5779:
3530:
910:) 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 '
668:
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 (
3090:("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
3544:
345:. 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.
3846:
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
675:
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
508:
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
631:
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,
492:
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.
942:
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
667:
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
926:. 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
497:). 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.
148:
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
504:
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.
2400:, 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.
943:
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.).
554:
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
2894:
1262:
3316:
2427:
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.
286:
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
157:). 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.
318:, 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
3369:. 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".
160:
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
546:
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
248:
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."
291:, 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
1322:
457:, 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.
3506:, 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
3002:
3847:
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)
551:; 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.
1501:
2385:
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.
2785:
877:
3514:
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
2342:
1908:
2144:
2622:
1175:
768:
2716:
1666:
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
226:, 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.
2928:
3240:
355:, 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).
4158:
2214:
1773:
1558:
1944:
1433:
3024:
3473:
3138:, 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
4833:
3182:
2452:
3406:
3367:
3343:
3202:
3450:
3430:
3268:
2948:
2294:
2190:
1964:
1860:
1749:
1106:
1082:
1062:
1042:
623:(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.
2097:
1509:, 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
1130:
3994:
1690:
1664:
1611:
3126:, 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
3068:
3044:
2662:
2642:
2274:
2254:
2234:
2068:
2048:
2028:
2004:
1984:
1840:
1816:
1793:
1710:
1638:
1582:
1391:
1371:
1351:
788:
702:
3699:"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a.
2791:
4916:
4057:
1181:
449:
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,
2566:
2364:, 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
3377:
3276:
3106:
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.
2443:
are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of
5230:
274:
The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (
5388:
4176:
581:
5243:
4566:
512:
Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and
4828:
2396:. 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
1268:
137:), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example
5248:
5238:
4975:
4181:
2957:
1618:
4726:
4172:
3370:
5384:
3961:
3681:
607:
but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern
894:
In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The '
5481:
5225:
4050:
39:
4786:
4479:
4220:
3626:
Hilbert also made explicit the assumptions that Euclid used in his proofs but did not list in his common notions and postulates.
3131:
2444:
3972:
988:
3662:
1692:(or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol
5742:
5444:
5207:
5202:
5027:
4448:
4132:
2464:
1438:
1712:
has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.
2356:
are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the
1503:
are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and
125:". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (
5737:
5520:
5437:
5150:
5081:
4958:
4200:
4808:
5803:
5662:
5488:
5174:
4407:
3617:
Although not complete; some of the stated results did not actually follow from the stated postulates and common notions.
2722:
972:
793:
4813:
3799:
2460:
2302:
1868:
608:
77:
5145:
4884:
4142:
4043:
5540:
5535:
2105:
1516:
Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.
3518:
was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.
5808:
5469:
5059:
4453:
4421:
4112:
493:
However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g.,
473:
35:
4186:
2575:
5759:
5708:
5605:
5103:
5064:
4541:
2007:
895:
240:
Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books,
162:
5600:
4215:
3486:. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.
5530:
5069:
4921:
4904:
4627:
4107:
2471:
is used, but in fact, most mathematicians can actually prove all they need in systems weaker than ZFC, such as
1142:
3871:
539:
Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.
707:
672:) the theory that the postulates install. A theory is considered valid as long as it has not been falsified.
237:
demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).
5432:
5409:
5370:
5256:
5197:
4843:
4763:
4607:
4551:
4164:
3897:
3507:
3499:
2674:
2472:
599:
It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of
5722:
5449:
5427:
5394:
5287:
5133:
5118:
5091:
5042:
4926:
4861:
4686:
4652:
4647:
4521:
4352:
4329:
3079:
2899:
2558:
2456:
1394:
935:
911:
641:
637:
403:
31:
3207:
5652:
5505:
5297:
5015:
4751:
4657:
4516:
4501:
4382:
4357:
3139:
2538:
2534:
2487:
2468:
612:
5778:
3529:
215:), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the
5625:
5587:
5464:
5268:
5108:
5032:
5010:
4838:
4796:
4695:
4662:
4526:
4314:
4225:
3772:
2448:
2195:
1754:
1561:
1539:
923:
915:
620:
570:
349:
5754:
5645:
5630:
5610:
5567:
5454:
5404:
5330:
5275:
5212:
5005:
5000:
4948:
4716:
4705:
4377:
4277:
4205:
4196:
4192:
4127:
4122:
3857:
3705:
3099:
2511:
2382:
1917:
1400:
1021:
574:
501:
494:
477:
450:
319:
3688:
a statement or proposition that is regarded as being established, accepted, or self-evidently true
3102:
geometries. If one also removes the second postulate ("a line can be extended indefinitely") then
3007:
676:
neither
Euclidean geometry or differential calculus that they imply. It became more apparent when
5783:
5552:
5515:
5500:
5493:
5476:
5262:
5128:
5054:
5037:
4990:
4803:
4712:
4546:
4531:
4491:
4443:
4428:
4416:
4372:
4347:
4117:
4066:
3762:
3535:
3455:
3127:
3087:
3083:
3047:
2507:
2483:
2479:
2412:
1520:
1085:
952:
880:
681:
556:
543:
395:
279:
154:
150:
95:
5280:
4736:
3511:
282:) was developed by the ancient Greeks, and has become the core principle of modern mathematics.
2010:.) In informal terms, this example allows us to state that, if we know that a certain property
588:, for example) to construct a statement whose truth is independent of that set of axioms. As a
402:
on the same side less than two right angles, the two straight lines, if produced indefinitely,
5718:
5525:
5335:
5325:
5217:
5098:
4933:
4909:
4690:
4674:
4579:
4556:
4433:
4402:
4367:
4262:
4097:
3988:
3957:
3677:
3549:
3167:
3135:
3103:
2370:
1819:
1796:
996:
980:
963:(somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).
931:
903:
884:
649:
525:
481:
283:
223:
60:
3391:
3352:
3328:
3187:
87:), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
5732:
5727:
5620:
5577:
5399:
5360:
5355:
5340:
5166:
5123:
5020:
4818:
4768:
4342:
4304:
3977:
3967:
3754:
3669:
3557:
3515:
2440:
2416:
645:
633:
593:
585:
521:
458:
454:
3673:
3435:
3415:
3253:
2933:
2279:
2175:
1949:
1845:
1734:
1091:
1067:
1047:
1027:
329:
assumption common to many branches of science. A good example would be the assertion that:
5713:
5703:
5657:
5640:
5595:
5557:
5459:
5379:
5186:
5113:
5086:
5074:
4980:
4894:
4868:
4823:
4791:
4592:
4394:
4337:
4287:
4252:
4210:
3567:
2495:
2491:
2365:
2073:
1115:
1000:
976:
677:
3583:
2889:{\displaystyle (\phi (0)\land \forall x.\,(\phi (x)\to \phi (Sx)))\to \forall x.\phi (x)}
1669:
1643:
1590:
2149:, that is, we must be able to give a "proof" of this fact, or more properly speaking, a
938:) in the Copenhagen and the Hidden variable case. The experiment was conducted first by
5698:
5677:
5635:
5615:
5510:
5365:
4963:
4953:
4943:
4938:
4872:
4746:
4622:
4511:
4506:
4484:
4085:
3593:
3573:
3503:
3053:
3029:
2665:
2647:
2627:
2526:
2522:
2357:
2259:
2239:
2219:
2053:
2033:
2013:
1989:
1969:
1825:
1801:
1778:
1695:
1623:
1567:
1376:
1356:
1336:
984:
773:
687:
669:
600:
466:
299:
hold a slightly different meaning for the present day mathematician, than they did for
216:
178:
5797:
5672:
5350:
4857:
4642:
4632:
4602:
4587:
4257:
3495:
3376:
Note that "completeness" has a different meaning here than it does in the context of
2562:
2515:
2436:
2408:
1257:{\displaystyle (\phi \to (\psi \to \chi ))\to ((\phi \to \psi )\to (\phi \to \chi ))}
1004:
922:. It was created so as to try to give deterministic explanation to phenomena such as
533:
529:
517:
367:
99:
72:
3098:
add up to exactly 180 degrees or less, respectively, and are known as
Euclidean and
1523:, but additional logical axioms are needed to include a quantifier in the calculus.
102:
or well-established, that it is accepted without controversy or question. In modern
5572:
5419:
5320:
5312:
5192:
5140:
5049:
4985:
4968:
4899:
4758:
4617:
4319:
4102:
3578:
3115:
2550:
2503:
1716:
1505:
1329:
939:
604:
563:
485:
374:
326:
4027:
3373:
establishes the completeness of a certain commonly used type of deductive system.
3311:{\displaystyle {\text{if }}\Sigma \models \phi {\text{ then }}\Sigma \vdash \phi }
2157:
of our theory of mathematical logic since we are dealing with the very concept of
1024:
it is common to take as logical axioms all formulae of the following forms, where
3973:
On an
Evolutionist Theory of Axioms: inaugural lecture delivered October 15, 1889
2070:
stands for a particular object in our structure, then we should be able to claim
5682:
5562:
4741:
4731:
4678:
4362:
4282:
4267:
4147:
4092:
3939:
Mendelson, "5. The Fixed Point
Theorem. Gödel's Incompleteness Theorem" of Ch. 2
3829:
3119:
2530:
2397:
934:
derived in 1964 a prediction that would lead to different experimental results (
927:
462:
388:
219:
192:
110:
577:
raised the possibility that any such system could turn out to be inconsistent.
472:
Structuralist mathematics goes further, and develops theories and axioms (e.g.
71:
or starting point for further reasoning and arguments. The word comes from the
4612:
4467:
4438:
4244:
4021:
4012:
3525:
2432:
1133:
919:
899:
616:
566:
548:
398:") It is true that, if a straight line falling on two straight lines make the
338:
275:
91:
3788:
3086:. The axioms are referred to as "4 + 1" because for nearly two millennia the
2561:. They are a set of axioms strong enough to prove many important facts about
2521:
This list could be expanded to include most fields of mathematics, including
205:), meaning "to deem worthy", but also "to require", which in turn comes from
5764:
5667:
4720:
4637:
4597:
4561:
4497:
4309:
4299:
4272:
3598:
3159:
2393:
888:
589:
300:
3981:
5749:
5547:
4995:
4700:
4294:
3095:
1109:
907:
513:
311:
250:
5345:
4137:
3766:
3588:
2361:
1333:, a rule for generating an infinite number of axioms. For example, if
562:
In a wider context, there was an attempt to base all of mathematics on
426:
Things which are equal to the same thing are also equal to one another.
399:
315:
288:
245:
241:
68:
4035:
3904:(Spring 2019 ed.), Metaphysics Research Lab, Stanford University
381:
304:
234:
82:
3758:
3412:, in the sense that there will always exist an arithmetic statement
559:, and the related demonstration of the consistency of those axioms.
3878:(Fall 2018 ed.), Metaphysics Research Lab, Stanford University
3510:
made elaborate efforts to derive them from traditional arithmetic.
3078:
Probably the oldest, and most famous, list of axioms are the 4 + 1
995:
some minimal set of tautologies that is sufficient for proving all
333:
When an equal amount is taken from equals, an equal amount results.
4889:
4235:
3562:
3091:
407:
261:
but in later manuscripts this usage was not always strictly kept.
103:
64:
20:
4008:
3718:"A proposition (whether true or false)" axiom, n., definition 2.
1317:{\displaystyle (\lnot \phi \to \lnot \psi )\to (\psi \to \phi ).}
341:
that were accepted without proof. Such a hypothesis was termed a
337:
At the foundation of the various sciences lay certain additional
2997:{\displaystyle {\mathfrak {N}}=\langle \mathbb {N} ,0,S\rangle }
435:
Things which coincide with one another are equal to one another.
24:
4039:
4017:
584:
that it is possible, for any sufficiently large set of axioms (
432:
If equals are subtracted from equals, the remainders are equal.
2478:
The study of topology in mathematics extends all over through
1003:
more logical axioms than that are required, in order to prove
2447:
with choice, abbreviated ZFC, or some very similar system of
955:, a clear distinction is made between two notions of axioms:
684:
where the invariant quantity is no more the
Euclidean length
580:
The formalist project suffered a setback a century ago, when
206:
196:
182:
3734:
Mendelson, "3. First-Order
Theories: Proper Axioms" of Ch. 2
3246:. A desirable property of a deductive system is that it be
596:
is an unprovable assertion within the scope of that theory.
2368:). Thus non-logical axioms, unlike logical axioms, are not
1084:
can be any formulae of the language and where the included
914:' approach was developed for some time by Albert Einstein,
453:, theorems) and definitions. One must concede the need for
106:, an axiom is a premise or starting point for reasoning.
3745:
Maddy, Penelope (June 1988). "Believing the Axioms, I".
3250:. A system is said to be complete if, for all formulas
3660:
Stevenson, Angus; Lindberg, Christine A., eds. (2015).
1496:{\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))}
652:
law, etc. These founding assertions are usually called
514:
mathematics itself can be regarded as a branch of logic
3819:, 1963, New York: New American Library, pp 47–48
3458:
3438:
3418:
3394:
3355:
3331:
3279:
3256:
3210:
3190:
3170:
3114:
The objectives of the study are within the domain of
3056:
3032:
3010:
2960:
2936:
2902:
2794:
2725:
2677:
2650:
2630:
2578:
2459:
of ZFC. Sometimes slightly stronger theories such as
2305:
2282:
2262:
2242:
2222:
2198:
2178:
2108:
2076:
2056:
2036:
2016:
1992:
1972:
1952:
1920:
1871:
1848:
1828:
1804:
1781:
1757:
1737:
1698:
1672:
1646:
1626:
1593:
1570:
1542:
1441:
1403:
1379:
1359:
1339:
1271:
1184:
1145:
1118:
1094:
1070:
1050:
1030:
796:
776:
710:
690:
592:, Gödel proved that the consistency of a theory like
542:
In the modern understanding, a set of axioms is any
429:
If equals are added to equals, the wholes are equal.
325:
An "axiom", in classical terminology, referred to a
5691:
5586:
5418:
5311:
5163:
4856:
4779:
4673:
4577:
4466:
4393:
4328:
4243:
4234:
4156:
4073:
2388:Non-logical axioms are often simply referred to as
3661:
3467:
3444:
3424:
3400:
3361:
3337:
3310:
3262:
3234:
3196:
3176:
3118:. The real numbers are uniquely picked out (up to
3062:
3038:
3018:
2996:
2942:
2922:
2888:
2780:{\displaystyle \forall x.\forall y.(Sx=Sy\to x=y)}
2779:
2710:
2656:
2636:
2616:
2336:
2288:
2268:
2248:
2228:
2208:
2184:
2138:
2091:
2062:
2042:
2022:
1998:
1978:
1958:
1938:
1902:
1854:
1834:
1810:
1787:
1767:
1743:
1719:, is that which provides us with what is known as
1704:
1684:
1658:
1632:
1605:
1576:
1552:
1495:
1427:
1385:
1365:
1345:
1316:
1256:
1169:
1124:
1100:
1076:
1056:
1036:
872:{\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}}
871:
782:
762:
696:
322:is a definitive exposition of the classical view.
30:Several terms redirect here. For other uses, see
2337:{\displaystyle \phi _{t}^{x}\to \exists x\,\phi }
1903:{\displaystyle \forall x\,\phi \to \phi _{t}^{x}}
883:where flat Minkowskian geometry is replaced with
611:for set theory. Furthermore, using techniques of
536:are some of the key figures in this development.
991:of values. Usually one takes as logical axioms
2565:and they allowed Gödel to establish his famous
2139:{\displaystyle \forall x\phi \to \phi _{t}^{x}}
3921:Mendelson, "6. Other Axiomatizations" of Ch. 1
3730:
3728:
3478:There is thus, on the one hand, the notion of
1112:of the immediately following proposition and "
1007:that are not tautologies in the strict sense.
348:The classical approach is well-illustrated by
4051:
3956:Belmont, California: Wadsworth & Brooks.
3930:Mendelson, "3. First-Order Theories" of Ch. 2
2490:, and all the related paraphernalia, such as
509:deal of extra information about this system.
8:
3703:Online, accessed 2012-04-28. Cf. Aristotle,
3475:can be proved from the given set of axioms.
3229:
3211:
2991:
2971:
2617:{\displaystyle {\mathfrak {L}}_{NT}=\{0,S\}}
2611:
2599:
2170:Axiom scheme for Existential Generalization.
1136:from antecedent to consequent propositions:
770:) > but the Minkowski spacetime interval
3789:"Axiom — Powszechna Encyklopedia Filozofii"
3484:completeness of a set of non-logical axioms
4877:
4472:
4240:
4058:
4044:
4036:
3993:: CS1 maint: location missing publisher (
3070:is naturally interpreted as the number 0.
2374:. Another name for a non-logical axiom is
2296:, the below formula is universally valid.
2161:itself. Aside from this, we can also have
1862:, the below formula is universally valid.
1584:, the below formula is universally valid.
1519:These axiom schemata are also used in the
3668:(3rd ed.). Oxford University Press.
3457:
3437:
3417:
3393:
3354:
3330:
3294:
3280:
3278:
3255:
3209:
3189:
3169:
3134:tell us that if we restrict ourselves to
3055:
3031:
3012:
3011:
3009:
2975:
2974:
2962:
2961:
2959:
2935:
2911:
2905:
2904:
2901:
2822:
2793:
2724:
2676:
2649:
2629:
2587:
2581:
2580:
2577:
2330:
2315:
2310:
2304:
2281:
2261:
2241:
2221:
2200:
2199:
2197:
2177:
2130:
2125:
2107:
2075:
2055:
2035:
2015:
1991:
1971:
1951:
1930:
1925:
1919:
1894:
1889:
1878:
1870:
1847:
1827:
1803:
1780:
1759:
1758:
1756:
1736:
1728:Axiom scheme for Universal Instantiation.
1697:
1671:
1645:
1625:
1592:
1569:
1544:
1543:
1541:
1440:
1402:
1378:
1358:
1338:
1270:
1183:
1170:{\displaystyle \phi \to (\psi \to \phi )}
1144:
1117:
1093:
1069:
1049:
1029:
863:
850:
837:
824:
814:
801:
795:
775:
754:
741:
728:
715:
709:
689:
3902:The Stanford Encyclopedia of Philosophy
3876:The Stanford Encyclopedia of Philosophy
3834:The Thirteen Books of Euclid's Elements
3639:
3610:
763:{\displaystyle l^{2}=x^{2}+y^{2}+z^{2}}
660:so as to distinguish from mathematical
122:
3986:
2711:{\displaystyle \forall x.\lnot (Sx=0)}
930:in 1935). Taking this idea seriously,
3321:that is, for any statement that is a
98:, an axiom is a statement that is so
7:
3805:from the original on 9 October 2022.
3796:Polskie Towarzystwo Tomasza z Akwinu
3783:
3781:
3674:10.1093/acref/9780195392883.001.0001
3378:Gödel's first incompleteness theorem
2923:{\displaystyle {\mathfrak {L}}_{NT}}
2453:Von Neumann–Bernays–Gödel set theory
3954:Introduction to mathematical logic.
3235:{\displaystyle \{(\Gamma ,\phi )\}}
2963:
2906:
2582:
2201:
1760:
1545:
438:The whole is greater than the part.
118:
3480:completeness of a deductive system
3459:
3395:
3356:
3332:
3299:
3285:
3217:
3191:
3171:
3153:Deductive systems and completeness
2865:
2813:
2735:
2726:
2687:
2678:
2324:
2109:
1872:
1715:Another, more interesting example
1481:
1451:
1284:
1275:
1095:
370:from any point to any other point.
94:varies across fields of study. In
16:Statement that is taken to be true
14:
3872:"Gödel's Incompleteness Theorems"
3570:, axiom in science and philosophy
3204:of non-logical axioms, and a set
5777:
3542:
3528:
2101:we are claiming that the formula
999:in the language; in the case of
469:were pioneers in this movement.
377:continuously in both directions.
40:Postulation (algebraic geometry)
3836:. New York: Dover. p. 200.
3408:of the Theory of Arithmetic is
3124:Dedekind complete ordered field
3026:is the set of natural numbers,
2209:{\displaystyle {\mathfrak {L}}}
1768:{\displaystyle {\mathfrak {L}}}
1553:{\displaystyle {\mathfrak {L}}}
640:in classical electromagnetism,
500:When mathematicians employ the
410:less than the two right angles.
384:with any center and any radius.
3664:New Oxford American Dictionary
3482:and on the other hand that of
3226:
3214:
2883:
2877:
2862:
2859:
2856:
2853:
2844:
2838:
2835:
2829:
2823:
2807:
2801:
2795:
2774:
2762:
2744:
2705:
2690:
2465:strongly inaccessible cardinal
2321:
2118:
2086:
2080:
1882:
1490:
1487:
1478:
1472:
1469:
1463:
1460:
1457:
1448:
1442:
1422:
1416:
1410:
1407:
1308:
1302:
1296:
1293:
1290:
1281:
1272:
1251:
1248:
1242:
1236:
1233:
1230:
1224:
1218:
1215:
1212:
1209:
1206:
1200:
1194:
1191:
1185:
1164:
1158:
1152:
1149:
1119:
406:on that side on which are the
310:The ancient Greeks considered
233:is to "demand"; for instance,
1:
5738:History of mathematical logic
3900:, in Zalta, Edward N. (ed.),
3874:, in Zalta, Edward N. (ed.),
2567:second incompleteness theorem
2407:is an elementary basis for a
1939:{\displaystyle \phi _{t}^{x}}
1428:{\displaystyle A\to (B\to A)}
1327:Each of these patterns is an
983:, that is, formulas that are
380:It is possible to describe a
229:The root meaning of the word
5663:Primitive recursive function
3817:Breakthroughs in Mathematics
3722:Online, accessed 2012-04-28.
3371:Gödel's completeness theorem
3019:{\displaystyle \mathbb {N} }
3468:{\displaystyle \lnot \phi }
2445:Zermelo–Fraenkel set theory
373:It is possible to extend a
141: + 0 =
5825:
4727:Schröder–Bernstein theorem
4454:Monadic predicate calculus
4113:Foundations of mathematics
3952:Mendelson, Elliot (1987).
3898:"The Continuum Hypothesis"
3870:Raatikainen, Panu (2018),
3646:Cf. axiom, n., etymology.
3388:set of non-logical axioms
3147:Role in mathematical logic
3088:fifth (parallel) postulate
2954:The standard structure is
2668:and the following axioms:
2256:that is substitutable for
2192:in a first-order language
2163:Existential Generalization
1751:in a first-order language
573:and similar antinomies of
253:translated 'postulate' as
207:
197:
183:
76:
36:Axiomatic (disambiguation)
29:
18:
5773:
5760:Philosophy of mathematics
5709:Automated theorem proving
4880:
4834:Von Neumann–Bernays–Gödel
4475:
3747:Journal of Symbolic Logic
3720:Oxford English Dictionary
3701:Oxford English Dictionary
3648:Oxford English Dictionary
3184:of logical axioms, a set
3132:Löwenheim–Skolem theorems
3122:) by the properties of a
2644:is a constant symbol and
2553:are the most widely used
2008:Substitution of variables
1617:This means that, for any
569:. Here, the emergence of
391:are equal to one another.
366:It is possible to draw a
163:philosophy of mathematics
3976:(1st ed.), Oxford,
3896:Koellner, Peter (2019),
3345:there actually exists a
3177:{\displaystyle \Lambda }
2431:Basic theories, such as
636:in classical mechanics,
619:) one can show that the
19:Not to be confused with
5410:Self-verifying theories
5231:Tarski's axiomatization
4182:Tarski's undefinability
4177:incompleteness theorems
3401:{\displaystyle \Sigma }
3380:, which states that no
3362:{\displaystyle \Sigma }
3338:{\displaystyle \Sigma }
3197:{\displaystyle \Sigma }
2950:with one free variable.
2473:second-order arithmetic
2461:Morse–Kelley set theory
2411:that together with the
1946:stands for the formula
1721:Universal Instantiation
1395:propositional variables
644:in general relativity,
609:Zermelo–Fraenkel axioms
314:as just one of several
257:and called the axioms
145:in integer arithmetic.
5784:Mathematics portal
5395:Proof of impossibility
5043:propositional variable
4353:Propositional calculus
3650:, accessed 2012-04-28.
3469:
3446:
3426:
3402:
3363:
3349:of the statement from
3339:
3312:
3264:
3236:
3198:
3178:
3064:
3040:
3020:
2998:
2944:
2924:
2890:
2781:
2712:
2658:
2638:
2618:
2559:first-order arithmetic
2467:allowing the use of a
2457:conservative extension
2338:
2290:
2270:
2250:
2230:
2210:
2186:
2153:. These examples are
2140:
2093:
2064:
2044:
2024:
2000:
1980:
1960:
1940:
1904:
1856:
1836:
1812:
1789:
1769:
1745:
1706:
1686:
1660:
1634:
1607:
1578:
1554:
1497:
1429:
1387:
1367:
1347:
1318:
1258:
1171:
1126:
1102:
1078:
1058:
1038:
873:
784:
764:
698:
648:of genetics, Darwin's
335:
265:Historical development
32:Axiom (disambiguation)
5653:Kolmogorov complexity
5606:Computably enumerable
5506:Model complete theory
5298:Principia Mathematica
4358:Propositional formula
4187:Banach–Tarski paradox
3470:
3447:
3445:{\displaystyle \phi }
3427:
3425:{\displaystyle \phi }
3403:
3364:
3340:
3313:
3265:
3263:{\displaystyle \phi }
3237:
3199:
3179:
3140:non-standard analysis
3065:
3041:
3021:
2999:
2945:
2943:{\displaystyle \phi }
2925:
2891:
2782:
2713:
2659:
2639:
2619:
2539:differential geometry
2535:representation theory
2498:. The development of
2488:differential topology
2469:Grothendieck universe
2463:or set theory with a
2339:
2291:
2289:{\displaystyle \phi }
2271:
2251:
2231:
2211:
2187:
2185:{\displaystyle \phi }
2141:
2094:
2065:
2045:
2025:
2001:
1981:
1961:
1959:{\displaystyle \phi }
1941:
1905:
1857:
1855:{\displaystyle \phi }
1837:
1813:
1790:
1770:
1746:
1744:{\displaystyle \phi }
1707:
1687:
1661:
1635:
1608:
1579:
1555:
1498:
1430:
1388:
1368:
1348:
1319:
1259:
1172:
1127:
1103:
1101:{\displaystyle \neg }
1086:primitive connectives
1079:
1077:{\displaystyle \psi }
1059:
1057:{\displaystyle \chi }
1039:
1037:{\displaystyle \phi }
874:
785:
765:
699:
331:
5601:Church–Turing thesis
5588:Computability theory
4797:continuum hypothesis
4315:Square of opposition
4173:Gödel's completeness
3456:
3436:
3416:
3392:
3353:
3329:
3277:
3254:
3208:
3188:
3168:
3054:
3030:
3008:
2958:
2934:
2900:
2792:
2723:
2675:
2648:
2628:
2576:
2502:brought with itself
2449:axiomatic set theory
2381:Almost every modern
2303:
2280:
2260:
2240:
2220:
2196:
2176:
2106:
2092:{\displaystyle P(t)}
2074:
2054:
2034:
2014:
1990:
1970:
1950:
1918:
1869:
1846:
1826:
1802:
1779:
1755:
1735:
1696:
1670:
1644:
1624:
1591:
1568:
1564:. For each variable
1562:first-order language
1540:
1439:
1401:
1377:
1357:
1337:
1269:
1182:
1143:
1125:{\displaystyle \to }
1116:
1092:
1068:
1048:
1028:
794:
774:
708:
688:
621:continuum hypothesis
387:It is true that all
63:that is taken to be
5804:Mathematical axioms
5755:Mathematical object
5646:P versus NP problem
5611:Computable function
5405:Reverse mathematics
5331:Logical consequence
5208:primitive recursive
5203:elementary function
4976:Free/bound variable
4829:Tarski–Grothendieck
4348:Logical connectives
4278:Logical equivalence
4128:Logical consequence
3706:Posterior Analytics
3323:logical consequence
3080:Euclid's postulates
2572:We have a language
2409:formal logic system
2383:mathematical theory
2320:
2135:
1935:
1899:
1685:{\displaystyle x=x}
1659:{\displaystyle x=x}
1606:{\displaystyle x=x}
1022:propositional logic
1016:Propositional logic
936:Bell's inequalities
887:geometry on curved
642:Einstein's equation
638:Maxwell's equations
495:hyperbolic geometry
320:posterior analytics
5553:Transfer principle
5516:Semantics of logic
5501:Categorical theory
5477:Non-standard model
4991:Logical connective
4118:Information theory
4067:Mathematical logic
3536:Mathematics portal
3500:axiomatic geometry
3490:Further discussion
3465:
3442:
3432:such that neither
3422:
3398:
3359:
3335:
3308:
3260:
3244:rules of inference
3232:
3194:
3174:
3164:consists of a set
3128:second-order logic
3074:Euclidean geometry
3060:
3048:successor function
3036:
3016:
2994:
2940:
2920:
2886:
2777:
2708:
2654:
2634:
2614:
2484:algebraic topology
2480:point set topology
2413:rules of inference
2354:Non-logical axioms
2349:Non-logical axioms
2334:
2306:
2286:
2266:
2246:
2226:
2206:
2182:
2136:
2121:
2089:
2060:
2040:
2020:
1996:
1976:
1956:
1936:
1921:
1900:
1885:
1852:
1832:
1808:
1785:
1765:
1741:
1702:
1682:
1656:
1630:
1603:
1574:
1550:
1533:Axiom of Equality.
1521:predicate calculus
1493:
1425:
1383:
1363:
1343:
1314:
1254:
1167:
1122:
1098:
1074:
1054:
1034:
971:These are certain
953:mathematical logic
947:Mathematical logic
881:general relativity
869:
780:
760:
694:
682:special relativity
557:Euclidean geometry
445:Modern development
396:Parallel postulate
280:rules of inference
155:Euclidean geometry
151:parallel postulate
96:classic philosophy
38:, and
5809:Concepts in logic
5791:
5790:
5723:Abstract category
5526:Theories of truth
5336:Rule of inference
5326:Natural deduction
5307:
5306:
4852:
4851:
4557:Cartesian product
4462:
4461:
4368:Many-valued logic
4343:Boolean functions
4226:Russell's paradox
4201:diagonal argument
4098:First-order logic
3550:Philosophy portal
3297:
3283:
3136:first-order logic
3104:elliptic geometry
3063:{\displaystyle 0}
3039:{\displaystyle S}
2657:{\displaystyle S}
2637:{\displaystyle 0}
2269:{\displaystyle x}
2249:{\displaystyle t}
2229:{\displaystyle x}
2063:{\displaystyle t}
2043:{\displaystyle x}
2023:{\displaystyle P}
1999:{\displaystyle x}
1979:{\displaystyle t}
1914:Where the symbol
1835:{\displaystyle x}
1811:{\displaystyle t}
1788:{\displaystyle x}
1705:{\displaystyle =}
1633:{\displaystyle x}
1577:{\displaystyle x}
1527:First-order logic
1386:{\displaystyle C}
1366:{\displaystyle B}
1346:{\displaystyle A}
981:universally valid
916:Erwin Schrödinger
904:Werner Heisenberg
896:Copenhagen school
885:pseudo-Riemannian
783:{\displaystyle s}
697:{\displaystyle l}
680:first introduced
650:Natural selection
571:Russell's paradox
455:primitive notions
259:notiones communes
123:non-logical axiom
5816:
5782:
5781:
5733:History of logic
5728:Category of sets
5621:Decision problem
5400:Ordinal analysis
5341:Sequent calculus
5239:Boolean algebras
5179:
5178:
5153:
5124:logical/constant
4878:
4864:
4787:Zermelo–Fraenkel
4538:Set operations:
4473:
4410:
4241:
4221:Löwenheim–Skolem
4108:Formal semantics
4060:
4053:
4046:
4037:
3998:
3992:
3984:
3968:John Cook Wilson
3940:
3937:
3931:
3928:
3922:
3919:
3913:
3912:
3911:
3909:
3893:
3887:
3886:
3885:
3883:
3867:
3861:
3858:Hilbert's axioms
3854:
3848:
3844:
3838:
3837:
3826:
3820:
3813:
3807:
3806:
3804:
3793:
3785:
3776:
3770:
3743:See for example
3741:
3735:
3732:
3723:
3716:
3710:
3697:
3691:
3690:
3667:
3657:
3651:
3644:
3627:
3624:
3618:
3615:
3558:Axiomatic system
3552:
3547:
3546:
3545:
3538:
3533:
3532:
3474:
3472:
3471:
3466:
3451:
3449:
3448:
3443:
3431:
3429:
3428:
3423:
3407:
3405:
3404:
3399:
3368:
3366:
3365:
3360:
3344:
3342:
3341:
3336:
3317:
3315:
3314:
3309:
3298:
3296: then
3295:
3284:
3281:
3269:
3267:
3266:
3261:
3241:
3239:
3238:
3233:
3203:
3201:
3200:
3195:
3183:
3181:
3180:
3175:
3148:
3069:
3067:
3066:
3061:
3045:
3043:
3042:
3037:
3025:
3023:
3022:
3017:
3015:
3003:
3001:
3000:
2995:
2978:
2967:
2966:
2949:
2947:
2946:
2941:
2929:
2927:
2926:
2921:
2919:
2918:
2910:
2909:
2895:
2893:
2892:
2887:
2786:
2784:
2783:
2778:
2717:
2715:
2714:
2709:
2663:
2661:
2660:
2655:
2643:
2641:
2640:
2635:
2623:
2621:
2620:
2615:
2595:
2594:
2586:
2585:
2500:abstract algebra
2441:complex analysis
2417:deductive system
2392:in mathematical
2343:
2341:
2340:
2335:
2319:
2314:
2295:
2293:
2292:
2287:
2275:
2273:
2272:
2267:
2255:
2253:
2252:
2247:
2235:
2233:
2232:
2227:
2215:
2213:
2212:
2207:
2205:
2204:
2191:
2189:
2188:
2183:
2172:Given a formula
2145:
2143:
2142:
2137:
2134:
2129:
2098:
2096:
2095:
2090:
2069:
2067:
2066:
2061:
2049:
2047:
2046:
2041:
2030:holds for every
2029:
2027:
2026:
2021:
2005:
2003:
2002:
1997:
1986:substituted for
1985:
1983:
1982:
1977:
1965:
1963:
1962:
1957:
1945:
1943:
1942:
1937:
1934:
1929:
1909:
1907:
1906:
1901:
1898:
1893:
1861:
1859:
1858:
1853:
1841:
1839:
1838:
1833:
1817:
1815:
1814:
1809:
1794:
1792:
1791:
1786:
1774:
1772:
1771:
1766:
1764:
1763:
1750:
1748:
1747:
1742:
1731:Given a formula
1711:
1709:
1708:
1703:
1691:
1689:
1688:
1683:
1665:
1663:
1662:
1657:
1639:
1637:
1636:
1631:
1612:
1610:
1609:
1604:
1583:
1581:
1580:
1575:
1559:
1557:
1556:
1551:
1549:
1548:
1502:
1500:
1499:
1494:
1434:
1432:
1431:
1426:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1352:
1350:
1349:
1344:
1323:
1321:
1320:
1315:
1263:
1261:
1260:
1255:
1176:
1174:
1173:
1168:
1131:
1129:
1128:
1123:
1107:
1105:
1104:
1099:
1083:
1081:
1080:
1075:
1063:
1061:
1060:
1055:
1043:
1041:
1040:
1035:
951:In the field of
912:hidden variables
878:
876:
875:
870:
868:
867:
855:
854:
842:
841:
829:
828:
819:
818:
806:
805:
789:
787:
786:
781:
769:
767:
766:
761:
759:
758:
746:
745:
733:
732:
720:
719:
703:
701:
700:
695:
594:Peano arithmetic
575:naïve set theory
459:Alessandro Padoa
210:
209:
200:
199:
186:
185:
86:
80:
67:, to serve as a
5824:
5823:
5819:
5818:
5817:
5815:
5814:
5813:
5794:
5793:
5792:
5787:
5776:
5769:
5714:Category theory
5704:Algebraic logic
5687:
5658:Lambda calculus
5596:Church encoding
5582:
5558:Truth predicate
5414:
5380:Complete theory
5303:
5172:
5168:
5164:
5159:
5151:
4871: and
4867:
4862:
4848:
4824:New Foundations
4792:axiom of choice
4775:
4737:Gödel numbering
4677: and
4669:
4573:
4458:
4408:
4389:
4338:Boolean algebra
4324:
4288:Equiconsistency
4253:Classical logic
4230:
4211:Halting problem
4199: and
4175: and
4163: and
4162:
4157:Theorems (
4152:
4069:
4064:
4005:
3985:
3966:
3949:
3947:Further reading
3944:
3943:
3938:
3934:
3929:
3925:
3920:
3916:
3907:
3905:
3895:
3894:
3890:
3881:
3879:
3869:
3868:
3864:
3855:
3851:
3845:
3841:
3828:
3827:
3823:
3814:
3810:
3802:
3791:
3787:
3786:
3779:
3759:10.2307/2274520
3744:
3742:
3738:
3733:
3726:
3717:
3713:
3698:
3694:
3684:
3659:
3658:
3654:
3645:
3641:
3636:
3631:
3630:
3625:
3621:
3616:
3612:
3607:
3568:First principle
3548:
3543:
3541:
3534:
3527:
3524:
3508:Boolean algebra
3492:
3454:
3453:
3434:
3433:
3414:
3413:
3390:
3389:
3351:
3350:
3327:
3326:
3319:
3275:
3274:
3252:
3251:
3206:
3205:
3186:
3185:
3166:
3165:
3155:
3150:
3146:
3112:
3076:
3052:
3051:
3028:
3027:
3006:
3005:
2956:
2955:
2932:
2931:
2903:
2898:
2897:
2790:
2789:
2721:
2720:
2673:
2672:
2646:
2645:
2626:
2625:
2579:
2574:
2573:
2547:
2496:homotopy theory
2492:homology theory
2425:
2358:natural numbers
2351:
2346:
2345:
2301:
2300:
2278:
2277:
2258:
2257:
2238:
2237:
2218:
2217:
2194:
2193:
2174:
2173:
2104:
2103:
2072:
2071:
2052:
2051:
2032:
2031:
2012:
2011:
1988:
1987:
1968:
1967:
1948:
1947:
1916:
1915:
1912:
1911:
1867:
1866:
1844:
1843:
1824:
1823:
1800:
1799:
1777:
1776:
1753:
1752:
1733:
1732:
1730:
1694:
1693:
1668:
1667:
1642:
1641:
1622:
1621:
1619:variable symbol
1615:
1614:
1589:
1588:
1566:
1565:
1538:
1537:
1535:
1529:
1437:
1436:
1399:
1398:
1375:
1374:
1355:
1354:
1335:
1334:
1267:
1266:
1180:
1179:
1141:
1140:
1114:
1113:
1090:
1089:
1066:
1065:
1046:
1045:
1026:
1025:
1018:
1013:
1001:predicate logic
977:formal language
969:
949:
859:
846:
833:
820:
810:
797:
792:
791:
772:
771:
750:
737:
724:
711:
706:
705:
686:
685:
678:Albert Einstein
629:
601:natural numbers
447:
400:interior angles
272:
267:
177:comes from the
171:
43:
28:
17:
12:
11:
5:
5822:
5820:
5812:
5811:
5806:
5796:
5795:
5789:
5788:
5774:
5771:
5770:
5768:
5767:
5762:
5757:
5752:
5747:
5746:
5745:
5735:
5730:
5725:
5716:
5711:
5706:
5701:
5699:Abstract logic
5695:
5693:
5689:
5688:
5686:
5685:
5680:
5678:Turing machine
5675:
5670:
5665:
5660:
5655:
5650:
5649:
5648:
5643:
5638:
5633:
5628:
5618:
5616:Computable set
5613:
5608:
5603:
5598:
5592:
5590:
5584:
5583:
5581:
5580:
5575:
5570:
5565:
5560:
5555:
5550:
5545:
5544:
5543:
5538:
5533:
5523:
5518:
5513:
5511:Satisfiability
5508:
5503:
5498:
5497:
5496:
5486:
5485:
5484:
5474:
5473:
5472:
5467:
5462:
5457:
5452:
5442:
5441:
5440:
5435:
5428:Interpretation
5424:
5422:
5416:
5415:
5413:
5412:
5407:
5402:
5397:
5392:
5382:
5377:
5376:
5375:
5374:
5373:
5363:
5358:
5348:
5343:
5338:
5333:
5328:
5323:
5317:
5315:
5309:
5308:
5305:
5304:
5302:
5301:
5293:
5292:
5291:
5290:
5285:
5284:
5283:
5278:
5273:
5253:
5252:
5251:
5249:minimal axioms
5246:
5235:
5234:
5233:
5222:
5221:
5220:
5215:
5210:
5205:
5200:
5195:
5182:
5180:
5161:
5160:
5158:
5157:
5156:
5155:
5143:
5138:
5137:
5136:
5131:
5126:
5121:
5111:
5106:
5101:
5096:
5095:
5094:
5089:
5079:
5078:
5077:
5072:
5067:
5062:
5052:
5047:
5046:
5045:
5040:
5035:
5025:
5024:
5023:
5018:
5013:
5008:
5003:
4998:
4988:
4983:
4978:
4973:
4972:
4971:
4966:
4961:
4956:
4946:
4941:
4939:Formation rule
4936:
4931:
4930:
4929:
4924:
4914:
4913:
4912:
4902:
4897:
4892:
4887:
4881:
4875:
4858:Formal systems
4854:
4853:
4850:
4849:
4847:
4846:
4841:
4836:
4831:
4826:
4821:
4816:
4811:
4806:
4801:
4800:
4799:
4794:
4783:
4781:
4777:
4776:
4774:
4773:
4772:
4771:
4761:
4756:
4755:
4754:
4747:Large cardinal
4744:
4739:
4734:
4729:
4724:
4710:
4709:
4708:
4703:
4698:
4683:
4681:
4671:
4670:
4668:
4667:
4666:
4665:
4660:
4655:
4645:
4640:
4635:
4630:
4625:
4620:
4615:
4610:
4605:
4600:
4595:
4590:
4584:
4582:
4575:
4574:
4572:
4571:
4570:
4569:
4564:
4559:
4554:
4549:
4544:
4536:
4535:
4534:
4529:
4519:
4514:
4512:Extensionality
4509:
4507:Ordinal number
4504:
4494:
4489:
4488:
4487:
4476:
4470:
4464:
4463:
4460:
4459:
4457:
4456:
4451:
4446:
4441:
4436:
4431:
4426:
4425:
4424:
4414:
4413:
4412:
4399:
4397:
4391:
4390:
4388:
4387:
4386:
4385:
4380:
4375:
4365:
4360:
4355:
4350:
4345:
4340:
4334:
4332:
4326:
4325:
4323:
4322:
4317:
4312:
4307:
4302:
4297:
4292:
4291:
4290:
4280:
4275:
4270:
4265:
4260:
4255:
4249:
4247:
4238:
4232:
4231:
4229:
4228:
4223:
4218:
4213:
4208:
4203:
4191:Cantor's
4189:
4184:
4179:
4169:
4167:
4154:
4153:
4151:
4150:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4089:
4088:
4077:
4075:
4071:
4070:
4065:
4063:
4062:
4055:
4048:
4040:
4034:
4033:
4025:
4015:
4004:
4003:External links
4001:
4000:
3999:
3964:
3948:
3945:
3942:
3941:
3932:
3923:
3914:
3888:
3862:
3856:For more, see
3849:
3839:
3821:
3808:
3777:
3753:(2): 481–511.
3736:
3724:
3711:
3692:
3682:
3652:
3638:
3637:
3635:
3632:
3629:
3628:
3619:
3609:
3608:
3606:
3603:
3602:
3601:
3596:
3594:Presupposition
3591:
3586:
3581:
3576:
3574:List of axioms
3571:
3565:
3560:
3554:
3553:
3539:
3523:
3520:
3516:modern algebra
3504:physical space
3502:as a model of
3496:mathematicians
3491:
3488:
3464:
3461:
3441:
3421:
3397:
3358:
3334:
3307:
3304:
3301:
3293:
3290:
3287:
3272:
3259:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3193:
3173:
3154:
3151:
3149:
3144:
3111:
3108:
3084:plane geometry
3075:
3072:
3059:
3035:
3014:
2993:
2990:
2987:
2984:
2981:
2977:
2973:
2970:
2965:
2952:
2951:
2939:
2917:
2914:
2908:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2787:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2718:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2666:unary function
2653:
2633:
2613:
2610:
2607:
2604:
2601:
2598:
2593:
2590:
2584:
2555:axiomatization
2546:
2543:
2527:ergodic theory
2523:measure theory
2424:
2421:
2350:
2347:
2333:
2329:
2326:
2323:
2318:
2313:
2309:
2298:
2285:
2265:
2245:
2225:
2203:
2181:
2167:
2133:
2128:
2124:
2120:
2117:
2114:
2111:
2088:
2085:
2082:
2079:
2059:
2039:
2019:
1995:
1975:
1966:with the term
1955:
1933:
1928:
1924:
1897:
1892:
1888:
1884:
1881:
1877:
1874:
1864:
1851:
1831:
1807:
1784:
1762:
1740:
1725:
1701:
1681:
1678:
1675:
1655:
1652:
1649:
1640:, the formula
1629:
1602:
1599:
1596:
1586:
1573:
1547:
1530:
1528:
1525:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1382:
1362:
1342:
1325:
1324:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1264:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1177:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1121:
1097:
1073:
1053:
1033:
1017:
1014:
1012:
1009:
1005:logical truths
968:
967:Logical axioms
965:
948:
945:
866:
862:
858:
853:
849:
845:
840:
836:
832:
827:
823:
817:
813:
809:
804:
800:
779:
757:
753:
749:
744:
740:
736:
731:
727:
723:
718:
714:
693:
628:
627:Other sciences
625:
586:Peano's axioms
467:Giuseppe Peano
446:
443:
442:
441:
440:
439:
436:
433:
430:
427:
423:
422:
420:
419:Common notions
414:
413:
412:
411:
392:
385:
378:
371:
363:
362:
271:
268:
266:
263:
244:remarks that "
224:mathematicians
195:from the verb
170:
167:
15:
13:
10:
9:
6:
4:
3:
2:
5821:
5810:
5807:
5805:
5802:
5801:
5799:
5786:
5785:
5780:
5772:
5766:
5763:
5761:
5758:
5756:
5753:
5751:
5748:
5744:
5741:
5740:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5696:
5694:
5690:
5684:
5681:
5679:
5676:
5674:
5673:Recursive set
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5623:
5622:
5619:
5617:
5614:
5612:
5609:
5607:
5604:
5602:
5599:
5597:
5594:
5593:
5591:
5589:
5585:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5542:
5539:
5537:
5534:
5532:
5529:
5528:
5527:
5524:
5522:
5519:
5517:
5514:
5512:
5509:
5507:
5504:
5502:
5499:
5495:
5492:
5491:
5490:
5487:
5483:
5482:of arithmetic
5480:
5479:
5478:
5475:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5453:
5451:
5448:
5447:
5446:
5443:
5439:
5436:
5434:
5431:
5430:
5429:
5426:
5425:
5423:
5421:
5417:
5411:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5390:
5389:from ZFC
5386:
5383:
5381:
5378:
5372:
5369:
5368:
5367:
5364:
5362:
5359:
5357:
5354:
5353:
5352:
5349:
5347:
5344:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5318:
5316:
5314:
5310:
5300:
5299:
5295:
5294:
5289:
5288:non-Euclidean
5286:
5282:
5279:
5277:
5274:
5272:
5271:
5267:
5266:
5264:
5261:
5260:
5258:
5254:
5250:
5247:
5245:
5242:
5241:
5240:
5236:
5232:
5229:
5228:
5227:
5223:
5219:
5216:
5214:
5211:
5209:
5206:
5204:
5201:
5199:
5196:
5194:
5191:
5190:
5188:
5184:
5183:
5181:
5176:
5170:
5165:Example
5162:
5154:
5149:
5148:
5147:
5144:
5142:
5139:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5116:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5093:
5090:
5088:
5085:
5084:
5083:
5080:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5057:
5056:
5053:
5051:
5048:
5044:
5041:
5039:
5036:
5034:
5031:
5030:
5029:
5026:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4993:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4951:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4932:
4928:
4925:
4923:
4922:by definition
4920:
4919:
4918:
4915:
4911:
4908:
4907:
4906:
4903:
4901:
4898:
4896:
4893:
4891:
4888:
4886:
4883:
4882:
4879:
4876:
4874:
4870:
4865:
4859:
4855:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4809:Kripke–Platek
4807:
4805:
4802:
4798:
4795:
4793:
4790:
4789:
4788:
4785:
4784:
4782:
4778:
4770:
4767:
4766:
4765:
4762:
4760:
4757:
4753:
4750:
4749:
4748:
4745:
4743:
4740:
4738:
4735:
4733:
4730:
4728:
4725:
4722:
4718:
4714:
4711:
4707:
4704:
4702:
4699:
4697:
4694:
4693:
4692:
4688:
4685:
4684:
4682:
4680:
4676:
4672:
4664:
4661:
4659:
4656:
4654:
4653:constructible
4651:
4650:
4649:
4646:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4585:
4583:
4581:
4576:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4539:
4537:
4533:
4530:
4528:
4525:
4524:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4499:
4495:
4493:
4490:
4486:
4483:
4482:
4481:
4478:
4477:
4474:
4471:
4469:
4465:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4423:
4420:
4419:
4418:
4415:
4411:
4406:
4405:
4404:
4401:
4400:
4398:
4396:
4392:
4384:
4381:
4379:
4376:
4374:
4371:
4370:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4335:
4333:
4331:
4330:Propositional
4327:
4321:
4318:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4289:
4286:
4285:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4258:Logical truth
4256:
4254:
4251:
4250:
4248:
4246:
4242:
4239:
4237:
4233:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4198:
4194:
4190:
4188:
4185:
4183:
4180:
4178:
4174:
4171:
4170:
4168:
4166:
4160:
4155:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4087:
4084:
4083:
4082:
4079:
4078:
4076:
4072:
4068:
4061:
4056:
4054:
4049:
4047:
4042:
4041:
4038:
4032:
4030:
4026:
4023:
4019:
4016:
4014:
4010:
4007:
4006:
4002:
3996:
3990:
3983:
3979:
3975:
3974:
3969:
3965:
3963:
3962:0-534-06624-0
3959:
3955:
3951:
3950:
3946:
3936:
3933:
3927:
3924:
3918:
3915:
3903:
3899:
3892:
3889:
3877:
3873:
3866:
3863:
3859:
3853:
3850:
3843:
3840:
3835:
3831:
3825:
3822:
3818:
3812:
3809:
3801:
3797:
3790:
3784:
3782:
3778:
3774:
3768:
3764:
3760:
3756:
3752:
3748:
3740:
3737:
3731:
3729:
3725:
3721:
3715:
3712:
3709:I.2.72a18-b4.
3708:
3707:
3702:
3696:
3693:
3689:
3685:
3683:9780199891535
3679:
3675:
3671:
3666:
3665:
3656:
3653:
3649:
3643:
3640:
3633:
3623:
3620:
3614:
3611:
3604:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3569:
3566:
3564:
3561:
3559:
3556:
3555:
3551:
3540:
3537:
3531:
3526:
3521:
3519:
3517:
3513:
3509:
3505:
3501:
3497:
3489:
3487:
3485:
3481:
3476:
3462:
3439:
3419:
3411:
3387:
3383:
3379:
3374:
3372:
3348:
3324:
3318:
3305:
3302:
3291:
3288:
3271:
3257:
3249:
3245:
3223:
3220:
3163:
3161:
3152:
3145:
3143:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3110:Real analysis
3109:
3107:
3105:
3101:
3097:
3093:
3089:
3085:
3081:
3073:
3071:
3057:
3049:
3033:
2988:
2985:
2982:
2979:
2968:
2937:
2915:
2912:
2880:
2874:
2871:
2868:
2850:
2847:
2841:
2832:
2826:
2819:
2816:
2810:
2804:
2798:
2788:
2771:
2768:
2765:
2759:
2756:
2753:
2750:
2747:
2741:
2738:
2732:
2729:
2719:
2702:
2699:
2696:
2693:
2684:
2681:
2671:
2670:
2669:
2667:
2651:
2631:
2608:
2605:
2602:
2596:
2591:
2588:
2570:
2568:
2564:
2563:number theory
2560:
2556:
2552:
2544:
2542:
2540:
2536:
2532:
2528:
2524:
2519:
2517:
2516:Galois theory
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2476:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2437:real analysis
2434:
2429:
2422:
2420:
2418:
2414:
2410:
2406:
2401:
2399:
2395:
2391:
2386:
2384:
2379:
2377:
2373:
2372:
2367:
2363:
2359:
2355:
2348:
2344:
2331:
2327:
2316:
2311:
2307:
2297:
2283:
2263:
2243:
2223:
2216:, a variable
2179:
2171:
2166:
2164:
2160:
2156:
2152:
2148:
2131:
2126:
2122:
2115:
2112:
2102:
2083:
2077:
2057:
2037:
2017:
2009:
1993:
1973:
1953:
1931:
1926:
1922:
1910:
1895:
1890:
1886:
1879:
1875:
1863:
1849:
1829:
1821:
1820:substitutable
1805:
1798:
1782:
1775:, a variable
1738:
1729:
1724:
1722:
1718:
1713:
1699:
1679:
1676:
1673:
1653:
1650:
1647:
1627:
1620:
1613:
1600:
1597:
1594:
1585:
1571:
1563:
1534:
1526:
1524:
1522:
1517:
1514:
1512:
1508:
1507:
1484:
1475:
1466:
1454:
1445:
1419:
1413:
1404:
1396:
1380:
1360:
1340:
1332:
1331:
1311:
1305:
1299:
1287:
1278:
1265:
1245:
1239:
1227:
1221:
1203:
1197:
1188:
1178:
1161:
1155:
1146:
1139:
1138:
1137:
1135:
1111:
1087:
1071:
1051:
1031:
1023:
1015:
1010:
1008:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
966:
964:
962:
958:
954:
946:
944:
941:
937:
933:
929:
925:
921:
917:
913:
909:
905:
901:
897:
892:
890:
886:
882:
864:
860:
856:
851:
847:
843:
838:
834:
830:
825:
821:
815:
811:
807:
802:
798:
777:
755:
751:
747:
742:
738:
734:
729:
725:
721:
716:
712:
691:
683:
679:
673:
671:
665:
663:
659:
655:
651:
647:
646:Mendel's laws
643:
639:
635:
634:Newton's laws
626:
624:
622:
618:
614:
610:
606:
602:
597:
595:
591:
587:
583:
578:
576:
572:
568:
565:
560:
558:
552:
550:
545:
540:
537:
535:
531:
527:
523:
519:
515:
510:
506:
503:
498:
496:
491:
487:
486:vector spaces
483:
479:
475:
470:
468:
464:
460:
456:
452:
444:
437:
434:
431:
428:
425:
424:
421:
418:
417:
416:
415:
409:
405:
401:
397:
393:
390:
386:
383:
379:
376:
372:
369:
368:straight line
365:
364:
360:
359:
358:
357:
356:
354:
353:
346:
344:
340:
334:
330:
328:
323:
321:
317:
313:
308:
306:
302:
298:
294:
290:
285:
281:
277:
269:
264:
262:
260:
256:
252:
247:
243:
238:
236:
232:
227:
225:
221:
218:
217:ancient Greek
214:
204:
194:
190:
180:
176:
168:
166:
164:
158:
156:
152:
146:
144:
140:
136:
132:
128:
124:
120:
119:logical axiom
116:
112:
107:
105:
101:
97:
93:
88:
85:
79:
74:
73:Ancient Greek
70:
66:
62:
58:
54:
50:
45:
41:
37:
33:
26:
22:
5775:
5573:Ultraproduct
5420:Model theory
5385:Independence
5321:Formal proof
5313:Proof theory
5296:
5269:
5226:real numbers
5198:second-order
5109:Substitution
4986:Metalanguage
4927:conservative
4900:Axiom schema
4844:Constructive
4814:Morse–Kelley
4780:Set theories
4759:Aleph number
4752:inaccessible
4658:Grothendieck
4542:intersection
4429:Higher-order
4417:Second-order
4363:Truth tables
4320:Venn diagram
4103:Formal proof
4080:
4028:
3971:
3953:
3935:
3926:
3917:
3906:, retrieved
3901:
3891:
3880:, retrieved
3875:
3865:
3852:
3842:
3833:
3830:Heath, T. L.
3824:
3816:
3811:
3795:
3750:
3746:
3739:
3719:
3714:
3704:
3700:
3695:
3687:
3663:
3655:
3647:
3642:
3622:
3613:
3584:Regulæ Juris
3579:Model theory
3493:
3483:
3479:
3477:
3409:
3385:
3381:
3375:
3346:
3322:
3320:
3273:
3247:
3243:
3158:
3156:
3123:
3116:real numbers
3113:
3077:
2953:
2571:
2554:
2551:Peano axioms
2548:
2520:
2504:group theory
2499:
2477:
2430:
2426:
2404:
2402:
2389:
2387:
2380:
2375:
2369:
2353:
2352:
2299:
2169:
2168:
2162:
2158:
2155:metatheorems
2154:
2150:
2146:
2100:
1913:
1865:
1727:
1726:
1720:
1717:axiom scheme
1714:
1616:
1587:
1532:
1531:
1518:
1515:
1511:modus ponens
1510:
1506:modus ponens
1504:
1330:axiom schema
1328:
1326:
1019:
992:
970:
960:
956:
950:
940:Alain Aspect
924:entanglement
893:
879:), and then
790:(defined as
704:(defined as
674:
666:
661:
657:
653:
630:
598:
582:Gödel showed
579:
561:
553:
541:
538:
511:
507:
499:
489:
478:group theory
474:field theory
471:
451:propositions
448:
389:right angles
375:line segment
351:
347:
342:
336:
332:
327:self-evident
324:
309:
296:
292:
273:
270:Early Greeks
258:
254:
239:
230:
228:
220:philosophers
212:
202:
188:
174:
172:
159:
147:
142:
138:
134:
130:
126:
114:
108:
90:The precise
89:
83:
56:
52:
48:
46:
44:
5683:Type theory
5631:undecidable
5563:Truth value
5450:equivalence
5129:non-logical
4742:Enumeration
4732:Isomorphism
4679:cardinality
4663:Von Neumann
4628:Ultrafilter
4593:Uncountable
4527:equivalence
4444:Quantifiers
4434:Fixed-point
4403:First-order
4283:Consistency
4268:Proposition
4245:Traditional
4216:Lindström's
4206:Compactness
4148:Type theory
4093:Cardinality
4031:axioms page
3120:isomorphism
2531:probability
2398:commutative
2371:tautologies
2236:and a term
1134:implication
997:tautologies
961:non-logical
928:EPR paradox
463:Mario Pieri
284:Tautologies
193:verbal noun
111:mathematics
5798:Categories
5494:elementary
5187:arithmetic
5055:Quantifier
5033:functional
4905:Expression
4623:Transitive
4567:identities
4552:complement
4485:hereditary
4468:Set theory
4022:PlanetMath
4013:PhilPapers
3908:19 October
3882:19 October
3815:Wolff, P.
3634:References
3386:consistent
3100:hyperbolic
2545:Arithmetic
2433:arithmetic
1088:are only "
989:assignment
920:David Bohm
900:Niels Bohr
658:postulates
654:principles
567:set theory
549:consistent
544:collection
488:) without
361:Postulates
339:hypotheses
276:syllogisms
133:) implies
117:may be a "
92:definition
57:assumption
5765:Supertask
5668:Recursion
5626:decidable
5460:saturated
5438:of models
5361:deductive
5356:axiomatic
5276:Hilbert's
5263:Euclidean
5244:canonical
5167:axiomatic
5099:Signature
5028:Predicate
4917:Extension
4839:Ackermann
4764:Operation
4643:Universal
4633:Recursive
4608:Singleton
4603:Inhabited
4588:Countable
4578:Types of
4562:power set
4532:partition
4449:Predicate
4395:Predicate
4310:Syllogism
4300:Soundness
4273:Inference
4263:Tautology
4165:paradoxes
3982:Q26720682
3599:Principle
3498:regarded
3463:ϕ
3460:¬
3440:ϕ
3420:ϕ
3396:Σ
3382:recursive
3357:Σ
3347:deduction
3333:Σ
3306:ϕ
3303:⊢
3300:Σ
3292:ϕ
3289:⊨
3286:Σ
3258:ϕ
3224:ϕ
3218:Γ
3192:Σ
3172:Λ
3160:deductive
2992:⟩
2972:⟨
2938:ϕ
2875:ϕ
2866:∀
2863:→
2842:ϕ
2839:→
2827:ϕ
2814:∀
2811:∧
2799:ϕ
2763:→
2736:∀
2727:∀
2688:¬
2679:∀
2415:define a
2403:Thus, an
2394:discourse
2376:postulate
2332:ϕ
2325:∃
2322:→
2308:ϕ
2284:ϕ
2180:ϕ
2151:metaproof
2123:ϕ
2119:→
2116:ϕ
2110:∀
2099:. Again,
2050:and that
1954:ϕ
1923:ϕ
1887:ϕ
1883:→
1880:ϕ
1873:∀
1850:ϕ
1739:ϕ
1482:¬
1479:→
1470:→
1461:→
1452:¬
1449:→
1417:→
1408:→
1306:ϕ
1303:→
1300:ψ
1294:→
1288:ψ
1285:¬
1282:→
1279:ϕ
1276:¬
1246:χ
1243:→
1240:ϕ
1234:→
1228:ψ
1225:→
1222:ϕ
1213:→
1204:χ
1201:→
1198:ψ
1192:→
1189:ϕ
1162:ϕ
1159:→
1156:ψ
1150:→
1147:ϕ
1120:→
1096:¬
1072:ψ
1052:χ
1032:ϕ
987:by every
985:satisfied
979:that are
932:John Bell
889:manifolds
857:−
844:−
831:−
670:falsified
590:corollary
404:intersect
350:Euclid's
343:postulate
301:Aristotle
297:postulate
231:postulate
173:The word
169:Etymology
61:statement
53:postulate
5750:Logicism
5743:timeline
5719:Concrete
5578:Validity
5548:T-schema
5541:Kripke's
5536:Tarski's
5531:semantic
5521:Strength
5470:submodel
5465:spectrum
5433:function
5281:Tarski's
5270:Elements
5257:geometry
5213:Robinson
5134:variable
5119:function
5092:spectrum
5082:Sentence
5038:variable
4981:Language
4934:Relation
4895:Automata
4885:Alphabet
4869:language
4723:-jection
4701:codomain
4687:Function
4648:Universe
4618:Infinite
4522:Relation
4305:Validity
4295:Argument
4193:theorem,
4029:Metamath
3989:citation
3978:Wikidata
3970:(1889),
3832:(1956).
3800:Archived
3522:See also
3410:complete
3282:if
3248:complete
3096:triangle
2930:formula
2896:for any
2423:Examples
2362:integers
2360:and the
2147:is valid
1818:that is
1110:negation
1011:Examples
993:at least
973:formulas
908:Max Born
605:infinite
564:Cantor's
526:Poincaré
482:topology
352:Elements
316:sciences
312:geometry
289:theorems
251:Boethius
121:" or a "
5692:Related
5489:Diagram
5387: (
5366:Hilbert
5351:Systems
5346:Theorem
5224:of the
5169:systems
4949:Formula
4944:Grammar
4860: (
4804:General
4517:Forcing
4502:Element
4422:Monadic
4197:paradox
4138:Theorem
4074:General
3773:realist
3767:2274520
3589:Theorem
3046:is the
2006:. (See
1397:, then
957:logical
613:forcing
530:Hilbert
522:Russell
255:petitio
246:Geminus
242:Proclus
203:axioein
198:ἀξιόειν
100:evident
69:premise
5455:finite
5218:Skolem
5171:
5146:Theory
5114:Symbol
5104:String
5087:atomic
4964:ground
4959:closed
4954:atomic
4910:ground
4873:syntax
4769:binary
4696:domain
4613:Finite
4378:finite
4236:Logics
4195:
4143:Theory
3980:
3960:
3771:for a
3765:
3680:
3512:Galois
3494:Early
3162:system
3130:. The
3092:angles
3004:where
2624:where
2537:, and
2514:, and
2512:fields
2390:axioms
2366:groups
1795:and a
1373:, and
1132:" for
1108:" for
1064:, and
662:axioms
532:, and
465:, and
408:angles
382:circle
305:Euclid
235:Euclid
189:axíōma
184:ἀξίωμα
84:axíōma
78:ἀξίωμα
34:,
5445:Model
5193:Peano
5050:Proof
4890:Arity
4819:Naive
4706:image
4638:Fuzzy
4598:Empty
4547:union
4492:Class
4133:Model
4123:Lemma
4081:Axiom
4018:Axiom
4009:Axiom
3803:(PDF)
3792:(PDF)
3775:view.
3763:JSTOR
3605:Notes
3563:Dogma
3094:of a
2664:is a
2508:rings
2451:like
2405:axiom
2159:proof
1560:be a
975:in a
617:Cohen
603:, an
534:Gödel
518:Frege
502:field
293:axiom
213:áxios
208:ἄξιος
191:), a
181:word
179:Greek
175:axiom
115:axiom
113:, an
104:logic
75:word
59:is a
55:, or
49:axiom
21:axion
5568:Type
5371:list
5175:list
5152:list
5141:Term
5075:rank
4969:open
4863:list
4675:Maps
4580:sets
4439:Free
4409:list
4159:list
4086:list
3995:link
3958:ISBN
3910:2019
3884:2019
3678:ISBN
3452:nor
3050:and
2549:The
2455:, a
2439:and
1822:for
1797:term
1536:Let
1435:and
1393:are
959:and
303:and
295:and
222:and
129:and
65:true
25:axon
5255:of
5237:of
5185:of
4717:Sur
4691:Map
4498:Ur-
4480:Set
4020:at
4011:at
3755:doi
3670:doi
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1020:In
898:' (
656:or
490:any
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5060:∃
5021:=
5016:↔
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5006:∧
5001:∨
4996:¬
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4715:/
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