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Let S be the structure/language of ZFC (including PL 1). Let CH refer to the well-known continuum hypothesis. My claims are as follows and could u just say if it's true or wrong and why?

  1. In S neither CH is true nor false because in S only tautologies and contradictions are already true/false and CH is not such.

  2. Let's say I assume only one axiom in S that says: there exists an empty set. Now in this scenario CH again is neither true nor false because we still can't talk about cardinalities of sets at all (and so we can't talk about CH).

  3. Let's say I assume ZFC in S. Now we can talk about cardinalities of sets. That means that here CH is a wff and so it is either true or false. But we cannot prove which one it is (Gödel, Cohen). But it means: CH is true xor false in ZFC in this very moment, we just don't know and we just will never know!

  4. If we just brutally assume CH to be true in ZFC (ZFC + CH), then there's no inconsistency (proof by Gödel), but if we take ZFC + ~CH we can prove there's no inconsistency either (Cohen), so ZFC is - loosely spoken - too general to catch CH's truth/falseness properly, just like a fisherman's net is sometimes too big to catch certain fishes.

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    Assuming ZFC is consistent, there exist models of ZFC where CH is true and models where CH is false. By Gödel's completeness theorem, this follows from the fact that CH is independent of ZFC. – Daniel Hast Nov 27 '20 at 21:17

1 Answers1

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There are several issues here, which may not feel important at first but over time will cloud the (already quite nuanced) picture.

First of all, you're conflating structures, theories, and languages. In increasing order of complexity:

  • A language (also called a signature or vocabulary) is a set of non-logical symbols, such as $\{\in\}$ or $\{+,\times,0,1,<\}$.

  • A theory is a set of first-order sentences, and for a language $\Sigma$ a $\Sigma$-theory is a theory consisting of sentences in the language $\Sigma$ - e.g. $\mathsf{ZFC}$ is an $\{\in\}$-theory and first-order $\mathsf{PA}$ is an $\{+,\times,0,1,<\}$-theory.

  • A structure in a given language is a set together with an interpretation of the various symbols in that language in a precise sense.

Whether or not a particular string of symbols is a wff depends only on the language involved, not on what axioms we're considering nor on what structure (if any) we're specifically focusing on. $\mathsf{CH}$ is a wff in the language $\{\in\}$. What the empty $\{\in\}$-theory (your "$S$") can't do is prove basic things about $\mathsf{CH}$ and related sentences. So $S$ can talk about $\mathsf{CH}$, it just doesn't have much to say. This issue is implicit in $(1)$ and $(2)$, and explicit in $(3)$.

Now on to the more subtle point: truth and falsity. The satisfaction relation $\models$ connects structures and sentences/theories, with "$\mathcal{A}\models\varphi$" (resp. "$\mathcal{A}\models\Gamma$") being read as "$\varphi$ is true in $\mathcal{A}$" (resp. "Every sentence in $\Gamma$ is true in $\mathcal{A}$"). But we use the term "true" only in this context; when talking about theories, the relevant term is provable.

The main reason for reserving terms like "true" and "false" for structures as opposed to theories is that the standard properties of truth such as bivalence only hold of truth-in-a-structure, not provability-in-a-theory. By separating the terms we make it easier to be precise and avoid subtle errors. This is an issue in your point $(3)$, where truth and provability get mixed up. In particular, the statement

CH is true xor false in ZFC in this very moment, we just don't know and we just will never know

doesn't parse.

OK, unfortunately you will find people say that things are true/false in $\mathsf{ZFC}$. The connection is that a sentence is provable in a theory $T$ iff it is true in all models of $T$, so this isn't totally unjustified. But this is an abuse of terminology, and should be avoided until the fundamentals of the topic are mastered.

After shifting from truth to provability, point $(4)$ then is correct with one slight additional hypothesis: assuming $\mathsf{ZFC}$ is consistent in the first place, both $\mathsf{ZFC+CH}$ and $\mathsf{ZFC+\neg CH}$ are consistent.

Noah Schweber
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  • I see. So if S is the structure on which ZFC is built then is CH true/false there (in S!)? I'd say: yes, since it's a wff, so it MUST be true/false in S, we just don't know which and therefore we create proof systems (you call it: theories). Am I right (else my next question doesn't make sense)? But doesn't that mean that e.g. ZFC+CH is a very risky theory/axiom-system since CH could be easily false in S (the same holds for ZFC+~CH)? –  Nov 29 '20 at 21:55
  • @Pippen What is "$\color{red}{\mbox{the}}$ structure on which ZFC is built"? There are many models of ZFC, some in which CH is true and others in which CH is false. – Noah Schweber Nov 29 '20 at 22:12
  • I begin to understand. I'll make a simple example to understand better. Let {+,=} be the language L. Let S be the structure {IN, +, =} where we interpret everything in the usual way. Then "1+1=2" is a true statement in S. Let S' be the structure {IN,+,=} where we interpret any number x as x+1 (so that 1+1=2 means actually 2+2=3), but else we interpret all like in S. Then 1+1=2 is false in S'. So the same statement "1+1=2" can be true and false, depending from the structures resulting from L. In the same way CH can be true and false in different structures on which ZFC is built. –  Nov 30 '20 at 07:09
  • @Pippen Yes, that's exactly right. I do still have a couple comments though. First, "on which ZFC is built" is an odd phrase: what you want to say is "which satisfy ZFC." Think of ZFC as a condition, or list of conditions, which a structure may or may not satisfy. Neither "builds" the other, but we can ask whether a given structure satisfies ZFC (or any other theory). Second, we have to be careful when we ask whether 1+1=2 in your various structures: 1 and 2 are elements of the structure, but we don't have constants naming them in the language. So "$1+1=2$" is a sentence with parameters. – Noah Schweber Nov 30 '20 at 07:12
  • This is fine, and your observation is exactly right, but it is important to keep track of when parameters are used or not. Here's a parameter-free example: $S$ satisfies the sentence $$\exists x\forall y(x+y=y)$$ (take $x=0$) but $S'$ does not. (For me, $0\in\mathbb{N}$.) – Noah Schweber Nov 30 '20 at 07:15
  • "which satisfy ZFC" means the same as "which make ZFC true statements" right? Also: Can I use the following picture: ZFC is like a fisher-net but to catch (prove) statements instead of fish. But ZFC's net is too raw so that CH and ~CH can both slip thru and ZFC can't catch (prove) them. To catch either CH or ~CH we have to make ZFC's net finer, e.g. by using CH or ~CH as axioms because then we can catch (prove) them. –  Nov 30 '20 at 07:29
  • Re: your first question, yes, but I recommend staying away from the word "true" at first. It sounds too absolute, whereas here what we want to emphasize is that each structure has its own notion of truth. (Granted, if we're Platonists we also posit a "genuine universe of mathematics," with a distinguished notion of truth - but that's sort of a separate issue.) Re: the metaphor, it oversimplifies a bit by suggesting that there's a specific "aspect" of a sentence which makes it $\mathsf{ZFC}$-undecidable ("size"), whereas things can be undecidable for very different reasons, but it's fine. – Noah Schweber Nov 30 '20 at 07:32
  • @NoahSchweber "Each structure has its own notion of truth." Would you please elaborate this? I'm self studying logic and struggling with that point. In my naive mind, "truth" of any mathematical statement is decided by nonlogical axioms and "proof" in "natural language" and this should be equivalent to a proof in formal langauge with formal axioms. I can't understand how the truth of a statement is decided in a specific structure's own way. I asked a question (https://math.stackexchange.com/q/4172852/674190) and got some answers but failed to understand the point. – MathSmurf Nov 07 '22 at 14:18