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Mathematical facts are necessary truths, either in a Platonic sense or by way of axioms. In the latter sense I mean that the Peano Axioms prove that 2+3=5, for example.

In other words, "PA ⊨ 2+3=5" is a necessary truth.

But what about mathematical statements such that they and their negation are both consistent? For example, is the Axiom of Choice a necessary truth (or necessarily impossible) or a contingent truth or something else?

Is it perhaps appropriate to say something like, "For every possible world w, such that the Axiom of Choice is true, there is another world w* which is exactly like w in every possible way except that the Axiom of Choice is false (and of course you lose what you can't prove without AC)".

Or is the entire question "is AC a necessary or contingent truth?" a metaphysical non-sequitur?

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    I believe this question heavily depends on the ontological status of infinite objects like e. g. the collection of all natural numbers. – მამუკა ჯიბლაძე May 6 '18 at 6:37
  • It is a reasonable interpretation; see Multiverse (set theory). – Mauro ALLEGRANZA May 6 '18 at 8:56
  • For the mathematical issues regarding AC you can see The Axiom of Choice. – Mauro ALLEGRANZA May 6 '18 at 10:08
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    @user4894 I don't think that I said "2+3 = 5" is a necessary truth unless you are a committed Platonist. Of course, it does seem everyone agrees that "PA ⊨ 2+3=5" is a necessary truth. I'm not sure if you meant to say that "2+2=5" or if that was a typo. – Squirtle May 7 '18 at 3:48
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    First, to clear the terminology, ontology is about what is, not what can be, the question is about the modal status of AC. Second, there are multiple notions of possibility, grouped under logical, metaphysical, physical, etc. If we pick the weakest, logical, possibility then AC is contingent, alternative set theories are logically possible. Metaphysical possibility only makes sense if one is a platonist. Since mathematics is kept fixed across metaphysical variations AC may be necessary or impossible, but we do not know which. It depends on whether it "metaphysically" holds in our actual world. – Conifold May 8 '18 at 3:24
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How one answers such questions obviously depends on ones philosophical views.

A realist in truth-value such as Quine or Putnam, will argue that AC has an objective truth value independent of the language, mind, or mathematician reflecting on the question.

On the other hand, a non-realist in truth-value will argue that AC is independent of set theory and therefore has no objective truth value.

More generally, the view of axioms as self-evident truths is one that is not in favour with mathematicians today. Contemporary mathematics now view axioms as "defining conditions" for a theory. For example, a modern set theorist is happy to study both ZF (set theory) with AC and ZF with ¬AC. Obviously one cannot view both AC and ¬AC as self-evident truths.

  • In modern times many sorts of perversion have spread. That is no reason to support them – Wilhelm May 6 '18 at 18:34
  • @Wilhelm Truer words have rarely been spoken! However, with regard to axioms, if you find set theory to be problematic, then consider geometry. Both Euclidean geometry with its five postulates, and non-Euclidean geometry with the negation of the parallel postulate - both of these geometries find wide ranging application. – Nick R May 6 '18 at 19:32
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    @ Nick R: Geometry in different spaces will obey different rules. As an introductory example for newbies I always use the geometry in the plane and on the surface of a sphere. In Euclidean space Euclid's axioms are true. But to apply an axiom that allows to "prove" that elements that by no means can be distinguished can be well-ordered is too much of perversion. It is not only by accident, that attempts have been made to use uncountable alphabets and infinite words, i.e., to use every real number as a letter, and further there are "proofs" contd – Wilhelm May 6 '18 at 20:01
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    There are "proofs" that all reals can be defined: "Is it consistent with the axioms of set theory that every real is definable in the language of set theory without parameters? The answer is Yes. Indeed, much more is true" (J.D. Hamkins). These points show that set theorists feel uncomfortable with the facts: Real numbers are ideas that have no existence unless they can be defind as individuals. – Wilhelm May 6 '18 at 20:04
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An approach to make sense of the necessary vs contingent truth distinction is by considering a theory $T$ and a model $M$ of that theory. Theorems of $T$ are the necessary truths in $T$. Statements that hold in $M$ but are not theories of $T$ are the contingent truths.

If we follow this approach, then the question of whether AC (if true) is a necessary or contingent truth essentially depends on the question what our chosen foundational theory is. We could consider all models of ZF as potential mathematical worlds, and just explore one where AC is true right now. But we could just as well start with ZFC as theory, and thus deem AC to be necessary.

Ultimately, this question does not really lead us anywhere, I believe.

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    "We could consider all models of ZF as potential mathematical worlds ..." What is special about ZF? It's consistent to deny the axiom of infinity. It's consistent to deny the axiom of powersets. Both of the resulting systems are studied and are the subject of serious papers you could look up. At the end of the day, ZF is historically contingent, dating from either 1904 or 1922 depending on how picky you want to be about Zermelo's papers. – user4894 May 6 '18 at 21:06
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    @user4894 ZF is the typical context in which the axiom of choice is discussed, nothing less, nothing more. – Arno May 6 '18 at 21:52
  • @Arno Of course ZF is the typical context in which AC comes up. But that's not what you wrote originally. You said "We could consider all models of ZF as potential mathematical worlds ...," which is simply not true. There are potential mathematical worlds far weaker than ZF and far stranger. – user4894 May 6 '18 at 22:12
  • @user4894 I am not saying that we have to use only ZF models als possible mathematical worlds. I'm not even saying that we should do that. I am saying we could do it, if it suits our purpose at any given moment. Obviously, if one wants to study very weak set theories as metatheories, it is a bad idea. – Arno May 6 '18 at 22:57
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The axiom of choice has gotten its bad reputation because it leads to contradictions with uncountable sets. But it is a very natural axiom and is frequently applied without notice in mathematics as Zermelo has correctly pointed out when defending his invention against objections of Borel, Peano, Poincaré, and others. [E. Zermelo: "Neuer Beweis für die Möglichkeit einer Wohlordnung", Math. Ann. 65 (1908) pp. 107-128]

The problem is only, as mentioned above, the application of the axiom to uncountable sets. In the history of mathematics, it was usual to fix factual conventions by an axiom, like "it is possible to draw a straight line from any point to any point" or "if n is a natural number, then n+1 is a natural number". The application of the axiom of choice claims for the first time a counterfactual convention, namely to choose an element without knowing what is chosen.

At least in 1904 it was clear that there are only countably many finite strings of letters, including strings defining mathematical objects. Cantor knew this theorem, as he wrote in a letter to Hilbert in 1906, although he did not believe that it is true. "If König's theorem was true, according to which all 'finitely definable' real numbers form a set of cardinality aleph_0, this would imply that the whole continuum was countable, and that is certainly false." [G. Cantor, letter to D. Hilbert (8 Aug 1906)]

Today there is no doubt that König's theorem is true. In order to maintain transfinite set theory, it is necessary to have the (in this realm) counterfactual axiom of choice for proving the basic theorem of set theory: Every set can be well-ordered. Otherwise a lot of ordinal theory would be unprovable. Therefore set theorists have agreed that the axiom is "not constructive", i.e., we can prove that we can choose every element, but we cannot choose every element. Although Zermelo used the axiom to prove that every set can be well-ordered, i.e., he thought it could be done and not only be proven that it could be done, knowing that in fact it cannot be done. [E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904)]

But what is the value of a counterfactual axiom? We could state many other axioms of same value like:

  • Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)

  • Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)

  • Axiom of prime number triples: There is a second triple of prime numbers, besides (3, 5, 7). (But provably this second triple is not arithmetically definable.)

  • Axiom of meagre sum: There is a set of n different positive natural numbers with sum n*n/2. (This axiom is not constructive. Provably no such set can be constructed.)

All theories based upon such axioms would have the same value as transfinite set theory, namely none.

Keeping this in mind and ignoring absurd attempts to apply uncountable alphabets or infinite definitions to define uncountably many elements, we can be sure that the axiom of choice is true in every world with correct mathematics and therefore without uncountable sets.

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    it leads to contradictions with uncountable sets. Source? Uncountable sets are part of ZF, so if there were a real contradiction here then AC would not be independent of ZF. This is a big discovery on your part! – Canyon May 6 '18 at 14:38
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    @Wilhelm: you clearly have a theoretical ax to grind here. Given that this site isn't the place for independent philosophizing, and that you give no signal that yours is a minority view, -1. Plus I think you're wrong. – Canyon May 6 '18 at 16:23
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    @Canyon: Only a small minority of intelligent humans supports your view in particular and finished infinity in general. So in fact it's you who belongs to a minority. Look: Atoms and photons exist independently of anyone knowing them. Ideas however can exist only if someone can think them. Real numbers are ideas. Undefinable "real" numbers are nonsense. Well-ordering undefinable ideas is the summit of nonsense. – Wilhelm May 6 '18 at 18:14
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    @Arno: Reasonable thought is usually called cranky in set theory because there are no arguments to refute them. But be sure that you belong to an absolute minority. The majority refuses finished infinity: "Set theory is wrong [L. Wittgenstein] The ordinary diagonal Verfahren ... I find it difficult to understand how such a situation should have been capable of persisting in mathematics. [P.W. Bridgman, Nobel laureate] ... demand that this disease, for which we are not responsible, be quarantined and kept out of our field [E.T. Jaynes] It is a Paradise of Fools [D. Zeilberger]. – Wilhelm May 6 '18 at 18:24
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    Contd: Cantor's second number class does not exist. [L.E.J. Brouwer]. I never met a more decided opponent of the Cantorian ideas (than Hermite) [Henri Poincaré]. There is no actual infinity. The Cantorians forgot this, and so have fallen into contradiction. [Henri Poincaré]. the idea of the totality of real numbers is no longer indispensable, and the axiom of choice is not at all evident." [P. Bernays]. For sources and context see hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf – Wilhelm May 6 '18 at 18:31

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