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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 – acb1516 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 – acb1516 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 – acb1516 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 – acb1516 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 – acb1516 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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  • Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Xander Henderson Apr 28 '24 at 13:14
  • in reading your answer once more, it just occured to me that the term "law of excluded middle" or "tertium non datur" is a huge misnomer and its explanations "every proposition is true or false" or even "every propisition is true or has a true negation" are not quite accurate. clearly, both of these statements are violated by the line $[\bot] ≠ [\mathsf{CH}] ≠ [\top]$ in that lindenbaum semantics of truth. it seems to me that the "law of excluded middle" is really just saying "for every proposition $p$, we have $p ∨ ¬ p$, whatever that means". – windfish Jun 21 '24 at 12:17