Continuum Hypothesis: Truth and Provability in FOL

[WWGD]

Hi,
Just to test my understanding. Is it the case that the Continuum Hypothesis( CH) is not considered true within ZFC because there are both models/interpretations of ZFC where CH holds , as well as models/interpretations of ZFC where it doesn't, whereas truth of a ( statement?)in FOL requires that the statement hold in all models/interpretations, here referring to interpretations of ZFC?
Thanks.
Edit,
Maybe
@stevendaryl Can chime in?

 

[andrewkirk]

I think the inferred meaning of terms like 'True' in relation to a sentence in a formal language will vary depending on which semantic theory we are using, and even on which textbook we are referring to.
Most semantics I've seen use the term True in relation to a specific interpretation. That is, they say a sentence S is True in interpretation R.

Given a theory T (a set of well-formed sentences closed under deduction) in language L, we call an interpretation of L a model of theory T if it assigns True to every sentence in T.

ZFC is a theory in the language of first order predicate logic (FOPL). The CH can be expressed as a well-formed sentence of FOPL but is not part of the theory ZFC, as it cannot be deduced from ZFC's axioms.

I see that wikipedia (I know!) suggests that we call a sentence S in a formal language L logically valid with respect to a theory T if every model of T in L assigns the value True to S, and consistent if at least one model of T in L assigns the value True to S; otherwise T is inconsistent. The status of being logically valid, consistent or inconsistent depends on all three of sentence S, theory T and language L.

Using that terminology, we'd say the sentence CH of language FOPL is consistent wrt theory ZFC but not logically valid.

 

[WWGD]

Thanks, Andrew, I believe too, there was a result linking truth in the above sense, with provability, right?

 

[pbuk]

Using that terminology, we'd say the sentence CH of language FOPL is consistent wrt theory ZFC but not logically valid.

You can of course say exactly the same about the sentence "Not CH".

 

[pbuk]

Thanks, Andrew, I believe too, there was a result linking truth in the above sense, with provability, right?

No, a sentence can be true but unprovable (incompleteness theorem).

 

[WWGD]

No, a sentence can be true but unprovable (incompleteness theorem).

Yes, thanks, I understand they're not equivalent, but I thought there were some results connecting the two, IIRC, soundness was one. Somehow I can't find straight answers from a search.
Edit: Well, here's something on soundness, from Wikipedia:

 

[SSequence]

In what follows consistency is assumed through-out (to avoid adding a qualification to every sentence). At the very first level the following can be thought of as an answer to the question:
https://www.physicsforums.com/threa...xiom-or-a-theorem.1053712/page-2#post-6913946
I will quote the relevant part here:

If we have an incomplete theory under consideration [as often happens to be the case] then (assuming consistency) every statement exactly falls into one of the following three categories:
(i) provable in the theory (or a theorem of the theory)
(ii) disproveable in the theory
(iii) independent

And once again the statements in category-(ii) that would be disproveable (in the theory) would have their negation as a theorem [theorem of the theory that is].

So in that sense CH just falls into category-(iii).==============================Here is a somewhat more detailed way of looking at it (and also sort of a heuristic way of thinking about it). But it is still quite basic of course and lacks depth and detailed justification for (too many) finer points (which I don't know either). It is more of a heuristic description as a way of looking at it. However, lacking detailed understanding, I still found it fairly useful.

Essentially we think of "world of sets" as given to us [or perhaps in other words "collection of all sets"]. This is denoted by ##V## (also called cumulative hierarchy). It can be thought of as the power-set operation running through the ordinals. We set ##V_{0}## as the empty set. Next we set ##V_{\alpha+1}=\mathcal{P}(V_{\alpha})##. For limit ordinals ##\alpha##, ##V_{\alpha}## is defined as the union of all lower levels. The following two points are quite basic but are worth mentioning because of their importance:
(a) For every set ##A##, there would exist some (smallest) ordinal ##\alpha## such that ##A \in V_{\alpha}##.
(b) For all ordinals ##\alpha##, ##V_{\alpha}## will be a set. However, ##V## itself is not a set.Now when we talk about models, there are two kinds of "models":
(i) set models
(ii) class models

A more complete description is well-beyond my own understanding/knowledge. However, knowing a few points about models is fairly useful (to get a very rough picture):
(a) A set model is, as the name implies, just a set.
(b) A class model can be thought of as a "collection" in some sense. However, they aren't sets. That's because class models pick elements from ##V_{\alpha}## with ##\alpha## taking arbitrarily large values in ##\mathrm{Ord}##. In an informal sense, they are too big to be sets.
(c) Normally texts often just "write" model with the context of whether a "set model" or "class model" is being talked about as understood from the context.
(d) There is a certain (precise) sense which makes qualifies a "set/class model" as a "model". Informally it is said that a model satisfies all the axioms of ZF(C) [and I think the sense is probably slightly different for set models and class models]. However, to be honest, I don't really know what that means in a more precise sense. I had quite vague sense of it few years back, but I have forgotten it. If you look up for it you might be able to find some descriptions at least.

Nevertheless, if you picked up any specific set from ##V##, it will either be a (set) model or it won't.

(e) Quite importantly, the "world of sets" ##V## itself is a class model.
Sorry this got a bit long. But now coming to the question, what does this have to do with CH? You already kind of described it in your question. Basically every statement/question that can posed in ZF(C) [and basically CH is one of them] has either a true or false value in a specific model [be it a set model or a class model]. Now this is how it relates to your initial question. When we think about any specific ##V##, the value of CH will be either true or false in the given ##V##. There will be no two ways about it.

For example, suppose that our ##V## is what is called "constructible universe" [note that word "universe" is a just a mathematical usage]. Then CH will be true in such a ##V##. In fact, GCH is also true in such a ##V##.

However, it is also possible to have a ##V## where ##CH## is false. But it is perfectly possible that we could have class models [that, very informally, select a sub-collection of the elements in ##V## using something like a additional axiom I think] for such a ##V## where ##CH## could be true.

And that is just about the limit of the depth of my understanding for the specific question at hand :P.