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alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.

joel david hamkins is a proponent of multiverse interpretation of set theory, where we should see ZFC and other formalisations of the concept of sets as theories of not one universe of sets, but of a multiverse of sets, in which there are many “universes” of sets, .. or let’s call them aliverses.

the reason he gives is that since we can perfectly model – within ZFC – different theories ZFC, so one in which CH is true and one in which it is false, we know how such “places” look like, so to say that .. we are in only “one true universe” in which CH is either true or it is false and we just don’t know .. would be to disregard either of these “places” as unreal – even though we can readily visit them. instead, he regards these “places” as just aliverses of a multiverse of sets.

that position is known as a pluralist view and i find his arguement compelling. but that’s not quite the way we think of set theory, is it? of course, prof. hamkins would propose we adapt our way of thinking here, but then, when doing any mathematics founded on set theory, we would never be in “one definite” of these aliverses, but rather always in “some arbitrary” aliverse, make an argument in them and arrive at statements, which by the arbitrariness of the aliverse, holds in any of them. just as when we're doing group theory, we are never in one definite group, but rather awalys in some arbitrary group. in fact, just as we prepend “let $G$ be a group” before arguing in some arbitrary group, we should prepend “let $U$ be a set-theoretic universe fulfilling ZFC” before doing any mathematics founded on set-theory at all.

however, i still think of sets as belonging to one universe of sets, just as i think of natural numbers as belonging to one structure of natural numbers. there may be, in some abstract sense, multipile competing definite concepts of sets or natural numbers, but i can’t ever quite get to know them. rather i accept that the one concept of sets and the one concept of natural numbers that i have in mind are perhaps amorphous and vague. taking that seriously, i would have to say that the independency of CH from ZFC and then from ZF, which i regard to be a fitting formalisation of my amorphous and vague concept of sets, tells me that, with respect to my concept of sets, CH just simply is neither true nor false, that is to say: classical logic is not appropriate.

ok, so i’m probably not the first to think of this. but i haven’t seen this argument so far. so i have to wonder: am i missing something? doesn’t the independency phenomenon make a case for non-classical logics?

remark. i realize that the axiom of choice implies LEM within ZF from say intuitionistic logic. so i’m ready give up full choice in favor of dependent choice in order to drop LEM.

windfish
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  • No reason to break with the classical logic. "True" and "false" are properties depending on the model in which we look at a statement. Goedel showed that only the universal true statements can be proven within a consistent theory. In ZFC , in fact the continuum hypothesis is neither true nor false because the theory is too weak to decide that. But classical logic still holds. – Peter Mar 29 '24 at 10:22
  • I strongly disagree to omit the law of the excluded middle (as some mathematicians suggest). Without this law , we lose many interesting results. Just that we cannot determine the truth or falseness of some statements does not mean that the law of the excluded middle does not hold. – Peter Mar 29 '24 at 10:25
  • We can extend ZFC such that RH is true and we can extend ZFC such that RH is false. A more powerful theory can decide RH. – Peter Mar 29 '24 at 10:27
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    @Peter “model” is a set-theoretic notion. not presupposing set-theory, i would replace that notion with that of an aliverse. i would say that classical logic holds in any of the aliverses. my argument is that i don’t have any specific aliverse in mind when doing set-theory. i also don't think that mathematics is about making statements about the set-theoretic multiverse, that is of the totality of all aliverses. one can do that, sure. that is to say: investigate all these aliverses out there. we have done that and i don't want to dismiss any of the mathematics. … – windfish Mar 29 '24 at 10:29
  • @Peter … it’s rather that what we’re doing here doesn’t correspond to the way we think of sets, i argue. at least that’s the case for me. as i said, if i’m honest, i would say that i do have one concept of sets in mind, but it’s indefinite. i can make statements about all the different ways of making that concept definite. that would be classical mathematics, but at some point i wouldn't make any statements of the one indefinite concept that i have in mind. and it seems you agree: if you say that the "continuum hypothesis is neither true nor false", that's a non-classical statement. – windfish Mar 29 '24 at 10:34
  • I said : "In ZFC , the continuum hypothesis is neither true nor false". This is an important restriction. Goedel destroyed the dream of provability of every true statement. Whether the continuum hypothesis is true depends which extension of ZFC we use. Or we restrict us to constructable sets. In this case, both the axiom of choice and the continuum hypothesis become provable theorems. – Peter Mar 29 '24 at 11:27
  • @Peter and i agree with “in ZFC, the continuum hypothesis is neither true nor false” and conclude from this “in ZFC, classical logic is inappropriate.” because we have found a statement which is neither true nor false, violating the law of the excluded middle. and whenever i do mathematics, i don’t use any extension of ZFC at all. restricting to constructible sets to my understanding means working in a model of ZFC within ZFC. but then the question remains what we are describing with the “outer” ZFC and whether classical logical is appropriate for it. – windfish Mar 29 '24 at 12:24
  • The bit "[...] just as we prepend “let G be a group” before arguing in some arbitrary group, we should prepend “let U be a set-theoretic universe fulfilling ZFC” before doing any mathematics founded on set-theory at all." is weird: people do this to/with groups/etc because they do not work only and directly with/in the first-order theory of groups, but rather look at a group 'from outside', consider subobjects and other constructions with higher types, etc, while when using a set theory for 'ordinary mathematics', one presumably works only in the theory – ac15 Mar 29 '24 at 12:29
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    also, there's no violation of excluded middle: "$CH \lor \neg CH$" is obviously provable – ac15 Mar 29 '24 at 12:31
  • @ac15 only if you assume law of excluded middle to begin with, i reckon? – windfish Mar 29 '24 at 12:31
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    yes, of course; the point is only that saying "because we have found a statement which is neither true nor false, violating the law of the excluded middle." is not adequate – ac15 Mar 29 '24 at 12:33
  • @ac15 let me rephrase that: assume classical logic. we have all the classical results about logic that we know about. i still think that we formalise one concept of sets with ZFC, so we are within one universe of sets. within that universe. that concept of set may or may not be specific, that is: it doesn't precisely clarify the nature of sets. by adding axioms to ZFC we can perhaps specify that concept a bit. the independence phenomenon now shows us that indeed that any concept formalised by ZFC is not specific: we can add both CH and its negation to ZFC to specify our concept a bit. – windfish Mar 29 '24 at 12:46
  • @ac15 so i don’t regard CH as true because we can specify our concept of sets in such a way that CH is false and i don’t regard CH as false because we can specify our concept of sets in such a way that CH is true. regarding to that unspecific concept of sets formalised by ZFC, we therefore have CH is neither true nor false. i therefore regard classical logic to be inapproriate – even though we may have used classical logic itself to arrive at that independency phenomenon; its application to the concept of sets formalised by ZFC is self-defeating. – windfish Mar 29 '24 at 12:48
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    "i still think that we formalise one concept of sets with ZFC, so we are within one universe of sets." this is not the case: syntax doesn't 'know' any of this – ac15 Mar 29 '24 at 12:49
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    Removing axioms doesn't let you prove more things. Independence is a stronger phenomenon than you think. – Asaf Karagila Mar 29 '24 at 13:51
  • @AsafKaragila not sure what you mean. i don’t want to “settle” CH or anything by dismissing of LEM if that’s what you’re thinking. – windfish Mar 29 '24 at 13:58
  • @ac15 sorry, that’s been unclear. i meant to say: “we intend to formalise one concept of sets”, that is: we start off with an informal notion of what a set is upon which we seem to agree and then try to formalise that notion. i think that’s also the historical viewpoint of the mathematicians that came up with and tried to formalise set theory. we can either give up on that programme (the pluralist view) or try to adhere to it (the monist view). i try to adhere to it by acknowledging that the concept of set i try to formalise is amorphous and vague and just doesn’t abide by classical logic. – windfish Mar 29 '24 at 14:01
  • @AsafKaragila oh, i also don’t want to get rid of independence by dismissing LEM. i interpret independent statements as being neither true nor false, therefore violating LEM, forcing me to dismiss it. i accept the phenomenon of independence in set theories under non-classical logics, as it doesn’t pose any problems for me there. – windfish Mar 29 '24 at 14:22
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    "i interpret independent statements as being neither true nor false, therefore violating LEM, forcing me to dismiss it": you're still mixing up syntax and semantics – ac15 Mar 29 '24 at 15:16
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    My point is that even in constructive logic, provability is not the same as truth value. – Asaf Karagila Mar 29 '24 at 15:22
  • @windfish: The proof of the relative independence of $\text{CH}$ from $\text{ZFC}$ is based in classical logic, $\text{LEM}$ included. It is inconsistent of you to reject $\text{LEM}$ yet accept the independence proof of $\text{CH}$. – Chad K Mar 29 '24 at 18:13
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    @ChadK That's just not true; Heyting arithmetic, for example, can prove the statement "If $\mathsf{ZFC}$ is consistent then it neither proves nor disproves $\mathsf{CH}$." – Noah Schweber Mar 29 '24 at 19:53
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    BTW I disagree with the vote to close. – Noah Schweber Mar 29 '24 at 19:54

1 Answers1

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As noted in the comments, you are - or at least are dangerously close to - mixing up syntax and semantics. That said, I think there is a way to make your feelings precise, and I at least sometimes share them; since I can't find an exact duplicate of this question (although similar things have been asked before, e.g. 1), I'll jot this approach down here.

I'll call this position ZFC-finalism. The idea is that, while we informally work in a universe of sets, our real stance is that the ZFC axioms and only the ZFC axioms are a priori justifiable.

  • A weaker version of this stance is that the mathematical community (or at least the set theoretic community) will never "canonize" any further axioms, independence phenomena and various proposals notwithstanding; an even weaker version adds "unless we reject ZFC-style foundations as a whole," and this much weaker version I think is likely true.

I personally think that ZFC-finalism is perfectly coherent, even if I don't share it usually. I suspect that it captures your own stance at least sometimes. The point is that there is a well-understood gadget set up to handle "ZFC-finalist truth values," namely the Lindenbaum algebra of $\mathsf{ZFC}$. This is the Boolean algebra whose elements are equivalence classes of sentences modulo $\mathsf{ZFC}$-provable-equivalence.

If we accept ZFC-finalism, then - I claim - the right logical shift is not really towards nonclassical logic but rather towards a Boolean algebra of truth values. In particular:

  • By definition, the truth value of $\mathsf{CH}$ is $[\mathsf{CH}]$ (where "$[\cdot]$" denotes the equivalence class in the Lindenbaum algebra).

  • Highly nontrivially, we have $[\perp]\not=[\mathsf{CH}]\not=[\top]$.

Set theorists of wildly different stripes can all agree with the points above (well, assuming they grant $\mathit{Con}(\mathsf{ZFC})$ for the second one); the only point of disagreement is how we deploy the phrase "truth value." Getting back to your original question, since the laws of Boolean algebra are exactly those of classical logic in a precise sense (i.e. every Boolean algebra has the same equational theory as the two-element Boolean algebra) in my opinion the philosophical takeaway from the preceding is that incompleteness does not in any serious way push us towards nonclassical logic. But of course mileage will vary on this point, and much better mathematicians than me will disagree with me (and with each other!).

Noah Schweber
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  • this is a really good answer, thanks! i don’t think that i mix up syntax with semantics, at least i don’t see how i do that. to me, the notion of a “truth value” represented by elements of some boolean algebra only makes sense within a formalised notion of sets, unless we are being circular, just like the notion of a “model” for a theory. (i admit that to this day it confuses me when people do that when discussing foundational matters. i suspect that i have misunderstood something very elementary and fundamental.) – windfish Mar 29 '24 at 21:43
  • I don't understand your "ZFC-finalism". If you believe ZFC is justifiable, then you must also believe that ZFC+Con(ZFC) is justifiable. You must believe that ZFC+(ZFC is arithmetically sound) is justifiable too. There cannot be a final point at which you can stop. – user21820 Mar 30 '24 at 10:37
  • @windfish: My above point notwithstanding, I hope you actually understand the general incompleteness theorems, to know that incompleteness has absolutely nothing to do with classical logic. – user21820 Mar 30 '24 at 10:40
  • @user21820 i can’t say that i understand the general incompleteness theorems, but i know and understand that the incompleteness phenomenon arises with arithmetic theories over non-classical logic as well and i have mentally reviewed a general outline of a proof of the completeness theorem with intuitionistic logic both as a base for formal derivability and as a base for meta-theoretic reasoning. in any case, i know that incompleteness doesn't rely on classical logic. and to my understanding, the independence phenomenon is yet again something somewhat different. – windfish Mar 30 '24 at 13:05
  • @user21820 i’m not quite sure that i subscribe to being a “ZFC finalist” as Noah put it, but he’s right that it mostly fits the bill. so speaking for me, you’re right in that i think that ZFC+Con(ZFC) is at least justifiable insofar as i assume the consistency of ZFC. but there’s a reason that i wouldn’t want to incorporate it into the theory: because Con(ZFC) is an “encoded” statement, it introduces some statement about sets that is not really about sets. instead, its incorporation introduces some set-theoretic artifacts. i therefore think it’s not actually justified. – windfish Mar 30 '24 at 15:14
  • @user21820 i also see no reason to incorporate it if i’m not really interested in model theory. i’m also very content with statements in ZFC of the form “Con(ZFC) → p” and i don’t think i would be limited in my set-theoretic constructions by doing so and forgoing Con(ZFC). perhaps the role of axioms is to provide us with (unprovable) root statements on which we rest the theory we want to explore and which we use all the time, perhaps implicitly so. i don’t regard Con(ZFC) to be such a statement. – windfish Mar 30 '24 at 15:16
  • @windfish: Con(ZFC) is not an encoded statement in any reasonable sense unless you say that even basic facts like ∀k,m∈ℕ ( k·m = m·k ) is also encoded when working in ZFC. Con(ZFC) does not introduce any set-theoretic artifacts; it is simply a concrete statement about all finite strings, that is computably checkable for each string. If you don't like finite strings of symbols, finite sequences from ℕ. Not even Gödel-coding is needed if you can use basic notions about finite strings or about finite sequences from ℕ. And all this has absolutely nothing to do with set theory. – user21820 Mar 31 '24 at 05:41
  • @windfish: It is fine if you want to work within ZFC but not ZFC+Con(ZFC), but there is no sense in which ZFC could be considered final that doesn't force you to expand it. There is no need to be overly defensive of ZFC; this fact applies to almost any foundational system and not just ZFC... – user21820 Mar 31 '24 at 05:42
  • @user21820 yes, of course i would say that $∀k,m ∈ ℕ (k·m = m·k)$ is an encoded statement. it is supposed to say something about natural numbers, but natural numbers are not a priori conceptionally sets. natural numbers are merely encoded as sets in ZFC and there are many ways in which we could do that and implement a set of natural numbers in ZFC. if that statement about natural numbers wasn’t provable from ZFC, i would say that adding it to ZFC as an axiom would most certainly lead to set-theoretic artifacts. – windfish Mar 31 '24 at 07:57
  • @user21820 another thing is that fully unpacking such a statement in the first-order language of set-theory (that is using only strings containing $∀, ∃, ∧, ∨, →, ¬, =, ∈$ and perhaps $⊂$ alongside variable names) would not only be cumbersome (as in the case of most of the other axioms of ZFC), but downright a hellish nightmare (and i would expect few if any people to be able to interpret it without being told how to do so). if you would do that, it would become obvious that such a statement doesn't say anything our concept of sets per se, but of something else that is encoded using sets. – windfish Mar 31 '24 at 08:02
  • @windfish: I think you have significant conceptual flaw in your understanding of logic and set theory. Let ZFC' be ZFC but with 4 sorts set, ℕ, ℕ×ℕ,ℕ→ℕ (where the last can be used to very directly represent sequences from ℕ) and function-symbol pair for ordered pairing plus axioms of PA for ℕ plus ℕ,ℕ×ℕ,ℕ→ℕ∈set plus ℕ→ℕ⊆ℕ×ℕ plus ∀k,m∈ℕ ( pair(k,m)∈ℕ×ℕ ) ∧ ∀i,j,km∈ℕ ( pair(i,j) = pair(k,m) ⇒ i=k ∧ j=m ) plus ∀p∈ℕ×ℕ ∃k,m∈ℕ ( pair(k,m) = p ) plus ∀f∈ℕ→ℕ ∀k∈ℕ ∃!m∈ℕ ( pair(k,m)∈f ). [cont] – user21820 Mar 31 '24 at 12:27
  • [cont] ZFC' can easily and clearly express Con(ZFC), because a proof is a sequence of finite strings, which can be clearly represented as a member of ℕ→ℕ by just shifting and delimiting (e.g. ⟨⟨3,1,4,1,5⟩,⟨⟩,⟨9,2,6⟩⟩ ↦ ⟨4,2,5,2,6,0,0,10,3,7,0⟩), and it is trivial to express basic notions like substring. Thus Con(ZFC) becomes a very clear statement about members of ℕ→ℕ that has absolutely nothing to do with set theory. [cont] – user21820 Mar 31 '24 at 12:30
  • [cont] And there are 2 programs TX and TP that any competent logician programmer can write down, that translate every sentence and proof respectively from ZFC' to ZFC (with only logical symbols and ∈), meaning that, for any strings p,x, if p is a ZFC'-proof of x then TP(p) is a ZFC-proof of TX(x). This fact is totally transparent to anyone who understands FOL and knows the axioms of ZFC and ZFC'. So there is an absolutely concrete basis for saying that ZFC' is no stronger than ZFC. [cont] – user21820 Mar 31 '24 at 12:30
  • [cont] Now we had Con(ZFC) being a clear non-set-theoretic sentence over ZFC', and it has a clear translation TX(Con(ZFC)) over ZFC. So ZFC+TX(Con(ZFC)) is an extension of ZFC that has no set-theoretic artifact beyond ZFC but that you are forced to accept if you accept ZFC. – user21820 Mar 31 '24 at 12:30
  • @user21820 i follow you and i of course understand that ZFC and ZFC' are equivalent theories. my objection would then be that ZFC' is no longer a “pure” set theory, but a theory of sets and natural numbers as it contains explicit axioms for PA. maybe this is odd to you, but i want an axiomatization of set theory to only contain axioms for the behavior of sets and nothing else. yes, this is only for moral reasons, but we are talking morality when speaking of “justifiable”. if you say to me “you are forced to accept ZFC+Con(ZFC)”, i say “no”. i don’t feel compelled to, it's not a theory of sets. – windfish Mar 31 '24 at 12:48
  • @windfish: That's a really odd position. Here is an orthogonal objection along the lines of my earlier comment. Since you want a theory that has only set-theoretic notions (which is weird if you don't want set-theoretic artifacts), let's at least include ∅ and the term-constructor {•,•} and also binary union/intersection and cartesian product (which are only about sets), and the axiom ∃I ( ∅∈I ∧ ∀x∈I ( x⋃{x}∈I ) ) and construct ω = { k : k∈I ∧ ∀J ( ∅∈J ∧ ∀x∈J ( x⋃{x}∈J ) ⇒ k∈J ) ) }. [cont] – user21820 Mar 31 '24 at 14:36
  • [cont] And now we can easily write down Con(ZFC) as a statement about ω that we can check for every actual term we can construct using {•,•} and prove is in ω. So this Con(ZFC) is now a sentence over a "pure set theory", which we can mechanically check for every term that is really in ω. Con(ZFC) is no longer susceptible to any artifacts unless you believe that something is wrong with the finite strings we can write down using only "{" and "}"! – user21820 Mar 31 '24 at 14:38
  • @user21820 alright, this resolves the point i made to illustrate my objection, but not the objection itself. the main point is: the consistency of ZFC is a statement about a formal theory of sets, not about sets. all the other axioms tell meː this is how sets behave and what you can do with sets. the one about the consistency doesn’t. i’m sorry for also having brought up the rather technical point of the consistency looking abstruse. that was only meant to illustrate that there’s an inherent difference. changing that amounts to cosmetics, but doesn’t change the moral. – windfish Mar 31 '24 at 15:54