomatization
91:Giuseppe Peano
26:
9:
6:
4:
3:
2:
9060:
9049:
9046:
9044:
9041:
9039:
9036:
9034:
9031:
9029:
9026:
9025:
9023:
9016:
9015:
9013:
9009:
9005:
8994:
8989:
8984:
8983:
8978:
8975:
8970:
8966:
8962:
8961:
8956:
8952:
8948:
8947:
8941:
8936:
8935:
8930:
8925:
8924:
8914:
8908:
8900:
8894:
8890:
8886:
8882:
8881:
8876:
8870:
8866:
8865:
8860:
8856:
8855:
8850:
8844:
8840:
8836:
8832:
8831:
8826:
8820:
8816:
8811:
8810:
8799:
8795:
8791:
8787:
8783:
8779:
8775:
8771:
8767:
8763:
8756:
8752:
8748:
8747:
8742:
8738:
8734:
8730:
8726:
8722:
8718:
8714:
8710:
8706:
8702:
8701:
8685:
8680:
8679:
8670:
8666:
8665:
8660:
8656:
8652:
8648:
8644:
8643:
8641:
8640:
8637:
8631:
8627:
8623:
8619:
8618:
8613:
8607:
8603:
8598:
8597:
8591:
8587:
8586:
8583:
8577:
8575:0-486-61630-4
8571:
8567:
8562:
8561:
8555:
8551:
8550:
8545:
8543:0-253-33020-3
8539:
8534:
8533:
8527:
8522:
8521:
8516:
8512:
8508:
8504:
8500:
8496:
8492:
8488:
8484:
8480:
8476:
8475:
8470:
8469:
8464:
8460:
8459:
8454:
8448:
8444:
8443:
8438:
8433:
8432:
8427:
8421:
8414:
8413:
8407:
8406:
8401:
8395:
8391:
8387:
8383:
8382:
8377:
8371:
8367:
8363:
8359:
8355:
8354:
8349:
8347:0-19-853213-X
8343:
8339:
8335:
8330:
8329:
8323:
8318:
8314:
8310:
8306:
8301:
8300:
8295:
8291:
8287:
8285:3-540-05819-2
8281:
8277:
8272:
8271:
8267:
8263:
8259:
8253:
8247:
8243:
8242:
8236:
8235:
8230:
8226:
8222:
8216:
8212:
8208:
8204:
8200:
8196:
8195:
8190:
8184:
8180:
8176:
8172:
8168:
8164:
8163:
8158:
8157:
8152:
8148:
8147:
8141:
8136:
8132:
8129:et al., eds.
8128:
8124:
8120:
8116:
8115:
8110:
8106:
8102:
8101:
8090:on 2018-04-11
8086:
8082:
8078:
8074:
8070:
8066:
8062:
8058:
8051:
8047:
8043:
8042:
8037:
8033:
8029:
8025:
8021:
8017:
8013:
8012:
8007:
8003:
8002:
7996:
7995:
7988:
7987:
7978:
7972:
7968:
7964:
7963:
7957:
7953:
7947:
7939:
7932:
7931:
7925:
7924:
7922:
7921:
7906:
7902:
7901:
7896:
7892:
7891:
7886:
7882:
7878:
7874:
7873:Davis, Martin
7870:
7869:
7856:
7851:
7844:
7839:
7832:
7827:
7821:
7817:
7812:
7805:
7800:
7793:
7788:
7781:
7776:
7769:
7764:
7757:
7752:
7746:
7741:
7735:
7730:
7723:
7718:
7711:
7706:
7697:
7693:
7688:
7669:
7666:
7661:
7657:
7653:
7650:
7647:
7642:
7638:
7634:
7629:
7625:
7621:
7616:
7612:
7601:
7597:
7592:
7585:
7580:
7574:
7570:
7565:
7558:
7552:
7545:
7540:
7534:
7530:
7529:
7525:
7522:
7515:
7509:
7505:
7504:
7497:
7491:, p. 27.
7490:
7485:
7478:
7473:
7467:, p. 83.
7466:
7461:
7459:
7451:
7446:
7440:, p. 94.
7439:
7434:
7427:
7422:
7415:
7410:
7403:
7399:
7394:
7387:
7382:
7375:
7374:
7369:
7364:
7360:
7329:
7326:
7323:
7320:
7317:
7297:
7294:
7291:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7267:
7247:
7244:
7241:
7238:
7235:
7232:
7229:
7226:
7223:
7220:
7217:
7214:
7211:
7191:
7188:
7185:
7182:
7179:
7159:
7156:
7153:
7150:
7147:
7144:
7141:
7138:
7135:
7132:
7129:
7109:
7106:
7103:
7100:
7097:
7094:
7091:
7088:
7085:
7082:
7076:
7073:
7070:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7014:
7011:
7008:
7005:
7002:
6993:
6980:
6970:
6961:
6957:
6947:
6944:
6942:
6939:
6937:
6934:
6932:
6929:
6927:
6924:
6922:
6919:
6917:
6914:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6894:
6893:
6889:
6883:
6878:
6875:
6864:
6842:
6830:
6824:
6821:
6815:
6812:
6809:
6802:
6801:
6800:
6798:
6794:
6790:
6782:
6778:
6753:
6747:
6744:
6738:
6735:
6732:
6725:
6724:
6723:
6700:
6694:
6691:
6685:
6677:
6655:
6644:
6640:
6636:
6624:
6622:
6618:
6614:
6610:
6606:
6602:
6598:
6594:
6590:
6586:
6582:
6578:
6574:
6570:
6560:
6557:
6553:
6549:
6544:
6540:
6536:
6532:
6528:
6524:
6520:
6516:
6511:
6509:
6505:
6500:
6497:
6493:
6489:
6485:
6481:
6475:
6465:
6463:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6431:
6427:
6423:
6419:
6414:
6412:
6408:
6404:
6400:
6396:
6393:According to
6386:
6384:
6379:
6377:
6373:
6335:
6315:
6307:
6284:
6281:
6278:
6269:
6259:
6256:
6237:
6234:
6231:
6225:
6222:
6219:
6210:
6204:
6201:
6198:
6195:
6188:
6167:
6164:
6161:
6158:
6155:
6146:
6137:
6134:
6131:
6122:
6119:
6116:
6106:
6088:
6085:
6082:
6079:
6076:
6073:
6070:
6064:
6061:
6058:
6055:
6052:
6049:
6046:
6037:
6034:
6031:
6028:
6025:
6015:
5997:
5994:
5991:
5988:
5985:
5982:
5979:
5973:
5970:
5967:
5958:
5955:
5952:
5949:
5946:
5936:
5933:
5914:
5911:
5908:
5905:
5902:
5899:
5896:
5893:
5890:
5887:
5884:
5875:
5872:
5869:
5859:
5856:
5834:
5831:
5828:
5813:
5803:
5800:
5781:
5778:
5775:
5769:
5766:
5763:
5760:
5757:
5754:
5751:
5742:
5739:
5736:
5733:
5730:
5720:
5717:
5698:
5695:
5692:
5689:
5686:
5677:
5667:
5664:
5660:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5602:
5592:
5589:
5567:
5564:
5561:
5555:
5549:
5546:
5543:
5537:
5531:
5528:
5525:
5519:
5516:
5507:
5504:
5501:
5498:
5495:
5485:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5440:
5437:
5434:
5424:
5403:
5400:
5397:
5391:
5388:
5385:
5382:
5379:
5373:
5370:
5367:
5355:
5352:
5349:
5346:
5343:
5333:
5330:
5311:
5308:
5305:
5302:
5299:
5296:
5293:
5284:
5281:
5278:
5268:
5265:
5243:
5240:
5237:
5231:
5228:
5225:
5222:
5219:
5213:
5210:
5207:
5195:
5192:
5189:
5186:
5183:
5173:
5172:
5171:
5169:
5158:
5156:
5151:
5147:
5140:
5118:
5076:
5070:
5067:
5061:
5058:
5026:
5020:
5014:
5008:
5002:
4987:
4981:
4978:
4972:
4962:
4956:
4944:
4938:
4935:
4929:
4906:
4893:
4892:
4891:
4889:
4885:
4878:
4874:
4867:
4863:
4859:
4854:
4850:
4828:
4825:
4822:
4819:
4816:
4813:
4807:
4801:
4798:
4795:
4786:
4783:
4780:
4770:
4753:
4750:
4747:
4744:
4741:
4732:
4722:
4702:
4699:
4696:
4690:
4687:
4681:
4675:
4672:
4669:
4660:
4657:
4654:
4644:
4627:
4624:
4621:
4618:
4615:
4606:
4596:
4579:
4576:
4573:
4564:
4558:
4555:
4549:
4543:
4534:
4531:
4528:
4518:
4498:
4492:
4489:
4486:
4477:
4467:
4466:
4465:
4463:
4458:
4456:
4452:
4447:
4444:
4443:
4438:
4434:
4430:
4420:
4404:
4390:
4386:
4382:
4381:ultrafinitism
4378:
4373:
4371:
4367:
4359:
4352:
4348:
4344:
4340:
4336:
4330:
4328:
4324:
4320:
4316:
4312:
4308:
4307:David Hilbert
4304:
4300:
4296:
4290:
4286:
4276:
4273:
4269:
4264:
4240:
4234:
4231:
4223:
4219:
4215:
4213:
4205:
4202:
4196:
4189:
4184:
4180:
4176:
4174:
4166:
4160:
4149:
4148:
4147:
4146:is such that
4142:
4138:
4133:
4128:
4124:
4120:
4116:
4108:
4104:
4099:
4094:
4086:
4082:
4076:
4072:
4068:
4060:
4051:
4047:
4043:
4038:
4034:
4031:
4024:
4018:
4013:
4007:
4003:
3999:
3994:
3989:
3985:
3980:
3975:
3970:
3966:
3961:
3956:
3952:
3948:
3945:
3940:
3936:
3931:
3927:
3921:
3916:
3910:
3903:
3899:
3892:
3888:
3883:
3878:
3872:
3864:
3863:
3862:
3860:
3852:
3847:
3842:
3838:
3834:
3830:
3820:
3818:
3814:
3810:
3806:
3802:
3798:
3793:
3790:
3786:
3782:
3778:
3774:
3748:
3745:
3742:
3739:
3736:
3730:
3721:
3718:
3715:
3709:
3706:
3703:
3700:
3694:
3685:
3682:
3679:
3670:
3664:
3661:
3658:
3652:
3643:
3640:
3637:
3628:
3625:
3619:
3613:
3610:
3608:
3603:
3593:
3590:
3587:
3581:
3572:
3566:
3563:
3557:
3548:
3539:
3533:
3527:
3518:
3509:
3506:
3500:
3494:
3491:
3489:
3484:
3474:
3468:
3456:
3444:
3438:
3426:
3423:
3417:
3411:
3408:
3406:
3401:
3391:
3389:
3384:
3373:
3372:
3371:
3369:
3365:
3361:
3339:
3333:
3330:
3327:
3321:
3315:
3308:
3307:
3306:
3304:
3300:
3296:
3292:
3288:
3287:set theoretic
3282:
3272:
3270:
3266:
3237:
3231:
3223:
3219:
3215:
3213:
3202:
3194:
3190:
3183:
3174:
3170:
3166:
3164:
3154:
3150:
3143:
3132:
3131:
3130:
3126:
3122:
3117:
3113:
3109:
3105:
3098:
3094:
3089:
3083:
3079:
3070:
3066:
3061:
3055:
3051:
3045:
3041:
3037:
3033:
3028:
3024:
3019:
3015:
3011:
3005:
2999:
2993:
2989:
2983:
2973:
2971:
2967:
2963:
2960:
2954:
2950:
2944:
2940:
2934:
2930:
2921:
2916:
2912:
2908:
2902:
2898:
2892:
2888:
2882:
2878:
2873:
2869:
2863:
2859:
2858:
2857:
2855:
2850:
2846:
2841:
2840:least element
2837:
2833:
2830:
2826:
2822:
2812:
2808:
2803:
2799:
2794:
2791:
2787:
2783:
2778:
2774:
2768:
2764:
2759:
2755:
2750:
2746:
2741:
2738:
2735:
2734:
2732:
2729:
2725:
2724:
2723:
2720:
2718:
2712:
2702:
2698:
2694:
2690:
2687:
2684:
2680:
2676:
2672:
2669:
2668:
2667:
2664:
2660:
2638:
2635:
2632:
2629:
2626:
2623:
2610:
2606:
2602:
2596:
2592:
2586:
2582:
2576:
2572:
2568:
2563:
2562:
2561:
2559:
2549:
2547:
2525:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2463:
2460:
2457:
2451:
2445:
2442:
2439:
2433:
2427:
2424:
2421:
2415:
2412:
2405:
2404:
2403:
2401:
2382:
2376:
2349:
2343:
2340:
2334:
2331:
2328:
2322:
2319:
2313:
2307:
2304:
2301:
2298:
2295:
2292:
2286:
2280:
2277:
2274:
2271:
2265:
2259:
2256:
2250:
2244:
2241:
2235:
2229:
2226:
2220:
2214:
2191:
2185:
2162:
2156:
2136:
2133:
2130:
2127:
2121:
2115:
2095:
2072:
2066:
2058:
2043:
2040:
2037:
2034:
2028:
2022:
1999:
1993:
1986:
1985:
1984:
1967:
1961:
1954:To show that
1938:
1935:
1932:
1929:
1926:
1923:
1917:
1914:
1911:
1905:
1902:
1899:
1893:
1887:
1884:
1881:
1874:
1873:
1872:
1870:
1851:
1845:
1818:
1812:
1809:
1806:
1800:
1797:
1794:
1792:
1784:
1778:
1775:
1772:
1765:
1762:
1759:
1757:
1752:
1749:
1746:
1735:
1734:
1733:
1731:
1721:
1719:
1715:
1711:
1707:
1703:
1700:
1694:
1688:
1685:
1679:
1674:
1658:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1600:
1577:
1574:
1571:
1565:
1562:
1559:
1556:
1550:
1544:
1524:
1521:
1518:
1515:
1512:
1469:
1463:
1457:
1454:
1451:
1448:
1445:
1424:
1418:
1412:
1406:
1403:
1401:
1381:
1378:
1375:
1369:
1366:
1364:
1355:by definition
1344:
1338:
1335:
1332:
1329:
1327:
1322:
1319:
1316:
1303:
1297:
1294:
1291:
1288:
1285:
1267:
1261:
1255:
1252:
1250:
1230:
1227:
1224:
1218:
1215:
1213:
1204:by definition
1193:
1187:
1184:
1181:
1178:
1176:
1171:
1168:
1165:
1146:
1140:
1134:
1131:
1129:
1109:
1106:
1103:
1097:
1094:
1092:
1083:by definition
1072:
1066:
1063:
1060:
1057:
1055:
1050:
1047:
1044:
1033:
1032:
1031:
1030:For example:
1001:
995:
992:
989:
983:
980:
978:
970:
964:
961:
958:
942:
939:
936:
934:
929:
926:
923:
912:
911:
910:
908:
904:
900:
896:
887:
885:
881:
873:
872:
867:
863:
859:
849:
847:
843:
839:
830:
826:
822:
815:
811:
807:
803:
799:
795:
791:
788:
785:
784:
782:
778:
774:
771:
764:
757:
753:
749:
746:implies that
745:
741:
737:
733:
730:
726:
725:
723:
719:
716:
714:
713:
708:
703:
701:
697:
693:
689:
681:
677:
673:
669:
664:
654:
650:
645:
641:
638:
634:
629:
625:
618:
614:
610:
606:
601:
597:
593:
590:
586:
582:
578:
574:
570:
567:
565:
562:
558:
549:
545:
540:
536:
531:
527:
523:
519:
516:
511:
507:
502:
498:
494:
490:
486:
482:
478:
474:
471:
466:
462:
456:
452:
447:
443:
439:
436:
431:
427:
422:
418:
415:
413:
410:
405:
403:
402:
392:
389:
386:
384:
380:
364:
351:
348:
344:
343:
337:
335:
331:
327:
326:Gottlob Frege
323:
322:
317:
313:
302:
299:
291:
280:
277:
273:
270:
266:
263:
259:
256:
252:
249: –
248:
244:
243:Find sources:
237:
233:
227:
226:
221:This article
219:
215:
210:
209:
201:
199:
195:
191:
187:
183:
179:
175:
170:
168:
167:
162:
158:
154:
150:
146:
142:
138:
134:
130:
125:
123:
119:
115:
110:
108:
104:
100:
99:number theory
96:
92:
88:
84:
80:
76:
71:
64:
37:
33:
19:
9001:
9000:
8980:
8958:
8945:
8932:
8889:Proof theory
8888:
8863:
8838:
8814:
8765:
8761:
8712:
8708:
8691:. Retrieved
8668:
8663:
8650:
8625:
8595:
8559:
8531:
8493:(1): 85–95.
8490:
8486:
8467:
8445:. Springer.
8441:
8411:
8389:
8361:
8333:
8312:
8308:
8275:
8257:
8244:. Elsevier.
8240:
8202:
8174:
8167:Gray, Jeremy
8155:
8122:
8118:
8112:
8092:. Retrieved
8085:the original
8060:
8056:
8019:
8015:
8009:
7993:
7961:
7929:
7913:. Retrieved
7904:
7899:
7876:
7850:
7838:
7826:
7811:
7799:
7787:
7775:
7763:
7756:Willard 2001
7751:
7745:Gentzen 1936
7740:
7729:
7717:
7710:Hilbert 1902
7705:
7687:
7591:
7579:
7573:Hatcher 2014
7564:
7551:
7539:
7519:
7514:
7502:
7500:Matt DeVos,
7496:
7484:
7479:, p. 1.
7472:
7452:, p. 2.
7445:
7433:
7426:Shields 1997
7421:
7409:
7393:
7381:
7371:
7363:
6979:
6969:
6960:
6911:Neo-logicism
6796:
6792:
6788:
6786:
6780:
6776:
6675:
6642:
6638:
6634:
6626:
6620:
6616:
6612:
6608:
6604:
6600:
6596:
6592:
6588:
6584:
6580:
6576:
6572:
6568:
6566:
6555:
6551:
6547:
6542:
6538:
6534:
6512:
6501:
6477:
6456:with higher
6453:
6449:
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6425:
6415:
6406:
6398:
6392:
6382:
6380:
6375:
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5107:
4887:
4883:
4876:
4872:
4865:
4861:
4857:
4846:
4459:
4448:
4442:axiom schema
4440:
4433:second-order
4426:
4374:
4331:
4302:
4292:
4271:
4267:
4262:
4259:
4140:
4136:
4131:
4126:
4122:
4118:
4114:
4106:
4102:
4097:
4092:
4084:
4080:
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4066:
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3992:
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3938:
3934:
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3925:
3919:
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3876:
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3794:
3788:
3784:
3780:
3772:
3770:
3367:
3359:
3357:
3302:
3284:
3263:and it is a
3262:
3124:
3120:
3115:
3111:
3107:
3104:homomorphism
3096:
3092:
3087:
3081:
3077:
3068:
3064:
3059:
3053:
3049:
3026:
3017:
3013:
3009:
3003:
2997:
2991:
2987:
2979:
2969:
2965:
2961:
2952:
2948:
2942:
2938:
2932:
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2925:
2919:
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2896:
2890:
2886:
2880:
2876:
2867:
2861:
2853:
2848:
2844:
2835:
2825:well-ordered
2820:
2818:
2810:
2806:
2801:
2797:
2789:
2785:
2781:
2776:
2772:
2766:
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2608:
2604:
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2590:
2584:
2580:
2574:
2570:
2566:
2555:
2552:Inequalities
2482:
2368:
1953:
1837:
1727:
1713:
1699:cancellative
1692:
1677:
1504:
1029:
902:
893:
870:
855:
835:
828:
824:
820:
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809:
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340:
338:
319:
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294:
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275:
268:
261:
254:
242:
230:Please help
225:verification
222:
197:
194:axiom schema
182:second-order
171:
156:
147:provided an
126:
121:
111:
78:
74:
36:Peano axioms
35:
29:
8719:: 145–158.
8693:2 September
8133:: 280–287.
8105:Gödel, Kurt
8046:Gödel, Kurt
7831:Hermes 1973
7816:Hermes 1973
7569:Suppes 1960
7414:Peirce 1881
6378:elements.
6376:nonstandard
5855:irreflexive
5588:distributes
5329:commutative
5264:associative
4435:, since it
4337:. In 1936,
4335:type theory
4325:proved his
4321:. In 1931,
4289:Consistency
4279:Consistency
3949:A morphism
3946:-morphisms.
3269:categorical
3129:satisfying
2959:contradicts
2792:)) is true,
2558:total order
1728:Similarly,
1704:, and thus
1684:commutative
1441:using
1281:using
907:recursively
842:first-order
816:)) is true,
783:such that:
779:is a unary
631:. That is,
196:. The term
178:first-order
143:. In 1881,
9022:Categories
9008:PlanetMath
8125:, Vol II.
8114:Dialectica
8094:2013-10-31
7734:Gödel 1958
7722:Gödel 1931
7596:Fritz 1952
7489:Peano 1908
7477:Peano 1889
7350:References
6799:such that
6617:proper cut
6531:order type
6527:computable
6506:, such as
6504:set theory
5932:trichotomy
5799:transitive
4437:quantifies
4323:Kurt Gödel
4311:finitistic
4303:consistent
3995:-morphism
3815:, and the
3044:isomorphic
2813:) is true.
2770:such that
2742:for every
2598:such that
2556:The usual
2402:addition:
1706:embeddable
1697:is also a
880:set theory
694:(0), 2 as
515:transitive
258:newspapers
129:arithmetic
118:arithmetic
103:consistent
8982:MathWorld
8965:EMS Press
8907:cite book
8705:Wang, Hao
8294:1431-4657
8264:and from
8229:121297669
8081:197663120
8036:122719892
7946:cite book
7855:Kaye 1991
7843:Kaye 1991
7833:, VI.3.1.
7804:Kaye 1991
7692:Gray 2013
7670:…
7651:…
7398:Wang 1957
7355:Citations
7321:⋅
7289:⋅
7271:⋅
7239:⋅
7227:⋅
7215:⋅
7183:⋅
7151:⋅
7145:∨
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7101:⋅
7089:⋅
7065:⋅
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7041:⋅
7006:⋅
6991:∀
6834:¯
6816:ϕ
6813:⊨
6757:¯
6739:ϕ
6736:⊨
6704:¯
6686:ϕ
6659:¯
6563:Overspill
6444:sets are
6422:algorithm
6336:≤
6282:≥
6267:∀
6235:≥
6229:⇒
6208:∀
6205:∧
6144:∃
6141:⇒
6114:∀
6086:⋅
6074:⋅
6068:⇒
6056:∧
6023:∀
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5944:∀
5906:∨
5894:∨
5867:∀
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5761:∧
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5000:⇒
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4957:∧
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4799:⋅
4778:∀
4745:⋅
4730:∀
4652:∀
4604:∀
4571:⇒
4526:∀
4490:≠
4475:∀
4455:signature
4401:Π
4349:up to an
4117: : (
3953: : (
3912: : 1
3813:empty set
3671:∪
3540:∪
3463:∅
3451:∅
3445:∪
3442:∅
3433:∅
3395:∅
3334:∪
3265:bijection
3032:‹See Tfd›
2728:predicate
2639:∈
2514:⋅
2461:⋅
2443:⋅
2416:⋅
2272:⋅
2227:⋅
2128:⋅
2108:(that is
2035:⋅
1915:⋅
1885:⋅
1810:⋅
1776:⋅
1750:⋅
1673:structure
1392:using (2)
1241:using (2)
1154:using (1)
1120:using (2)
781:predicate
742:being in
707:successor
678:) limits
676:induction
637:injection
557:successor
470:symmetric
435:reflexive
141:induction
8887:(2013).
8753:(2001).
8741:26896458
8661:(1889).
8649:(1890).
8624:(1967).
8556:(1960).
8481:(1881).
8465:(1908).
8360:(1965).
8169:(2013).
8153:(1861).
8107:(1958).
8048:(1931).
7911:. Vieweg
7897:(1888).
7875:(1974).
7524:Archived
6860:See also
6779:∈
6775:for all
6607:∈
6599:∈
6583:so that
6490:"); the
6372:standard
5716:identity
5659:identity
4000: :
3933: :
3837:category
3783: :
3110: :
3023:Dedekind
2996:, where
2957:, which
2946:. Thus,
2874:For any
2829:nonempty
2726:For any
2666:, then:
2564:For all
2546:semiring
2149:), then
1718:integers
895:Addition
890:Addition
858:addition
754:) is in
727:0 is in
561:function
520:For all
412:relation
409:equality
334:Schröder
288:May 2024
174:equality
107:complete
85:for the
8967:, 2001
8798:2822314
8790:1833464
8782:2695030
8733:2964176
8515:1507856
8507:2369151
7865:Sources
7368:"Peano"
6255:covered
4871:, ...,
4353:called
4351:ordinal
4317:of his
3991:) is a
2968:. Thus
2904:. Then
2827:—every
2780:, then
1716:is the
848:below.
621:, then
542:, then
503:, then
458:, then
272:scholar
77:or the
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8034:
7973:
7915:4 July
7907:]
7696:p. 133
7600:p. 137
6997:
6974:piece.
6611:) and
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4664:
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4538:
4481:
4315:second
3904:, and
3896:where
3831:. Let
3366:under
3364:closed
3036:German
2976:Models
2838:has a
2832:subset
2715:is an
2483:Thus,
1687:monoid
1671:. The
864:, and
635:is an
587:under
585:closed
548:closed
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267:
260:
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81:, are
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8794:S2CID
8778:JSTOR
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8667:[
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7909:(PDF)
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6952:Notes
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4385:total
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2990:, 0,
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2656:, if
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1708:in a
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1682:is a
819:then
761:then
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655:) = 0
602:, if
528:, if
487:, if
448:, if
330:Boole
279:JSTOR
265:books
161:Latin
8913:link
8893:ISBN
8869:ISBN
8843:ISBN
8819:ISBN
8695:2023
8630:ISBN
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8570:ISBN
8538:ISBN
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8420:ISBN
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7917:2016
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524:and
495:and
483:and
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377:The
332:and
251:news
139:and
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105:and
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8770:doi
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8317:doi
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5882:(
5876:y
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5639:=
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5559:(
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5502:y
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3216:=
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2383:0
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2353:)
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2335:0
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2320:=
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2305:+
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2266:0
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2257:+
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2248:(
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2242:=
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2236:a
2233:(
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2224:)
2221:0
2218:(
2215:S
2195:)
2192:a
2189:(
2186:S
2166:)
2163:0
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2125:)
2122:0
2119:(
2116:S
2096:a
2076:)
2073:0
2070:(
2067:S
2056:.
2044:0
2041:=
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2032:)
2029:0
2026:(
2023:S
2003:)
2000:0
1997:(
1994:S
1971:)
1968:0
1965:(
1962:S
1939:a
1936:=
1933:0
1930:+
1927:a
1924:=
1921:)
1918:0
1912:a
1909:(
1906:+
1903:a
1900:=
1897:)
1894:0
1891:(
1888:S
1882:a
1855:)
1852:0
1849:(
1846:S
1819:.
1816:)
1813:b
1807:a
1804:(
1801:+
1798:a
1795:=
1788:)
1785:b
1782:(
1779:S
1773:a
1766:,
1763:0
1760:=
1753:0
1747:a
1714:N
1693:N
1691:(
1678:N
1676:(
1659:b
1639:a
1636:+
1633:b
1630:=
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1624:+
1621:a
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1581:)
1578:b
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1572:a
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1566:S
1563:=
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1557:+
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1522:=
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1516:+
1513:0
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1473:)
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1467:(
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1461:(
1458:S
1455:=
1452:2
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1446:a
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1431:)
1428:)
1425:a
1422:(
1419:S
1416:(
1413:S
1410:(
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1404:=
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1382:2
1379:+
1376:a
1373:(
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1367:=
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1345:2
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1336:+
1333:a
1330:=
1323:3
1320:+
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1304:a
1301:(
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1295:=
1292:1
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1286:a
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1271:)
1268:a
1265:(
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1259:(
1256:S
1253:=
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1216:=
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1194:1
1191:(
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1185:+
1182:a
1179:=
1172:2
1169:+
1166:a
1147:,
1144:)
1141:a
1138:(
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1132:=
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1110:0
1107:+
1104:a
1101:(
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1095:=
1076:)
1073:0
1070:(
1067:S
1064:+
1061:a
1058:=
1051:1
1048:+
1045:a
1002:.
999:)
996:b
993:+
990:a
987:(
984:S
981:=
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971:b
968:(
965:S
962:+
959:a
943:,
940:a
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930:0
927:+
924:a
903:N
871:N
831:.
829:n
825:n
823:(
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814:n
812:(
810:S
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800:(
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777:φ
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744:K
740:n
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729:K
722:K
700:S
698:(
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680:N
668:N
653:n
651:(
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644:n
639:.
633:S
628:n
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617:n
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607:(
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591:.
589:S
581:n
579:(
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564:S
544:a
539:b
535:a
530:b
526:b
522:a
517:.
510:z
506:x
501:z
497:y
493:y
489:x
485:z
481:y
477:x
472:.
465:x
461:y
455:y
451:x
446:y
442:x
437:.
430:x
426:x
421:x
383:S
365:.
361:N
350:N
301:)
295:(
290:)
286:(
276:·
269:·
262:·
255:·
228:.
159:(
63:/
57:n
51:ˈ
48:i
45:p
42:/
38:(
20:)
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