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The original title of this question was supposed to be "Do sets exist?", but it was too short. In philosophy of mathematics we sometimes ask whether mathematical objects exist. I think this question will be solved if we can show that sets exist, because all mathematical objects can be constructed out of sets. So, then, do sets exist?

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  • I am not at all a mathematician, but it seems to me that the power of set theory is its fundamental and intuitive appeal. It is not difficult for most people to think of "sets" of things. So in part the question becomes are "thoughts" real? More directly, the set bracketing, like all mathematics, raises the ancient issue of whether "reality" is continuous or discrete. As far as I know, this is not and probably never will be a settled matter in physics, let alone philosophy. Discrete bits seem the "least thinkable" units. But their infinite possible content suggest a more basic continuity. Nov 20 '20 at 5:54
  • Yes, in the "mathematical universe" we have sets and spaces and numbers. Nov 20 '20 at 7:04
  • "this question will be solved if we can show that sets exist, because all mathematical objects can be constructed out of sets." But from a "philosophical" point of view, the assertion that numbers exist is much more reasonable that the assertion that sets do. Nov 20 '20 at 9:30
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    Does this answer your question? What is mathematical existence?
    – Conifold
    Nov 20 '20 at 12:53
  • Please be aware that questions and answers are subject to editing and closure, and that reflects the site's policies on acceptable questions and NOT a personal attack. What to avoid in questions. Anything closed can be edited to bring it within guidelines. Keeping questions on-topic. Additional clarification at MetaPhil. Is a variant of 'what is mathematical existence'.
    – J D
    Nov 20 '20 at 15:31
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People frequently bring up "indispensability arguments", but the most popular of those is completely fallacious. For instance, it claims that "reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories". This is plainly false. It is well-known to all real experts in logic that very weak subsystems of second-order arithmetic suffice for all empirically backed real-world applications of mathematics to date. For reference look up Reverse Mathematics.

In particular, our best scientific theories only rely on mathematics that can be carried out in such weak systems, apparently not needing reasoning about anything beyond arithmetical sets, and this can be handled easily by ACA0 or at most ACA (i.e. ACA0 plus full induction). So the popular notion that set theory is indispensable to science is just bogus.

There is no doubt that ZFC is an elegant set theory that is capable of supporting practically all modern mathematics, but that fact does not imply anything about its real-world relevance. For all we know ZFC may one day be found to prove itself inconsistent, which would not at all matter to reality. On the other hand, logicians have no doubt that ACA will never be found to be like that.

This shows that your question is not well-defined, because "set" is ill-defined until you specify a set theory and provide some kind of real-world or ontological interpretation, which you have not done. In fact, ACA can be interpreted to have nothing to do with sets at all. On the surface, ACA is about ℕ and its arithmetical subsets. However, that is mere appearance. We could use the obvious interpretation of ℕ in terms of finite binary encodings in some specific physical medium with the appropriate operations on them, and we could interpret those 'subsets' as simply arithmetical formulae with one free variable. For any such set S = { x : x∈ℕ ∧ Q(x) }, the truth-value of "k∈S" for k∈ℕ is simply the truth-value of "Q(k)", which is already well-defined once you believe the meaningfulness of PA.

Furthermore, your question includes a wrong premise, namely the claim that all mathematical objects can be constructed from sets. In the first place, that claim is nebulous. What does "construct" mean? Even if you allow an object to be 'constructed' via any well-ordered sequence of steps, and assuming that there is an empty set ∅ and one can construct any set whose elements have previously been constructed, it is still impossible to construct all objects from ∅ unless every object is a well-founded set. But that assumes the Regularity axiom, yet even the mathematician who came up with it did not believe it was true...

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  • So are you predicting in the next 50 years most people will use ACA instead of ZFC in most math areas, or do you think there's some specific area in math which ZFC would still dominate? May 20 at 20:21
  • The answer to your question is already evident from the history of ZFC as well as my post. ZFC was invented to support modern mathematics and set theory, not just mathematics with real-world applications. Obviously, set theorists will still use ZFC, but no real-world application ever needs Replacement, Regularity or Choice. And just because people will continue to use ZFC does not mean it is philosophically justifiable. Remember that Frege did do lots of correct mathematics within his inconsistent system...
    – user21820
    May 20 at 20:34
  • @user21820, the Law of Large Numbers is one of those mathematical generalisations that very much depends on being a corollary of the more contentious ZFC axioms. I don’t think it’s a simple matter of just taking ACA and induction when it comes to establishing statements about ensembles. Oct 18 at 7:40
  • @SofieSelnes: False. LLN can be proven in a very weak system. This paper states that Weak LLN can be proven in RCA0, which is the weakest of the Big 5, and seems to imply that Strong LLN can be proven in RCA0+WWKL, which is weaker than WKL0, which is weaker than ACA0. Furthermore, all ordinary mathematics including measure theory can be done in bounded ZC set theory, which is much weaker than ZC, not to say ZFC.
    – user21820
    Oct 18 at 9:32
  • @user21820, even at this level though there’s still contention that something fundamentally non-constructive is at work. Konig’s lemma and it’s variations are often understood as “choice” axioms - in as much as there is controversy in ZFC, this is where a lot of the sparks are! Oct 19 at 21:16
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Indispensability arguments for the existence of mathematical entities can be used to argue for the existence not just of finite elementary sets, but infinite elementary ones as well as complex finite and infinite examples. Roughly, the idea is that mathematics quantifies over sets; we have reason to believe that what our best empirical theories quantify over exists; mathematics is indispensable in formulating our best empirical theories; therefore, we have reason to believe that sets (or numbers or what) exist. Since our best theories of mathematics ultimately involve various infinite sets, presto, the existence of these is (albeit indirectly) implicated in our best science, wherefore...

IIRC this argument originated with Quine, whose "new foundations" include a universal set, wherefore even that extravagant structure might find purchase in our web of belief as such...

Now from a Platonic perspective, it would be easier to "decide" whether sets exist: just consult your Platonic memory of the world of Forms, and if you remember Forms of sets, well... I speak rather flippantly, here, though. Put a better way, the point would be that, "Do sets exist?" has an abstract meaning, and a concrete one, such that the question of V can evolve into a question of a multi-V, and then if V might not be counted as its own concrete world, after all, among many no less?

The structuralist, on the other hand, will want to say that sets only exist wherefrom structures. Then whether and where these structures exist, decides whether and where the sets (usually numbers, here, though) do.

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  • Just some side note: Indispensability arguments for the existence of mathematical entities proposed by Quine and Putnam was argued against by Putnam's own student Hartry Field (en.wikipedia.org/wiki/Philosophy_of_mathematics#Fictionalism). Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real, there is no reason to treat parts of mathematics that involve reference to or quantification as true. Math is just by our conventions, similar but more radical to Poincare's conventionalism... Apr 19 at 21:48
  • @Double knot: Given that Field showed the 'dispensibility' of mathematics by 'giving a complete axiomatisation of Newtons laws' through Hilberts axioms of synthetic geometry, I don't think he showed the 'dispensibility' of mathematics at all. Unless of course you think synthetic geometry is not a part of mathematics at all ... Oct 19 at 7:29
  • @MoziburUllah in Field's Science without Numbers, he tried not to use arithmetic numbers in synthetic geometry instead of focus on betweeness, congruence, etc, though I agree with you his definition of math/number is very confusing. His main thesis there in his Principle C which states math has no causal contribution to physics, only as a medium of metaphorical (false) explanation, only physical entities are true. Sets definitely is remote there since even arithmetic systems like PA, ACA0 are only metaphors. However, ironically later Field's interest moved to study set theory mainly... Oct 19 at 19:01
  • @Double Knot: Well, those are part of Hilbert's axioms for synthetic geometry. Fields book is subtitled as A defence of nominalism. This simply means that mathematics has no actual existence and hence no 'causal' influence. Fields also states that maths remains useful and hence has truth content, even if ultimately the abstract entities it talks about have no real existence. They are not 'falsehoods'. The falsehood is that mathematical entities - according to Fields and garden variety Nominalism - they have no ontological weight. Of course, if you are a Platonist, things work out ... Oct 22 at 0:00
  • @Double Knot: ... very differently. Oct 22 at 0:00
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Sets are a subcategory of something simpler: rules. The empty set is a thing that obeys the rule that nothing is contained in it by reference. So, to what extent do rules 'exist'?

They clearly do not exist as physical objects. There is no physical object that keeps you from killing people. If you write down the rule that one should not kill people, that does not make the rule any more real than it already was. Nor are they actual states of nature. People do in fact kill other people all the time. Physical rules take a different form. They only allow predictions, you cannot really break them. If you find that one has an exception, it really already had that exception, you just did not know that. Nothing is broken.

But in an ideal world, one does not kill people. Except that we do not all accept that this 'ideal' world is truly ideal. We can disagree that war is necessarily always bad, because we like some of its consequences, like the existence of our country, and the end of oppressive regimes. So for some people, the ideal world actually involves some killing.

In an equally ideal world, there is a set with no elements. We can imagine violating this rule, just shoving something in there out of spite. But that puts us outside that ideal world. But this ideal world is so devoid of real references that we cannot disagree about it. We can imagine different versions of it, but they are not worth fighting over. Unlike 'shouldness', this world has only 'wouldness'. If we could create it, we believe we know how it would behave. But we can't create it, except in a very peculiar sense where we all seem to know exactly what it is like, and that it isn't real.

These are the three Hermetic modes: Fixed physical realities, Cardinal moral realities, and Mutable thought-only realities, or alternately, the three worlds of Lacan: the Real, the Symbolic, and the Ideal. We can create amalgams of these: Fictional realities which are thought-only in actually, but physical in interpretation. Religious realities, which are made up of documents and institutions as part of the physical world, but have primarily symbolic moral force. Game realities, which are purely Cardinal rules one should obey, and does not have to, but played out in approximation with symbolic pieces...

To say something 'exists' means something very different in the worlds of these different kinds. If someone just injects a new rule in the middle of a game, you can argue over whether it 'exists' by looking at the history of the game, and whether they might actually have been taught to play it that way, and feel that behavior is the correct one. If someone finds a new moral principle, you can question whether it really exists in the sense that they actually believe that it is real and good for the world, rather than just being a political manipulation meant to control the ignorant.

So it is not sophistry to ask the 'Bill Clinton' question. We need an inventory of what the different definitions of 'is' is -- one for each 'mode'. In the purely 'Mutable', 'Ideal' world of mathematics, there is no question that the rule of containment is a thing, and therefore the empty set exists. But in the 'Fixed', 'Real' world, we know equally well that it cannot -- because this ideal notion of containment is a figment of language with no physical embodiment.

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