5790:
3541:
921:) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another '
679:
contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (
3555:
3101:("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior
356:. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.
3857:
Aristotle, Metaphysics Bk IV, Chapter 3, 1005b "Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come
686:
Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite
519:
It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great
642:
Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance,
503:
particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses.
953:
in the early 1980s, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be
678:
As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In
937:. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the
508:). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.
159:
Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the
515:
axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.
2411:, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.
954:
considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc.).
565:
It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was
Hilbert's formalization of
2905:
1273:
3327:
2438:
This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.
297:
excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions
168:). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.
329:, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's
3380:. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation".
171:
Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the
557:
of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be
259:
held that this
Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property."
302:, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms
1333:
468:, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts.
3517:, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as
3013:
3858:
to a special study, and not be inquiring into them while they are listening to lectures on it." W.D. Ross translation, in The Basic Works of
Aristotle, ed. Richard McKeon, (Random House, New York, 1941)
562:; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.
1512:
2396:
starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.
2796:
888:
3525:
showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and
2353:
1919:
2155:
2633:
1186:
779:
2727:
1677:
can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by
237:, axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof.
2939:
3251:
366:, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).
2225:
1784:
1569:
1955:
1444:
3035:
3484:
3149:, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in
3193:
3417:
3378:
3354:
3213:
3461:
3441:
3279:
2959:
2305:
2201:
1975:
1871:
1760:
1117:
1093:
1073:
1053:
634:(Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.
2108:
1520:, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with
1141:
1701:
1675:
1622:
3137:, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use of
3079:
3055:
2673:
2653:
2285:
2265:
2245:
2079:
2059:
2039:
2015:
1995:
1851:
1827:
1804:
1721:
1649:
1593:
1402:
1382:
1362:
799:
713:
3710:"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a.
2802:
1192:
460:
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates,
4169:
2375:, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as
3287:
3117:
arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.
2454:
are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of
285:
The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (
4844:
2463:
4005:
4927:
4068:
2577:
523:
Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and
2407:. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is
1279:
148:), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example
3388:
2968:
618:
but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern
905:
In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The '
5241:
3637:
Hilbert also made explicit the assumptions that Euclid used in his proofs but did not list in his common notions and postulates.
3983:
3673:
1703:(or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol
5399:
1449:
4187:
1723:
has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.
592:
5254:
4577:
2367:
are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the
1514:
are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and
136:". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (
3628:
Although not complete; some of the stated results did not actually follow from the stated postulates and common notions.
2733:
804:
5259:
5249:
4986:
4839:
4192:
3810:
2313:
1879:
1629:
88:
4183:
3381:
5395:
3972:
3692:
2116:
1527:
Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.
4737:
3529:
was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.
5492:
5236:
4061:
504:
However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g.,
50:
2586:
4797:
4490:
2455:
999:
251:
Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books,
4231:
3497:. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.
3142:
5753:
5455:
5218:
5213:
5038:
4459:
4143:
2482:
is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as
2475:
1153:
3882:
550:
Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.
5748:
5531:
5448:
5161:
5092:
4969:
4211:
718:
683:) the theory that the postulates install. A theory is considered valid as long as it has not been falsified.
248:
demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).
3908:
2685:
610:
It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of
5814:
5673:
5499:
5185:
4819:
4418:
983:
2910:
5551:
5546:
5156:
4895:
4824:
4153:
4054:
3218:
2471:
619:
1830:
5819:
5480:
5070:
4464:
4432:
4123:
484:
226:), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the
46:
5770:
5719:
5616:
5114:
5075:
4552:
4197:
2206:
2018:
1765:
1550:
906:
173:
4226:
5611:
5541:
5080:
4932:
4915:
4638:
4118:
1928:
1411:
3699:
a statement or proposition that is regarded as being established, accepted, or self-evidently true
3113:
geometries. If one also removes the second postulate ("a line can be extended indefinitely") then
3018:
687:
neither
Euclidean geometry or differential calculus that they imply. It became more apparent when
5443:
5420:
5381:
5267:
5208:
4854:
4774:
4618:
4562:
4175:
3518:
3510:
3466:
2483:
293:) was developed by the ancient Greeks, and has become the core principle of modern mathematics.
5733:
5460:
5438:
5405:
5298:
5144:
5129:
5102:
5053:
4937:
4872:
4697:
4663:
4658:
4532:
4363:
4340:
3090:
2569:
2467:
2021:.) In informal terms, this example allows us to state that, if we know that a certain property
1405:
946:
922:
652:
648:
599:, for example) to construct a statement whose truth is independent of that set of axioms. As a
414:
413:
on the same side less than two right angles, the two straight lines, if produced indefinitely,
42:
17:
3178:
974:(somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).
5663:
5516:
5308:
5026:
4762:
4668:
4527:
4512:
4393:
4368:
3402:
3363:
3339:
3198:
3150:
2549:
2545:
2498:
2479:
623:
98:), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
2419:
5636:
5598:
5475:
5279:
5119:
5043:
5021:
4849:
4807:
4706:
4673:
4537:
4325:
4236:
3783:
3684:
3446:
3426:
3264:
2944:
2459:
2290:
2186:
1960:
1856:
1807:
1745:
1572:
1102:
1078:
1058:
1038:
934:
631:
581:
360:
340:
assumption common to many branches of science. A good example would be the assertion that:
2084:
1126:
926:
8:
5765:
5656:
5641:
5621:
5578:
5465:
5415:
5341:
5286:
5223:
5016:
5011:
4959:
4727:
4716:
4388:
4288:
4216:
4207:
4203:
4138:
4133:
3868:
3716:
3110:
2900:{\displaystyle (\phi (0)\land \forall x.\,(\phi (x)\to \phi (Sx)))\to \forall x.\phi (x)}
2522:
2393:
1680:
1654:
1601:
1032:
512:
505:
488:
461:
330:
2160:, that is, we must be able to give a "proof" of this fact, or more properly speaking, a
949:) in the Copenhagen and the Hidden variable case. The experiment was conducted first by
585:
5794:
5563:
5526:
5511:
5504:
5487:
5291:
5273:
5139:
5065:
5048:
5001:
4814:
4723:
4557:
4542:
4502:
4454:
4439:
4427:
4383:
4358:
4128:
4077:
3773:
3546:
3138:
3098:
3094:
3064:
3058:
3040:
2658:
2638:
2518:
2494:
2490:
2423:
2270:
2250:
2230:
2064:
2044:
2024:
2000:
1980:
1836:
1812:
1789:
1706:
1634:
1578:
1531:
1387:
1367:
1347:
1096:
963:
891:
784:
698:
692:
567:
554:
406:
310:
hold a slightly different meaning for the present day mathematician, than they did for
290:
165:
161:
106:
4747:
3522:
5789:
5729:
5536:
5346:
5336:
5228:
5109:
4944:
4920:
4701:
4685:
4590:
4567:
4444:
4413:
4378:
4273:
4108:
3999:
3968:
3688:
3560:
3540:
3387:
Note that "completeness" has a different meaning here than it does in the context of
3146:
3114:
2381:
1268:{\displaystyle (\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi ))}
1007:
991:
942:
933:. It was created so as to try to give deterministic explanation to phenomena such as
914:
895:
660:
492:
294:
234:
71:
3109:
add up to exactly 180 degrees or less, respectively, and are known as
Euclidean and
1534:, but additional logical axioms are needed to include a quantifier in the calculus.
536:
113:
or well-established, that it is accepted without controversy or question. In modern
5743:
5738:
5631:
5588:
5410:
5371:
5366:
5351:
5177:
5134:
5031:
4829:
4779:
4353:
4315:
3988:
3978:
3765:
3680:
3568:
3526:
2451:
2427:
656:
644:
604:
596:
532:
469:
465:
4038:
3384:
establishes the completeness of a certain commonly used type of deductive system.
3322:{\displaystyle {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }
2168:
of our theory of mathematical logic since we are dealing with the very concept of
1035:
it is common to take as logical axioms all formulae of the following forms, where
5724:
5714:
5668:
5651:
5606:
5568:
5470:
5390:
5197:
5124:
5097:
5085:
4991:
4905:
4879:
4834:
4802:
4603:
4405:
4348:
4298:
4263:
4221:
3984:
On an
Evolutionist Theory of Axioms: inaugural lecture delivered October 15, 1889
3578:
2506:
2502:
2376:
2081:
stands for a particular object in our structure, then we should be able to claim
1011:
987:
688:
3950:
Mendelson, "5. The Fixed Point
Theorem. Gödel's Incompleteness Theorem" of Ch. 2
3594:
945:
derived in 1964 a prediction that would lead to different experimental results (
5709:
5688:
5646:
5626:
5521:
5376:
4974:
4964:
4954:
4949:
4883:
4757:
4633:
4522:
4517:
4495:
4096:
3604:
3584:
3514:
2676:
2537:
2533:
2368:
995:
680:
611:
588:
raised the possibility that any such system could turn out to be inconsistent.
483:
Structuralist mathematics goes further, and develops theories and axioms (e.g.
477:
227:
189:
82:
or starting point for further reasoning and arguments. The word comes from the
409:") It is true that, if a straight line falling on two straight lines make the
5808:
5683:
5361:
4868:
4653:
4643:
4613:
4598:
4268:
3799:
3506:
3097:. The axioms are referred to as "4 + 1" because for nearly two millennia the
2573:
2572:. They are a set of axioms strong enough to prove many important facts about
2532:
This list could be expanded to include most fields of mathematics, including
2526:
2447:
1015:
540:
528:
378:
216:), meaning "to deem worthy", but also "to require", which in turn comes from
110:
83:
544:
5583:
5430:
5331:
5323:
5203:
5151:
5060:
4996:
4979:
4910:
4769:
4628:
4330:
4113:
3589:
3126:
2561:
2514:
1727:
1516:
1340:
950:
615:
574:
496:
385:
337:
3992:
5693:
5573:
4752:
4742:
4689:
4373:
4293:
4278:
4158:
4103:
3840:
3130:
2541:
2408:
938:
473:
399:
230:
203:
121:
4623:
4478:
4449:
4255:
4032:
4023:
3777:
2443:
1344:, a rule for generating an infinite number of axioms. For example, if
1144:
930:
910:
627:
577:
573:
In a wider context, there was an attempt to base all of mathematics on
559:
437:
Things which are equal to the same thing are also equal to one another.
349:
286:
102:
5775:
5678:
4731:
4648:
4608:
4572:
4508:
4320:
4310:
4283:
4046:
3915:(Spring 2019 ed.), Metaphysics Research Lab, Stanford University
3609:
3170:
2404:
899:
600:
311:
93:
3769:
3423:, in the sense that there will always exist an arithmetic statement
570:, and the related demonstration of the consistency of those axioms.
5760:
5558:
5006:
4711:
4305:
3889:(Fall 2018 ed.), Metaphysics Research Lab, Stanford University
3521:
made elaborate efforts to derive them from traditional arithmetic.
3106:
3089:
Probably the oldest, and most famous, list of axioms are the 4 + 1
1120:
1006:
some minimal set of tautologies that is sufficient for proving all
918:
524:
344:
When an equal amount is taken from equals, an equal amount results.
322:
261:
272:
but in later manuscripts this usage was not always strictly kept.
5356:
4148:
4019:
3729:"A proposition (whether true or false)" axiom, n., definition 2.
3599:
2372:
1328:{\displaystyle (\lnot \phi \to \lnot \psi )\to (\psi \to \phi ).}
410:
352:
that were accepted without proof. Such a hypothesis was termed a
348:
At the foundation of the various sciences lay certain additional
326:
299:
256:
252:
79:
3008:{\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle }
446:
Things which coincide with one another are equal to one another.
4028:
595:
that it is possible, for any sufficiently large set of axioms (
443:
If equals are subtracted from equals, the remainders are equal.
392:
315:
245:
2489:
The study of topology in mathematics extends all over through
1014:
more logical axioms than that are required, in order to prove
4900:
4246:
3573:
3102:
2458:
with choice, abbreviated ZFC, or some very similar system of
966:, a clear distinction is made between two notions of axioms:
695:
where the invariant quantity is no more the
Euclidean length
591:
The formalist project suffered a setback a century ago, when
418:
217:
207:
193:
114:
75:
31:
3745:
Mendelson, "3. First-Order
Theories: Proper Axioms" of Ch. 2
3257:. A desirable property of a deductive system is that it be
607:
is an unprovable assertion within the scope of that theory.
2379:). Thus non-logical axioms, unlike logical axioms, are not
1095:
can be any formulae of the language and where the included
925:' approach was developed for some time by Albert Einstein,
464:, theorems) and definitions. One must concede the need for
35:
117:, an axiom is a premise or starting point for reasoning.
3756:
Maddy, Penelope (June 1988). "Believing the Axioms, I".
3261:. A system is said to be complete if, for all formulas
3671:
Stevenson, Angus; Lindberg, Christine A., eds. (2015).
1507:{\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))}
663:
law, etc. These founding assertions are usually called
525:
mathematics itself can be regarded as a branch of logic
3830:, 1963, New York: New American Library, pp 47–48
3469:
3449:
3429:
3405:
3366:
3342:
3290:
3267:
3221:
3201:
3181:
3163:
3125:
The objectives of the study are within the domain of
3067:
3043:
3021:
2971:
2947:
2913:
2805:
2736:
2688:
2661:
2641:
2589:
2470:
of ZFC. Sometimes slightly stronger theories such as
2316:
2293:
2273:
2253:
2233:
2209:
2189:
2119:
2087:
2067:
2047:
2027:
2003:
1983:
1963:
1931:
1882:
1859:
1839:
1815:
1792:
1768:
1748:
1709:
1683:
1657:
1637:
1604:
1581:
1553:
1452:
1414:
1390:
1370:
1350:
1282:
1195:
1156:
1129:
1105:
1081:
1061:
1041:
807:
787:
721:
701:
603:, Gödel proved that the consistency of a theory like
3536:
553:
In the modern understanding, a set of axioms is any
440:
If equals are added to equals, the wholes are equal.
336:
An "axiom", in classical terminology, referred to a
2399:Non-logical axioms are often simply referred to as
3672:
3478:
3455:
3435:
3411:
3372:
3348:
3321:
3273:
3245:
3207:
3187:
3129:. The real numbers are uniquely picked out (up to
3073:
3049:
3029:
3007:
2953:
2933:
2899:
2791:{\displaystyle \forall x.\forall y.(Sx=Sy\to x=y)}
2790:
2721:
2667:
2647:
2627:
2347:
2299:
2279:
2259:
2239:
2219:
2195:
2149:
2102:
2073:
2053:
2033:
2009:
1989:
1969:
1949:
1913:
1865:
1845:
1821:
1798:
1778:
1754:
1730:, is that which provides us with what is known as
1715:
1695:
1669:
1643:
1616:
1587:
1563:
1506:
1438:
1396:
1376:
1356:
1327:
1267:
1180:
1135:
1111:
1087:
1067:
1047:
883:{\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}}
882:
793:
773:
707:
333:is a definitive exposition of the classical view.
41:Several terms redirect here. For other uses, see
3670:
2348:{\displaystyle \phi _{t}^{x}\to \exists x\,\phi }
1914:{\displaystyle \forall x\,\phi \to \phi _{t}^{x}}
894:where flat Minkowskian geometry is replaced with
622:for set theory. Furthermore, using techniques of
547:are some of the key figures in this development.
5806:
3977:
1002:of values. Usually one takes as logical axioms
2576:and they allowed Gödel to establish his famous
2150:{\displaystyle \forall x\phi \to \phi _{t}^{x}}
3932:Mendelson, "6. Other Axiomatizations" of Ch. 1
3741:
3739:
3489:There is thus, on the one hand, the notion of
1123:of the immediately following proposition and "
1018:that are not tautologies in the strict sense.
359:The classical approach is well-illustrated by
4062:
3967:Belmont, California: Wadsworth & Brooks.
3941:Mendelson, "3. First-Order Theories" of Ch. 2
3156:
2501:, and all the related paraphernalia, such as
520:deal of extra information about this system.
3714:Online, accessed 2012-04-28. Cf. Aristotle,
3486:can be proved from the given set of axioms.
3240:
3222:
3002:
2982:
2628:{\displaystyle {\mathfrak {L}}_{NT}=\{0,S\}}
2622:
2610:
2181:Axiom scheme for Existential Generalization.
1147:from antecedent to consequent propositions:
781:) > but the Minkowski spacetime interval
3880:
3800:"Axiom — Powszechna Encyklopedia Filozofii"
3736:
3495:completeness of a set of non-logical axioms
4254:
4069:
4055:
4004:: CS1 maint: location missing publisher (
3081:is naturally interpreted as the number 0.
2385:. Another name for a non-logical axiom is
2307:, the below formula is universally valid.
2172:itself. Aside from this, we can also have
1873:, the below formula is universally valid.
1595:, the below formula is universally valid.
1530:These axiom schemata are also used in the
3679:(3rd ed.). Oxford University Press.
3145:tell us that if we restrict ourselves to
3023:
2986:
2833:
2341:
1889:
1739:Axiom scheme for Universal Instantiation.
1181:{\displaystyle \phi \to (\psi \to \phi )}
275:
3906:
3913:The Stanford Encyclopedia of Philosophy
3887:The Stanford Encyclopedia of Philosophy
3845:The Thirteen Books of Euclid's Elements
774:{\displaystyle l^{2}=x^{2}+y^{2}+z^{2}}
671:so as to distinguish from mathematical
133:
14:
5807:
4076:
2722:{\displaystyle \forall x.\lnot (Sx=0)}
1026:
941:in 1935). Taking this idea seriously,
4050:
3839:
3755:
3500:
3332:that is, for any statement that is a
3084:
2359:
957:
455:
109:, an axiom is a statement that is so
3816:from the original on 9 October 2022.
3807:Polskie Towarzystwo Tomasza z Akwinu
3794:
3792:
3685:10.1093/acref/9780195392883.001.0001
3389:Gödel's first incompleteness theorem
2934:{\displaystyle {\mathfrak {L}}_{NT}}
2464:Von Neumann–Bernays–Gödel set theory
1537:
3965:Introduction to mathematical logic.
3246:{\displaystyle \{(\Gamma ,\phi )\}}
2974:
2917:
2593:
2212:
1771:
1556:
449:The whole is greater than the part.
129:
24:
3957:
3491:completeness of a deductive system
3470:
3406:
3367:
3343:
3310:
3296:
3228:
3202:
3182:
3164:Deductive systems and completeness
2876:
2824:
2746:
2737:
2698:
2689:
2335:
2120:
1883:
1726:Another, more interesting example
1492:
1462:
1295:
1286:
1106:
381:from any point to any other point.
105:varies across fields of study. In
27:Statement that is taken to be true
25:
5831:
4013:
3883:"Gödel's Incompleteness Theorems"
3789:
3581:, axiom in science and philosophy
3215:of non-logical axioms, and a set
977:
637:
5788:
3553:
3539:
3120:
2112:we are claiming that the formula
1010:in the language; in the case of
480:were pioneers in this movement.
388:continuously in both directions.
51:Postulation (algebraic geometry)
3944:
3935:
3926:
3900:
3874:
3861:
3851:
3847:. New York: Dover. p. 200.
3631:
3622:
3419:of the Theory of Arithmetic is
3135:Dedekind complete ordered field
3037:is the set of natural numbers,
2220:{\displaystyle {\mathfrak {L}}}
1779:{\displaystyle {\mathfrak {L}}}
1564:{\displaystyle {\mathfrak {L}}}
651:in classical electromagnetism,
511:When mathematicians employ the
421:less than the two right angles.
395:with any center and any radius.
280:
3833:
3820:
3748:
3723:
3704:
3675:New Oxford American Dictionary
3664:
3651:
3493:and on the other hand that of
3237:
3225:
2894:
2888:
2873:
2870:
2867:
2864:
2855:
2849:
2846:
2840:
2834:
2818:
2812:
2806:
2785:
2773:
2755:
2716:
2701:
2476:strongly inaccessible cardinal
2332:
2129:
2097:
2091:
1893:
1501:
1498:
1489:
1483:
1480:
1474:
1471:
1468:
1459:
1453:
1433:
1427:
1421:
1418:
1319:
1313:
1307:
1304:
1301:
1292:
1283:
1262:
1259:
1253:
1247:
1244:
1241:
1235:
1229:
1226:
1223:
1220:
1217:
1211:
1205:
1202:
1196:
1175:
1169:
1163:
1160:
1130:
417:on that side on which are the
321:The ancient Greeks considered
244:is to "demand"; for instance,
13:
1:
5749:History of mathematical logic
3911:, in Zalta, Edward N. (ed.),
3885:, in Zalta, Edward N. (ed.),
3644:
2578:second incompleteness theorem
2555:
2433:
2418:is an elementary basis for a
1950:{\displaystyle \phi _{t}^{x}}
1439:{\displaystyle A\to (B\to A)}
1338:Each of these patterns is an
994:, that is, formulas that are
391:It is possible to describe a
240:The root meaning of the word
5674:Primitive recursive function
3828:Breakthroughs in Mathematics
3733:Online, accessed 2012-04-28.
3382:Gödel's completeness theorem
3030:{\displaystyle \mathbb {N} }
179:
7:
3532:
3479:{\displaystyle \lnot \phi }
2456:Zermelo–Fraenkel set theory
1021:
384:It is possible to extend a
152: + 0 =
10:
5836:
4738:Schröder–Bernstein theorem
4465:Monadic predicate calculus
4124:Foundations of mathematics
3963:Mendelson, Elliot (1987).
3909:"The Continuum Hypothesis"
3881:Raatikainen, Panu (2018),
3657:Cf. axiom, n., etymology.
3399:set of non-logical axioms
3158:Role in mathematical logic
3099:fifth (parallel) postulate
2965:The standard structure is
2679:and the following axioms:
2267:that is substitutable for
2203:in a first-order language
2174:Existential Generalization
1762:in a first-order language
584:and similar antinomies of
264:translated 'postulate' as
218:
208:
194:
87:
47:Axiomatic (disambiguation)
40:
29:
5784:
5771:Philosophy of mathematics
5720:Automated theorem proving
5702:
5597:
5429:
5322:
5174:
4891:
4867:
4845:Von Neumann–Bernays–Gödel
4790:
4684:
4588:
4486:
4477:
4404:
4339:
4245:
4167:
4084:
3758:Journal of Symbolic Logic
3731:Oxford English Dictionary
3712:Oxford English Dictionary
3659:Oxford English Dictionary
3195:of logical axioms, a set
3157:
3143:Löwenheim–Skolem theorems
3133:) by the properties of a
2655:is a constant symbol and
2564:are the most widely used
2019:Substitution of variables
1628:This means that, for any
580:. Here, the emergence of
402:are equal to one another.
377:It is possible to draw a
174:philosophy of mathematics
3987:(1st ed.), Oxford,
3907:Koellner, Peter (2019),
3615:
3356:there actually exists a
3188:{\displaystyle \Lambda }
2442:Basic theories, such as
647:in classical mechanics,
630:) one can show that the
30:Not to be confused with
5421:Self-verifying theories
5242:Tarski's axiomatization
4193:Tarski's undefinability
4188:incompleteness theorems
3412:{\displaystyle \Sigma }
3391:, which states that no
3373:{\displaystyle \Sigma }
3349:{\displaystyle \Sigma }
3208:{\displaystyle \Sigma }
2961:with one free variable.
2484:second-order arithmetic
2472:Morse–Kelley set theory
2422:that together with the
1957:stands for the formula
1732:Universal Instantiation
1406:propositional variables
655:in general relativity,
620:Zermelo–Fraenkel axioms
325:as just one of several
268:and called the axioms
156:in integer arithmetic.
5795:Mathematics portal
5406:Proof of impossibility
5054:propositional variable
4364:Propositional calculus
3661:, accessed 2012-04-28.
3480:
3457:
3437:
3413:
3374:
3360:of the statement from
3350:
3323:
3275:
3247:
3209:
3189:
3075:
3051:
3031:
3009:
2955:
2935:
2901:
2792:
2723:
2669:
2649:
2629:
2570:first-order arithmetic
2478:allowing the use of a
2468:conservative extension
2349:
2301:
2281:
2261:
2241:
2221:
2197:
2164:. These examples are
2151:
2104:
2075:
2055:
2035:
2011:
1991:
1971:
1951:
1915:
1867:
1847:
1823:
1800:
1780:
1756:
1717:
1697:
1671:
1645:
1618:
1589:
1565:
1508:
1440:
1398:
1378:
1358:
1329:
1269:
1182:
1137:
1113:
1089:
1069:
1049:
884:
795:
775:
709:
659:of genetics, Darwin's
346:
276:Historical development
43:Axiom (disambiguation)
5664:Kolmogorov complexity
5617:Computably enumerable
5517:Model complete theory
5309:Principia Mathematica
4369:Propositional formula
4198:Banach–Tarski paradox
3481:
3458:
3456:{\displaystyle \phi }
3438:
3436:{\displaystyle \phi }
3414:
3375:
3351:
3324:
3276:
3274:{\displaystyle \phi }
3248:
3210:
3190:
3151:non-standard analysis
3076:
3052:
3032:
3010:
2956:
2954:{\displaystyle \phi }
2936:
2902:
2793:
2724:
2670:
2650:
2630:
2550:differential geometry
2546:representation theory
2509:. The development of
2499:differential topology
2480:Grothendieck universe
2474:or set theory with a
2350:
2302:
2300:{\displaystyle \phi }
2282:
2262:
2242:
2222:
2198:
2196:{\displaystyle \phi }
2152:
2105:
2076:
2056:
2036:
2012:
1992:
1972:
1970:{\displaystyle \phi }
1952:
1916:
1868:
1866:{\displaystyle \phi }
1848:
1824:
1801:
1781:
1757:
1755:{\displaystyle \phi }
1718:
1698:
1672:
1646:
1619:
1590:
1566:
1509:
1441:
1399:
1379:
1359:
1330:
1270:
1183:
1138:
1114:
1112:{\displaystyle \neg }
1097:primitive connectives
1090:
1088:{\displaystyle \psi }
1070:
1068:{\displaystyle \chi }
1050:
1048:{\displaystyle \phi }
885:
796:
776:
710:
342:
5612:Church–Turing thesis
5599:Computability theory
4808:continuum hypothesis
4326:Square of opposition
4184:Gödel's completeness
3467:
3447:
3427:
3403:
3364:
3340:
3288:
3265:
3219:
3199:
3179:
3065:
3041:
3019:
2969:
2945:
2911:
2803:
2734:
2686:
2659:
2639:
2587:
2513:brought with itself
2460:axiomatic set theory
2392:Almost every modern
2314:
2291:
2271:
2251:
2231:
2207:
2187:
2117:
2103:{\displaystyle P(t)}
2085:
2065:
2045:
2025:
2001:
1981:
1961:
1929:
1880:
1857:
1837:
1813:
1790:
1766:
1746:
1707:
1681:
1655:
1635:
1602:
1579:
1575:. For each variable
1573:first-order language
1551:
1450:
1412:
1388:
1368:
1348:
1280:
1193:
1154:
1136:{\displaystyle \to }
1127:
1103:
1079:
1059:
1039:
805:
785:
719:
699:
632:continuum hypothesis
398:It is true that all
74:that is taken to be
5815:Mathematical axioms
5766:Mathematical object
5657:P versus NP problem
5622:Computable function
5416:Reverse mathematics
5342:Logical consequence
5219:primitive recursive
5214:elementary function
4987:Free/bound variable
4840:Tarski–Grothendieck
4359:Logical connectives
4289:Logical equivalence
4139:Logical consequence
3717:Posterior Analytics
3334:logical consequence
3091:Euclid's postulates
2583:We have a language
2420:formal logic system
2394:mathematical theory
2331:
2146:
1946:
1910:
1696:{\displaystyle x=x}
1670:{\displaystyle x=x}
1617:{\displaystyle x=x}
1033:propositional logic
1027:Propositional logic
947:Bell's inequalities
898:geometry on curved
653:Einstein's equation
649:Maxwell's equations
506:hyperbolic geometry
331:posterior analytics
5564:Transfer principle
5527:Semantics of logic
5512:Categorical theory
5488:Non-standard model
5002:Logical connective
4129:Information theory
4078:Mathematical logic
3547:Mathematics portal
3511:axiomatic geometry
3501:Further discussion
3476:
3453:
3443:such that neither
3433:
3409:
3370:
3346:
3319:
3271:
3255:rules of inference
3243:
3205:
3185:
3175:consists of a set
3139:second-order logic
3085:Euclidean geometry
3071:
3059:successor function
3047:
3027:
3005:
2951:
2931:
2897:
2788:
2719:
2665:
2645:
2625:
2495:algebraic topology
2491:point set topology
2424:rules of inference
2365:Non-logical axioms
2360:Non-logical axioms
2345:
2317:
2297:
2277:
2257:
2237:
2217:
2193:
2147:
2132:
2100:
2071:
2051:
2031:
2007:
1987:
1967:
1947:
1932:
1911:
1896:
1863:
1843:
1819:
1796:
1776:
1752:
1713:
1693:
1667:
1641:
1614:
1585:
1561:
1544:Axiom of Equality.
1532:predicate calculus
1504:
1436:
1394:
1374:
1354:
1325:
1265:
1178:
1133:
1109:
1085:
1065:
1045:
982:These are certain
964:mathematical logic
958:Mathematical logic
892:general relativity
880:
791:
771:
705:
693:special relativity
568:Euclidean geometry
456:Modern development
407:Parallel postulate
291:rules of inference
166:Euclidean geometry
162:parallel postulate
107:classic philosophy
49:, and
5820:Concepts in logic
5802:
5801:
5734:Abstract category
5537:Theories of truth
5347:Rule of inference
5337:Natural deduction
5318:
5317:
4863:
4862:
4568:Cartesian product
4473:
4472:
4379:Many-valued logic
4354:Boolean functions
4237:Russell's paradox
4212:diagonal argument
4109:First-order logic
3561:Philosophy portal
3308:
3294:
3147:first-order logic
3115:elliptic geometry
3074:{\displaystyle 0}
3050:{\displaystyle S}
2668:{\displaystyle S}
2648:{\displaystyle 0}
2280:{\displaystyle x}
2260:{\displaystyle t}
2240:{\displaystyle x}
2074:{\displaystyle t}
2054:{\displaystyle x}
2034:{\displaystyle P}
2010:{\displaystyle x}
1990:{\displaystyle t}
1925:Where the symbol
1846:{\displaystyle x}
1822:{\displaystyle t}
1799:{\displaystyle x}
1716:{\displaystyle =}
1644:{\displaystyle x}
1588:{\displaystyle x}
1538:First-order logic
1397:{\displaystyle C}
1377:{\displaystyle B}
1357:{\displaystyle A}
992:universally valid
927:Erwin Schrödinger
915:Werner Heisenberg
907:Copenhagen school
896:pseudo-Riemannian
794:{\displaystyle s}
708:{\displaystyle l}
691:first introduced
661:Natural selection
582:Russell's paradox
466:primitive notions
270:notiones communes
134:non-logical axiom
16:(Redirected from
5827:
5793:
5792:
5744:History of logic
5739:Category of sets
5632:Decision problem
5411:Ordinal analysis
5352:Sequent calculus
5250:Boolean algebras
5190:
5189:
5164:
5135:logical/constant
4889:
4888:
4875:
4798:Zermelo–Fraenkel
4549:Set operations:
4484:
4483:
4421:
4252:
4251:
4232:Löwenheim–Skolem
4119:Formal semantics
4071:
4064:
4057:
4048:
4047:
4009:
4003:
3995:
3979:John Cook Wilson
3951:
3948:
3942:
3939:
3933:
3930:
3924:
3923:
3922:
3920:
3904:
3898:
3897:
3896:
3894:
3878:
3872:
3869:Hilbert's axioms
3865:
3859:
3855:
3849:
3848:
3837:
3831:
3824:
3818:
3817:
3815:
3804:
3796:
3787:
3781:
3754:See for example
3752:
3746:
3743:
3734:
3727:
3721:
3708:
3702:
3701:
3678:
3668:
3662:
3655:
3638:
3635:
3629:
3626:
3569:Axiomatic system
3563:
3558:
3557:
3556:
3549:
3544:
3543:
3485:
3483:
3482:
3477:
3462:
3460:
3459:
3454:
3442:
3440:
3439:
3434:
3418:
3416:
3415:
3410:
3379:
3377:
3376:
3371:
3355:
3353:
3352:
3347:
3328:
3326:
3325:
3320:
3309:
3307: then
3306:
3295:
3292:
3280:
3278:
3277:
3272:
3252:
3250:
3249:
3244:
3214:
3212:
3211:
3206:
3194:
3192:
3191:
3186:
3159:
3080:
3078:
3077:
3072:
3056:
3054:
3053:
3048:
3036:
3034:
3033:
3028:
3026:
3014:
3012:
3011:
3006:
2989:
2978:
2977:
2960:
2958:
2957:
2952:
2940:
2938:
2937:
2932:
2930:
2929:
2921:
2920:
2906:
2904:
2903:
2898:
2797:
2795:
2794:
2789:
2728:
2726:
2725:
2720:
2674:
2672:
2671:
2666:
2654:
2652:
2651:
2646:
2634:
2632:
2631:
2626:
2606:
2605:
2597:
2596:
2511:abstract algebra
2452:complex analysis
2428:deductive system
2403:in mathematical
2354:
2352:
2351:
2346:
2330:
2325:
2306:
2304:
2303:
2298:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2226:
2224:
2223:
2218:
2216:
2215:
2202:
2200:
2199:
2194:
2183:Given a formula
2156:
2154:
2153:
2148:
2145:
2140:
2109:
2107:
2106:
2101:
2080:
2078:
2077:
2072:
2060:
2058:
2057:
2052:
2041:holds for every
2040:
2038:
2037:
2032:
2016:
2014:
2013:
2008:
1997:substituted for
1996:
1994:
1993:
1988:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1945:
1940:
1920:
1918:
1917:
1912:
1909:
1904:
1872:
1870:
1869:
1864:
1852:
1850:
1849:
1844:
1828:
1826:
1825:
1820:
1805:
1803:
1802:
1797:
1785:
1783:
1782:
1777:
1775:
1774:
1761:
1759:
1758:
1753:
1742:Given a formula
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1676:
1674:
1673:
1668:
1650:
1648:
1647:
1642:
1623:
1621:
1620:
1615:
1594:
1592:
1591:
1586:
1570:
1568:
1567:
1562:
1560:
1559:
1513:
1511:
1510:
1505:
1445:
1443:
1442:
1437:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1363:
1361:
1360:
1355:
1334:
1332:
1331:
1326:
1274:
1272:
1271:
1266:
1187:
1185:
1184:
1179:
1142:
1140:
1139:
1134:
1118:
1116:
1115:
1110:
1094:
1092:
1091:
1086:
1074:
1072:
1071:
1066:
1054:
1052:
1051:
1046:
962:In the field of
923:hidden variables
889:
887:
886:
881:
879:
878:
866:
865:
853:
852:
840:
839:
830:
829:
817:
816:
800:
798:
797:
792:
780:
778:
777:
772:
770:
769:
757:
756:
744:
743:
731:
730:
714:
712:
711:
706:
605:Peano arithmetic
586:naïve set theory
470:Alessandro Padoa
221:
220:
211:
210:
197:
196:
97:
91:
78:, to serve as a
21:
5835:
5834:
5830:
5829:
5828:
5826:
5825:
5824:
5805:
5804:
5803:
5798:
5787:
5780:
5725:Category theory
5715:Algebraic logic
5698:
5669:Lambda calculus
5607:Church encoding
5593:
5569:Truth predicate
5425:
5391:Complete theory
5314:
5183:
5179:
5175:
5170:
5162:
4882: and
4878:
4873:
4859:
4835:New Foundations
4803:axiom of choice
4786:
4748:Gödel numbering
4688: and
4680:
4584:
4469:
4419:
4400:
4349:Boolean algebra
4335:
4299:Equiconsistency
4264:Classical logic
4241:
4222:Halting problem
4210: and
4186: and
4174: and
4173:
4168:Theorems (
4163:
4080:
4075:
4016:
3997:
3996:
3960:
3958:Further reading
3955:
3954:
3949:
3945:
3940:
3936:
3931:
3927:
3918:
3916:
3905:
3901:
3892:
3890:
3879:
3875:
3866:
3862:
3856:
3852:
3838:
3834:
3825:
3821:
3813:
3802:
3798:
3797:
3790:
3770:10.2307/2274520
3753:
3749:
3744:
3737:
3728:
3724:
3709:
3705:
3695:
3669:
3665:
3656:
3652:
3647:
3642:
3641:
3636:
3632:
3627:
3623:
3618:
3579:First principle
3559:
3554:
3552:
3545:
3538:
3535:
3519:Boolean algebra
3503:
3468:
3465:
3464:
3448:
3445:
3444:
3428:
3425:
3424:
3404:
3401:
3400:
3365:
3362:
3361:
3341:
3338:
3337:
3330:
3305:
3291:
3289:
3286:
3285:
3266:
3263:
3262:
3220:
3217:
3216:
3200:
3197:
3196:
3180:
3177:
3176:
3166:
3161:
3123:
3087:
3066:
3063:
3062:
3042:
3039:
3038:
3022:
3020:
3017:
3016:
2985:
2973:
2972:
2970:
2967:
2966:
2946:
2943:
2942:
2922:
2916:
2915:
2914:
2912:
2909:
2908:
2804:
2801:
2800:
2735:
2732:
2731:
2687:
2684:
2683:
2660:
2657:
2656:
2640:
2637:
2636:
2598:
2592:
2591:
2590:
2588:
2585:
2584:
2558:
2507:homotopy theory
2503:homology theory
2436:
2369:natural numbers
2362:
2357:
2356:
2326:
2321:
2315:
2312:
2311:
2292:
2289:
2288:
2272:
2269:
2268:
2252:
2249:
2248:
2232:
2229:
2228:
2211:
2210:
2208:
2205:
2204:
2188:
2185:
2184:
2141:
2136:
2118:
2115:
2114:
2086:
2083:
2082:
2066:
2063:
2062:
2046:
2043:
2042:
2026:
2023:
2022:
2002:
1999:
1998:
1982:
1979:
1978:
1962:
1959:
1958:
1941:
1936:
1930:
1927:
1926:
1923:
1922:
1905:
1900:
1881:
1878:
1877:
1858:
1855:
1854:
1838:
1835:
1834:
1814:
1811:
1810:
1791:
1788:
1787:
1770:
1769:
1767:
1764:
1763:
1747:
1744:
1743:
1741:
1708:
1705:
1704:
1682:
1679:
1678:
1656:
1653:
1652:
1636:
1633:
1632:
1630:variable symbol
1626:
1625:
1603:
1600:
1599:
1580:
1577:
1576:
1555:
1554:
1552:
1549:
1548:
1546:
1540:
1451:
1448:
1447:
1413:
1410:
1409:
1389:
1386:
1385:
1369:
1366:
1365:
1349:
1346:
1345:
1281:
1278:
1277:
1194:
1191:
1190:
1155:
1152:
1151:
1128:
1125:
1124:
1104:
1101:
1100:
1080:
1077:
1076:
1060:
1057:
1056:
1040:
1037:
1036:
1029:
1024:
1012:predicate logic
988:formal language
980:
960:
874:
870:
861:
857:
848:
844:
835:
831:
825:
821:
812:
808:
806:
803:
802:
786:
783:
782:
765:
761:
752:
748:
739:
735:
726:
722:
720:
717:
716:
700:
697:
696:
689:Albert Einstein
640:
612:natural numbers
458:
411:interior angles
283:
278:
188:comes from the
182:
54:
39:
28:
23:
22:
15:
12:
11:
5:
5833:
5823:
5822:
5817:
5800:
5799:
5785:
5782:
5781:
5779:
5778:
5773:
5768:
5763:
5758:
5757:
5756:
5746:
5741:
5736:
5727:
5722:
5717:
5712:
5710:Abstract logic
5706:
5704:
5700:
5699:
5697:
5696:
5691:
5689:Turing machine
5686:
5681:
5676:
5671:
5666:
5661:
5660:
5659:
5654:
5649:
5644:
5639:
5629:
5627:Computable set
5624:
5619:
5614:
5609:
5603:
5601:
5595:
5594:
5592:
5591:
5586:
5581:
5576:
5571:
5566:
5561:
5556:
5555:
5554:
5549:
5544:
5534:
5529:
5524:
5522:Satisfiability
5519:
5514:
5509:
5508:
5507:
5497:
5496:
5495:
5485:
5484:
5483:
5478:
5473:
5468:
5463:
5453:
5452:
5451:
5446:
5439:Interpretation
5435:
5433:
5427:
5426:
5424:
5423:
5418:
5413:
5408:
5403:
5393:
5388:
5387:
5386:
5385:
5384:
5374:
5369:
5359:
5354:
5349:
5344:
5339:
5334:
5328:
5326:
5320:
5319:
5316:
5315:
5313:
5312:
5304:
5303:
5302:
5301:
5296:
5295:
5294:
5289:
5284:
5264:
5263:
5262:
5260:minimal axioms
5257:
5246:
5245:
5244:
5233:
5232:
5231:
5226:
5221:
5216:
5211:
5206:
5193:
5191:
5172:
5171:
5169:
5168:
5167:
5166:
5154:
5149:
5148:
5147:
5142:
5137:
5132:
5122:
5117:
5112:
5107:
5106:
5105:
5100:
5090:
5089:
5088:
5083:
5078:
5073:
5063:
5058:
5057:
5056:
5051:
5046:
5036:
5035:
5034:
5029:
5024:
5019:
5014:
5009:
4999:
4994:
4989:
4984:
4983:
4982:
4977:
4972:
4967:
4957:
4952:
4950:Formation rule
4947:
4942:
4941:
4940:
4935:
4925:
4924:
4923:
4913:
4908:
4903:
4898:
4892:
4886:
4869:Formal systems
4865:
4864:
4861:
4860:
4858:
4857:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4817:
4812:
4811:
4810:
4805:
4794:
4792:
4788:
4787:
4785:
4784:
4783:
4782:
4772:
4767:
4766:
4765:
4758:Large cardinal
4755:
4750:
4745:
4740:
4735:
4721:
4720:
4719:
4714:
4709:
4694:
4692:
4682:
4681:
4679:
4678:
4677:
4676:
4671:
4666:
4656:
4651:
4646:
4641:
4636:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4595:
4593:
4586:
4585:
4583:
4582:
4581:
4580:
4575:
4570:
4565:
4560:
4555:
4547:
4546:
4545:
4540:
4530:
4525:
4523:Extensionality
4520:
4518:Ordinal number
4515:
4505:
4500:
4499:
4498:
4487:
4481:
4475:
4474:
4471:
4470:
4468:
4467:
4462:
4457:
4452:
4447:
4442:
4437:
4436:
4435:
4425:
4424:
4423:
4410:
4408:
4402:
4401:
4399:
4398:
4397:
4396:
4391:
4386:
4376:
4371:
4366:
4361:
4356:
4351:
4345:
4343:
4337:
4336:
4334:
4333:
4328:
4323:
4318:
4313:
4308:
4303:
4302:
4301:
4291:
4286:
4281:
4276:
4271:
4266:
4260:
4258:
4249:
4243:
4242:
4240:
4239:
4234:
4229:
4224:
4219:
4214:
4202:Cantor's
4200:
4195:
4190:
4180:
4178:
4165:
4164:
4162:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4101:
4100:
4099:
4088:
4086:
4082:
4081:
4074:
4073:
4066:
4059:
4051:
4045:
4044:
4036:
4026:
4015:
4014:External links
4012:
4011:
4010:
3975:
3959:
3956:
3953:
3952:
3943:
3934:
3925:
3899:
3873:
3867:For more, see
3860:
3850:
3832:
3819:
3788:
3764:(2): 481–511.
3747:
3735:
3722:
3703:
3693:
3663:
3649:
3648:
3646:
3643:
3640:
3639:
3630:
3620:
3619:
3617:
3614:
3613:
3612:
3607:
3605:Presupposition
3602:
3597:
3592:
3587:
3585:List of axioms
3582:
3576:
3571:
3565:
3564:
3550:
3534:
3531:
3527:modern algebra
3515:physical space
3513:as a model of
3507:mathematicians
3502:
3499:
3475:
3472:
3452:
3432:
3408:
3369:
3345:
3318:
3315:
3312:
3304:
3301:
3298:
3283:
3270:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3204:
3184:
3165:
3162:
3160:
3155:
3122:
3119:
3095:plane geometry
3086:
3083:
3070:
3046:
3025:
3004:
3001:
2998:
2995:
2992:
2988:
2984:
2981:
2976:
2963:
2962:
2950:
2928:
2925:
2919:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2798:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2729:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2677:unary function
2664:
2644:
2624:
2621:
2618:
2615:
2612:
2609:
2604:
2601:
2595:
2566:axiomatization
2557:
2554:
2538:ergodic theory
2534:measure theory
2435:
2432:
2361:
2358:
2344:
2340:
2337:
2334:
2329:
2324:
2320:
2309:
2296:
2276:
2256:
2236:
2214:
2192:
2178:
2144:
2139:
2135:
2131:
2128:
2125:
2122:
2099:
2096:
2093:
2090:
2070:
2050:
2030:
2006:
1986:
1977:with the term
1966:
1944:
1939:
1935:
1908:
1903:
1899:
1895:
1892:
1888:
1885:
1875:
1862:
1842:
1818:
1795:
1773:
1751:
1736:
1712:
1692:
1689:
1686:
1666:
1663:
1660:
1651:, the formula
1640:
1613:
1610:
1607:
1597:
1584:
1558:
1541:
1539:
1536:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1393:
1373:
1353:
1336:
1335:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1275:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1188:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1132:
1108:
1084:
1064:
1044:
1028:
1025:
1023:
1020:
1016:logical truths
979:
978:Logical axioms
976:
959:
956:
877:
873:
869:
864:
860:
856:
851:
847:
843:
838:
834:
828:
824:
820:
815:
811:
790:
768:
764:
760:
755:
751:
747:
742:
738:
734:
729:
725:
704:
639:
638:Other sciences
636:
597:Peano's axioms
478:Giuseppe Peano
457:
454:
453:
452:
451:
450:
447:
444:
441:
438:
434:
433:
431:
430:Common notions
425:
424:
423:
422:
403:
396:
389:
382:
374:
373:
282:
279:
277:
274:
255:remarks that "
235:mathematicians
206:from the verb
181:
178:
26:
9:
6:
4:
3:
2:
5832:
5821:
5818:
5816:
5813:
5812:
5810:
5797:
5796:
5791:
5783:
5777:
5774:
5772:
5769:
5767:
5764:
5762:
5759:
5755:
5752:
5751:
5750:
5747:
5745:
5742:
5740:
5737:
5735:
5731:
5728:
5726:
5723:
5721:
5718:
5716:
5713:
5711:
5708:
5707:
5705:
5701:
5695:
5692:
5690:
5687:
5685:
5684:Recursive set
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5634:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5604:
5602:
5600:
5596:
5590:
5587:
5585:
5582:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5553:
5550:
5548:
5545:
5543:
5540:
5539:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5506:
5503:
5502:
5501:
5498:
5494:
5493:of arithmetic
5491:
5490:
5489:
5486:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5458:
5457:
5454:
5450:
5447:
5445:
5442:
5441:
5440:
5437:
5436:
5434:
5432:
5428:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5401:
5400:from ZFC
5397:
5394:
5392:
5389:
5383:
5380:
5379:
5378:
5375:
5373:
5370:
5368:
5365:
5364:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5329:
5327:
5325:
5321:
5311:
5310:
5306:
5305:
5300:
5299:non-Euclidean
5297:
5293:
5290:
5288:
5285:
5283:
5282:
5278:
5277:
5275:
5272:
5271:
5269:
5265:
5261:
5258:
5256:
5253:
5252:
5251:
5247:
5243:
5240:
5239:
5238:
5234:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5205:
5202:
5201:
5199:
5195:
5194:
5192:
5187:
5181:
5176:Example
5173:
5165:
5160:
5159:
5158:
5155:
5153:
5150:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5127:
5126:
5123:
5121:
5118:
5116:
5113:
5111:
5108:
5104:
5101:
5099:
5096:
5095:
5094:
5091:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5068:
5067:
5064:
5062:
5059:
5055:
5052:
5050:
5047:
5045:
5042:
5041:
5040:
5037:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5004:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4962:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4939:
4936:
4934:
4933:by definition
4931:
4930:
4929:
4926:
4922:
4919:
4918:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4893:
4890:
4887:
4885:
4881:
4876:
4870:
4866:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4820:Kripke–Platek
4818:
4816:
4813:
4809:
4806:
4804:
4801:
4800:
4799:
4796:
4795:
4793:
4789:
4781:
4778:
4777:
4776:
4773:
4771:
4768:
4764:
4761:
4760:
4759:
4756:
4754:
4751:
4749:
4746:
4744:
4741:
4739:
4736:
4733:
4729:
4725:
4722:
4718:
4715:
4713:
4710:
4708:
4705:
4704:
4703:
4699:
4696:
4695:
4693:
4691:
4687:
4683:
4675:
4672:
4670:
4667:
4665:
4664:constructible
4662:
4661:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4596:
4594:
4592:
4587:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4550:
4548:
4544:
4541:
4539:
4536:
4535:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4514:
4510:
4506:
4504:
4501:
4497:
4494:
4493:
4492:
4489:
4488:
4485:
4482:
4480:
4476:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4438:
4434:
4431:
4430:
4429:
4426:
4422:
4417:
4416:
4415:
4412:
4411:
4409:
4407:
4403:
4395:
4392:
4390:
4387:
4385:
4382:
4381:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4346:
4344:
4342:
4341:Propositional
4338:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4300:
4297:
4296:
4295:
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4275:
4272:
4270:
4269:Logical truth
4267:
4265:
4262:
4261:
4259:
4257:
4253:
4250:
4248:
4244:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4209:
4205:
4201:
4199:
4196:
4194:
4191:
4189:
4185:
4182:
4181:
4179:
4177:
4171:
4166:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4098:
4095:
4094:
4093:
4090:
4089:
4087:
4083:
4079:
4072:
4067:
4065:
4060:
4058:
4053:
4052:
4049:
4043:
4041:
4037:
4034:
4030:
4027:
4025:
4021:
4018:
4017:
4007:
4001:
3994:
3990:
3986:
3985:
3980:
3976:
3974:
3973:0-534-06624-0
3970:
3966:
3962:
3961:
3947:
3938:
3929:
3914:
3910:
3903:
3888:
3884:
3877:
3870:
3864:
3854:
3846:
3842:
3836:
3829:
3823:
3812:
3808:
3801:
3795:
3793:
3785:
3779:
3775:
3771:
3767:
3763:
3759:
3751:
3742:
3740:
3732:
3726:
3720:I.2.72a18-b4.
3719:
3718:
3713:
3707:
3700:
3696:
3694:9780199891535
3690:
3686:
3682:
3677:
3676:
3667:
3660:
3654:
3650:
3634:
3625:
3621:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3580:
3577:
3575:
3572:
3570:
3567:
3566:
3562:
3551:
3548:
3542:
3537:
3530:
3528:
3524:
3520:
3516:
3512:
3508:
3498:
3496:
3492:
3487:
3473:
3450:
3430:
3422:
3398:
3394:
3390:
3385:
3383:
3359:
3335:
3329:
3316:
3313:
3302:
3299:
3282:
3268:
3260:
3256:
3234:
3231:
3174:
3172:
3154:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3121:Real analysis
3118:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3082:
3068:
3060:
3044:
2999:
2996:
2993:
2990:
2979:
2948:
2926:
2923:
2891:
2885:
2882:
2879:
2861:
2858:
2852:
2843:
2837:
2830:
2827:
2821:
2815:
2809:
2799:
2782:
2779:
2776:
2770:
2767:
2764:
2761:
2758:
2752:
2749:
2743:
2740:
2730:
2713:
2710:
2707:
2704:
2695:
2692:
2682:
2681:
2680:
2678:
2662:
2642:
2619:
2616:
2613:
2607:
2602:
2599:
2581:
2579:
2575:
2574:number theory
2571:
2567:
2563:
2553:
2551:
2547:
2543:
2539:
2535:
2530:
2528:
2527:Galois theory
2524:
2520:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2487:
2485:
2481:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2449:
2448:real analysis
2445:
2440:
2431:
2429:
2425:
2421:
2417:
2412:
2410:
2406:
2402:
2397:
2395:
2390:
2388:
2384:
2383:
2378:
2374:
2370:
2366:
2355:
2342:
2338:
2327:
2322:
2318:
2308:
2294:
2274:
2254:
2234:
2227:, a variable
2190:
2182:
2177:
2175:
2171:
2167:
2163:
2159:
2142:
2137:
2133:
2126:
2123:
2113:
2094:
2088:
2068:
2048:
2028:
2020:
2004:
1984:
1964:
1942:
1937:
1933:
1921:
1906:
1901:
1897:
1890:
1886:
1874:
1860:
1840:
1832:
1831:substitutable
1816:
1809:
1793:
1786:, a variable
1749:
1740:
1735:
1733:
1729:
1724:
1710:
1690:
1687:
1684:
1664:
1661:
1658:
1638:
1631:
1624:
1611:
1608:
1605:
1596:
1582:
1574:
1545:
1535:
1533:
1528:
1525:
1523:
1519:
1518:
1495:
1486:
1477:
1465:
1456:
1430:
1424:
1415:
1407:
1391:
1371:
1351:
1343:
1342:
1322:
1316:
1310:
1298:
1289:
1276:
1256:
1250:
1238:
1232:
1214:
1208:
1199:
1189:
1172:
1166:
1157:
1150:
1149:
1148:
1146:
1122:
1098:
1082:
1062:
1042:
1034:
1019:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
985:
975:
973:
969:
965:
955:
952:
948:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
903:
901:
897:
893:
875:
871:
867:
862:
858:
854:
849:
845:
841:
836:
832:
826:
822:
818:
813:
809:
788:
766:
762:
758:
753:
749:
745:
740:
736:
732:
727:
723:
702:
694:
690:
684:
682:
676:
674:
670:
666:
662:
658:
657:Mendel's laws
654:
650:
646:
645:Newton's laws
635:
633:
629:
625:
621:
617:
613:
608:
606:
602:
598:
594:
589:
587:
583:
579:
576:
571:
569:
563:
561:
556:
551:
548:
546:
542:
538:
534:
530:
526:
521:
517:
514:
509:
507:
502:
498:
497:vector spaces
494:
490:
486:
481:
479:
475:
471:
467:
463:
448:
445:
442:
439:
436:
435:
432:
429:
428:
427:
426:
420:
416:
412:
408:
404:
401:
397:
394:
390:
387:
383:
380:
379:straight line
376:
375:
371:
370:
369:
368:
367:
365:
364:
357:
355:
351:
345:
341:
339:
334:
332:
328:
324:
319:
317:
313:
309:
305:
301:
296:
292:
288:
273:
271:
267:
263:
258:
254:
249:
247:
243:
238:
236:
232:
229:
228:ancient Greek
225:
215:
205:
201:
191:
187:
177:
175:
169:
167:
163:
157:
155:
151:
147:
143:
139:
135:
131:
130:logical axiom
127:
123:
118:
116:
112:
108:
104:
99:
96:
90:
85:
84:Ancient Greek
81:
77:
73:
69:
65:
61:
56:
52:
48:
44:
37:
33:
19:
5786:
5584:Ultraproduct
5431:Model theory
5396:Independence
5332:Formal proof
5324:Proof theory
5307:
5280:
5237:real numbers
5209:second-order
5120:Substitution
4997:Metalanguage
4938:conservative
4911:Axiom schema
4855:Constructive
4825:Morse–Kelley
4791:Set theories
4770:Aleph number
4763:inaccessible
4669:Grothendieck
4553:intersection
4440:Higher-order
4428:Second-order
4374:Truth tables
4331:Venn diagram
4114:Formal proof
4091:
4039:
3982:
3964:
3946:
3937:
3928:
3917:, retrieved
3912:
3902:
3891:, retrieved
3886:
3876:
3863:
3853:
3844:
3841:Heath, T. L.
3835:
3827:
3822:
3806:
3761:
3757:
3750:
3730:
3725:
3715:
3711:
3706:
3698:
3674:
3666:
3658:
3653:
3633:
3624:
3595:Regulæ Juris
3590:Model theory
3504:
3494:
3490:
3488:
3420:
3396:
3392:
3386:
3357:
3333:
3331:
3284:
3258:
3254:
3169:
3167:
3134:
3127:real numbers
3124:
3088:
2964:
2582:
2565:
2562:Peano axioms
2559:
2531:
2515:group theory
2510:
2488:
2441:
2437:
2415:
2413:
2400:
2398:
2391:
2386:
2380:
2364:
2363:
2310:
2180:
2179:
2173:
2169:
2166:metatheorems
2165:
2161:
2157:
2111:
1924:
1876:
1738:
1737:
1731:
1728:axiom scheme
1725:
1627:
1598:
1543:
1542:
1529:
1526:
1522:modus ponens
1521:
1517:modus ponens
1515:
1341:axiom schema
1339:
1337:
1030:
1003:
981:
971:
967:
961:
951:Alain Aspect
935:entanglement
904:
890:), and then
801:(defined as
715:(defined as
685:
677:
672:
668:
664:
641:
609:
593:Gödel showed
590:
572:
564:
552:
549:
522:
518:
510:
500:
489:group theory
485:field theory
482:
462:propositions
459:
400:right angles
386:line segment
362:
358:
353:
347:
343:
338:self-evident
335:
320:
307:
303:
284:
281:Early Greeks
269:
265:
250:
241:
239:
231:philosophers
223:
213:
199:
185:
183:
170:
158:
153:
149:
145:
141:
137:
125:
119:
101:The precise
100:
94:
67:
63:
59:
57:
55:
5694:Type theory
5642:undecidable
5574:Truth value
5461:equivalence
5140:non-logical
4753:Enumeration
4743:Isomorphism
4690:cardinality
4674:Von Neumann
4639:Ultrafilter
4604:Uncountable
4538:equivalence
4455:Quantifiers
4445:Fixed-point
4414:First-order
4294:Consistency
4279:Proposition
4256:Traditional
4227:Lindström's
4217:Compactness
4159:Type theory
4104:Cardinality
4042:axioms page
3131:isomorphism
2542:probability
2409:commutative
2382:tautologies
2247:and a term
1145:implication
1008:tautologies
972:non-logical
939:EPR paradox
474:Mario Pieri
295:Tautologies
204:verbal noun
122:mathematics
5809:Categories
5505:elementary
5198:arithmetic
5066:Quantifier
5044:functional
4916:Expression
4634:Transitive
4578:identities
4563:complement
4496:hereditary
4479:Set theory
4033:PlanetMath
4024:PhilPapers
3919:19 October
3893:19 October
3826:Wolff, P.
3645:References
3397:consistent
3111:hyperbolic
2556:Arithmetic
2444:arithmetic
1099:are only "
1000:assignment
931:David Bohm
911:Niels Bohr
669:postulates
665:principles
578:set theory
560:consistent
555:collection
499:) without
372:Postulates
350:hypotheses
287:syllogisms
144:) implies
128:may be a "
103:definition
68:assumption
5776:Supertask
5679:Recursion
5637:decidable
5471:saturated
5449:of models
5372:deductive
5367:axiomatic
5287:Hilbert's
5274:Euclidean
5255:canonical
5178:axiomatic
5110:Signature
5039:Predicate
4928:Extension
4850:Ackermann
4775:Operation
4654:Universal
4644:Recursive
4619:Singleton
4614:Inhabited
4599:Countable
4589:Types of
4573:power set
4543:partition
4460:Predicate
4406:Predicate
4321:Syllogism
4311:Soundness
4284:Inference
4274:Tautology
4176:paradoxes
3993:Q26720682
3610:Principle
3509:regarded
3474:ϕ
3471:¬
3451:ϕ
3431:ϕ
3407:Σ
3393:recursive
3368:Σ
3358:deduction
3344:Σ
3317:ϕ
3314:⊢
3311:Σ
3303:ϕ
3300:⊨
3297:Σ
3269:ϕ
3235:ϕ
3229:Γ
3203:Σ
3183:Λ
3171:deductive
3003:⟩
2983:⟨
2949:ϕ
2886:ϕ
2877:∀
2874:→
2853:ϕ
2850:→
2838:ϕ
2825:∀
2822:∧
2810:ϕ
2774:→
2747:∀
2738:∀
2699:¬
2690:∀
2426:define a
2414:Thus, an
2405:discourse
2387:postulate
2343:ϕ
2336:∃
2333:→
2319:ϕ
2295:ϕ
2191:ϕ
2162:metaproof
2134:ϕ
2130:→
2127:ϕ
2121:∀
2110:. Again,
2061:and that
1965:ϕ
1934:ϕ
1898:ϕ
1894:→
1891:ϕ
1884:∀
1861:ϕ
1750:ϕ
1493:¬
1490:→
1481:→
1472:→
1463:¬
1460:→
1428:→
1419:→
1317:ϕ
1314:→
1311:ψ
1305:→
1299:ψ
1296:¬
1293:→
1290:ϕ
1287:¬
1257:χ
1254:→
1251:ϕ
1245:→
1239:ψ
1236:→
1233:ϕ
1224:→
1215:χ
1212:→
1209:ψ
1203:→
1200:ϕ
1173:ϕ
1170:→
1167:ψ
1161:→
1158:ϕ
1131:→
1107:¬
1083:ψ
1063:χ
1043:ϕ
998:by every
996:satisfied
990:that are
943:John Bell
900:manifolds
868:−
855:−
842:−
681:falsified
601:corollary
415:intersect
361:Euclid's
354:postulate
312:Aristotle
308:postulate
242:postulate
184:The word
180:Etymology
72:statement
64:postulate
5761:Logicism
5754:timeline
5730:Concrete
5589:Validity
5559:T-schema
5552:Kripke's
5547:Tarski's
5542:semantic
5532:Strength
5481:submodel
5476:spectrum
5444:function
5292:Tarski's
5281:Elements
5268:geometry
5224:Robinson
5145:variable
5130:function
5103:spectrum
5093:Sentence
5049:variable
4992:Language
4945:Relation
4906:Automata
4896:Alphabet
4880:language
4734:-jection
4712:codomain
4698:Function
4659:Universe
4629:Infinite
4533:Relation
4316:Validity
4306:Argument
4204:theorem,
4040:Metamath
4000:citation
3989:Wikidata
3981:(1889),
3843:(1956).
3811:Archived
3533:See also
3421:complete
3293:if
3259:complete
3107:triangle
2941:formula
2907:for any
2434:Examples
2373:integers
2371:and the
2158:is valid
1829:that is
1121:negation
1022:Examples
1004:at least
984:formulas
919:Max Born
616:infinite
575:Cantor's
537:Poincaré
493:topology
363:Elements
327:sciences
323:geometry
300:theorems
262:Boethius
132:" or a "
5703:Related
5500:Diagram
5398: (
5377:Hilbert
5362:Systems
5357:Theorem
5235:of the
5180:systems
4960:Formula
4955:Grammar
4871: (
4815:General
4528:Forcing
4513:Element
4433:Monadic
4208:paradox
4149:Theorem
4085:General
3784:realist
3778:2274520
3600:Theorem
3057:is the
2017:. (See
1408:, then
968:logical
624:forcing
541:Hilbert
533:Russell
266:petitio
257:Geminus
253:Proclus
214:axioein
209:ἀξιόειν
111:evident
80:premise
5466:finite
5229:Skolem
5182:
5157:Theory
5125:Symbol
5115:String
5098:atomic
4975:ground
4970:closed
4965:atomic
4921:ground
4884:syntax
4780:binary
4707:domain
4624:Finite
4389:finite
4247:Logics
4206:
4154:Theory
3991:
3971:
3782:for a
3776:
3691:
3523:Galois
3505:Early
3173:system
3141:. The
3103:angles
3015:where
2635:where
2548:, and
2525:, and
2523:fields
2401:axioms
2377:groups
1806:and a
1384:, and
1143:" for
1119:" for
1075:, and
673:axioms
543:, and
476:, and
419:angles
393:circle
316:Euclid
246:Euclid
200:axíōma
195:ἀξίωμα
95:axíōma
89:ἀξίωμα
45:,
18:Axioms
5456:Model
5204:Peano
5061:Proof
4901:Arity
4830:Naive
4717:image
4649:Fuzzy
4609:Empty
4558:union
4503:Class
4144:Model
4134:Lemma
4092:Axiom
4029:Axiom
4020:Axiom
3814:(PDF)
3803:(PDF)
3786:view.
3774:JSTOR
3616:Notes
3574:Dogma
3105:of a
2675:is a
2519:rings
2462:like
2416:axiom
2170:proof
1571:be a
986:in a
628:Cohen
614:, an
545:Gödel
529:Frege
513:field
304:axiom
224:áxios
219:ἄξιος
202:), a
192:word
190:Greek
186:axiom
126:axiom
124:, an
115:logic
86:word
70:is a
66:, or
60:axiom
32:axion
5579:Type
5382:list
5186:list
5163:list
5152:Term
5086:rank
4980:open
4874:list
4686:Maps
4591:sets
4450:Free
4420:list
4170:list
4097:list
4006:link
3969:ISBN
3921:2019
3895:2019
3689:ISBN
3463:nor
3061:and
2560:The
2466:, a
2450:and
1833:for
1808:term
1547:Let
1446:and
1404:are
970:and
314:and
306:and
233:and
140:and
76:true
36:axon
5266:of
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4702:Map
4509:Ur-
4491:Set
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667:or
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5007:¬
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