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In "Professor Stewart's Incredible Numbers," by Prof. Ian Stewart, it is claimed that $\mathfrak{c}$, the number of real numbers, is bigger than $\aleph_0$, the number of natural numbers, but

"[h]ow much bigger is moot: it depends on which axiom system you use to formalise mathematics".

Would someone explain this to me, please? I couldn't find anything on it.

Shaun
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    Maybe he is alluding to the Continuum hypothesis. – Mauro ALLEGRANZA Sep 19 '17 at 13:22
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    It's consistent, for example,that $2^{\aleph_0}=\aleph_{201832957}$. – Noah Schweber Sep 19 '17 at 13:22
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    See Easton's theorem, which shows that there are remarkably few restrictions on the relationship between the cardinality of a set and the cardinality of its power set. – Arthur Sep 19 '17 at 13:23
  • @Stella: Model theory?!? Where? – Asaf Karagila Sep 19 '17 at 13:59
  • @Stella: I don't know any model theoretic methods which give a proof of independence of CH from ZFC. – Asaf Karagila Sep 19 '17 at 14:02
  • @AsafKaragila 1) I didn't say that there was a model theoretic independence proof, I said that I felt like to sufficiently answer the question you need to talk about model theory and that I did so and that the linked post has none of those details. 2) Unramified forcing explicitly uses models and was something that I learned in my first grad model theory course 3) forcing is equivalent to the method of Boolean-valued models which therefore can prove independence and is explicitly about model theory – Stella Biderman Sep 19 '17 at 14:16
  • @Stella: Wow, I envy you for having a model theory course that starts with forcing. That is just awesome! But using models does not make something model theoretic. That is exactly my argument against people who put a set theory tag on any question which involves sets, or number theoretic tags on questions which involve "numbers". Ultimately, the question is about the statement "how much larger is the continuum from $\aleph_0$ depends on your assumptions". The other question explains exactly that this is it. If you want to suggest another duplicate, I am open to hear your suggestions. – Asaf Karagila Sep 19 '17 at 14:20
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    @AsafKaragila hmmm okay. On rereading the question I realize that I had misread it... I thought it said "construct mathematics" not "formalize mathematics" which is why I started talking about model theory in the first place. I agree with you now. – Stella Biderman Sep 19 '17 at 14:27

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The following assumes that ZFC is consistent.

As mentioned in the comments, there are a number of remarkable theorems that show that $\mathfrak{c}$ can take on a huge swath of cardinal values and still be consistent with $ZFC$.

There are a number of axioms that do put real restrictions on the size of $\mathfrak{c}$, but they're not (almost) universally used the way that $ZFC$ is.

I also feel that it is an under appreciated fact that in specifying a particular collection of objects that satisfy the axioms of set theory, you actually decide what aleph number $\mathfrak{c}$ is, along with every other question that's independent of $ZFC$ for the model in question. These constructions are called models, and $ZFC$ has many different models that have different properties. For example, there are models in which $\mathfrak{c}=\aleph_{111}$ and models in which it is $\aleph_1$. For any statement, $S$, if $S$ is consistent with $ZFC$ then there is a model of $ZFC$ in which $S$ is true.

Most of mathematics takes place in the realm where all of these different models agree. This is precisely the realm of questions that can be proven true or false with $ZFC$. However, it's not uncommon for questions to crop up that don't fall within this realm. Most mathematicians are currently happy to say that that's a question that doesn't have an answer, though others are pushing for the acceptance of more axioms. Accepting more axioms (notably large cardinal axioms) rules out any model on which those axioms fail to hold as places where you can do "the right kind of mathematics" (there is mathematics done in models that don't conform to $ZFC$ but it's a very very niche field) and so increases the set on which all "acceptable" models (meaning ones that agree with our axioms) agree.

We can ask the philosophical question "is there one true model of $ZFC$ that we should be using," but I think most mathematicians (and definitely most philosophers of science) would say no.

But yes, it is a factual statement to say "a particular construction of the set theoretic universe answers the question of the size of $\mathfrak{c}$" and it is a factual statement to say "the size of $\mathfrak{c}$ is not settled by a set of axioms accepted by the majority of the mathematics community" as most people are agnostic on these stronger axioms I have mentioned. Prof Stewart's statement seems like an immediate corollary of these two assertions.

Stella Biderman
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