How can different models of set theory be constructed from the same set of axioms?

Question

A beginner question here, sorry if it seems obvious.

From my understanding, a set theory like ZFC strictly dictates what can and cannot exist, e.g. the Axiom of Infinity implies the existence of an infinite set, while the Axiom of Foundation prohibits the existence of non-well-founded sets.

Then how can there be different models of the same set theory, based on the same set of axioms?

If a model contains set A, then ZFC must allow the existence of A, so why don't all the other models also contain A?

Edit: Do all models need something like axioms of their own? E.g. ZFC + CH is a model? But then how does a model differ from just a stronger theory?

Answer 1

Then how can there be different models of the same set theory, based on the same set of axioms?

This turns out to be a very general feature of models of first-order theories, due to the compactness theorem and the upward and downward Löwenheim-Skolem theorems. The upward and downward Löwenheim-Skolem theorems in particular guarantee that if ZFC has a model (which is necessarily infinite) then it has a model of every infinite cardinality (including a countable model, funnily enough; this is Skolem's paradox), so ZFC can't even pin down how big its models are.

If a model contains set A, then ZFC must allow the existence of A, so why don't all the other models also contain A?

This is a pretty subtle question as written, because it's not clear that given a set in a model of ZFC, you can make sense of what it would mean for that set to also exist or not exist in some other model of ZFC. That is: what does it mean for a set in a model of ZFC to be "the same" as a set in a different model of ZFC? I am not a set theorist but it's not clear a priori that this makes sense in general.

It's also not entirely clear what you mean by "why" but the simplest thing to say is that there is simply a difference between "it is consistent with ZFC that (some stuff happens)" and "ZFC requires (some stuff to happen)," and by the completeness theorem this is precisely equivalent to the difference between a statement being true in some model vs. it being true in every model. ZFC simply has different models in which different statements are true (by the incompleteness theorem) and that's just something we all have to live with.

Edit: Do all models need something like axioms of their own? E.g. ZFC + CH is a model? But then how does a model differ from just a stronger theory?

Associated to every model $M$ of ZFC is its theory $\text{Th}(M)$, which is the set of all statements true in $M$. This is always a complete theory containing ZFC. Unfortunately, this means that (by the incompleteness theorem) none of these theories can be computably axiomatizable. So in a very strong sense we can't write any of them down. ZFC, for its faults, is at least something we can write down!

Answer 2

It's helpful to think of the axioms of ZFC as being exactly like the axioms for groups. It just happens that the ZFC axioms are more complicated. Then, different "models" of the ZFC axioms are exactly like different "models" for the group axioms! That is, different groups!

Then just like group axioms can posit the existence of certain elements that must exist in every group (the identity, for instance), the ZFC axioms posit the existence of certain sets that must exist in every model of ZFC (the empty set, for instance).

Similarly, just like two groups can look very different, even though they're both "models of group axioms", two models of ZFC can look very different! In this lens, the continuum hypothesis is analogous to abelian-ness. You can ask "do the group axioms imply that $ab=ba$?" just like you can ask "do the ZFC axioms imply $\aleph_1 = 2^{\aleph_0}$?". In both cases, the answer is "this is independent". The axioms don't decide the question one way or the other. In groups, this is because there exist both abelian and nonabelian groups, and in ZFC this is because there exist both "CH models" and $\lnot$CH models".

When you realize that set theorists study "models of ZFC" in the same way that group theorist study "models of the group axioms", I think the whole situation becomes more clear. Part of the reason people don't understand this comes from the fact that ZFC is a rich enough theory that it can "talk about itself" whereas the theory of groups can't (This is a very slight fib, but at least the theory of groups can't easily talk about itself). Another reason this gets complicated is that we can look at "toy models" of the theory of groups. Things like cyclic groups, symmetric groups, etc. are small models of the group axioms which we can play with to learn how groups work. But unfortunately there are no toy models of the theory of ZFC. So it's much harder to get a handle on how they work.

Hopefully this all helps explain why set theorists ask the kinds of questions they ask as well. When someone proves "PFA implies $2^{\aleph_0} = \aleph_2$", they're proving that every ZFC-model that satisfies PFA satisfies $2^{\aleph_0} = \aleph_2$. This is entirely analogous to a group theorist proving that, say, "every 2-step nilpotent group is also 2-step solvable".

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I hope this helps ^_^

Answer 3

The theory of mammals includes several axioms:

• Female mammals produce milk
• Mammals are warm-blooded
• Mammals have hair

There are many creatures that satisfy these axioms and are therefore mammals. These include the giraffe, the elephant, and the tree shrew. We can call these models for the theory of mammals.

We can ask questions like Do all mammals have hair?. Answer: yes, because this follows (trivially) from the axioms. There might be other questions where the answer is yes but not quite so trivially. For example: Are all female mammals warm-blooded? Answer: yes, because every female mammal is a mammal. Et cetera.

There are other questions for which the answer is certainly "No", like Are there any cold-blooded mammals?

Those are questions for which the answer follows logically from the axioms. But there are other kinds of questions we could ask, like: Do mammals have trunks? Answer: There are some mammals (i.e. models of the theory of mammals) that have trunks and others that don't.

ZFC is a theory about universes of sets --- it is a list of axioms, like "There exists an infinite set". There are many different universes of sets, each of which is called a model of ZFC if it satisfies those axioms.

We can ask questions about those universes, like Does every universe contain an infinite set?. Answer, yes, because that follows from the axioms.

We can also ask questions regarding which the axioms are silent, like Is there a set with cardinality greater than the natural numbers but smaller than the set of subsets of natural numbers? (This is a form of the continuum hypothesis.) It is not immediately obvious whether the axioms dictate the answer to this question. But Kurt Godel gave an example of a universe of sets (that is, a model of ZFC, where all the axioms are satisfied) where the answer is "no", whereas Paul Cohen gave an example in which the answer is "yes". Those are two models among a vast array of others.

(This is just like proving that the theory of mammals can't decide whether all mammals have trunks by pointing first to an elephant and then to a zebra.)

Bottom line: Just because you have a bunch of axioms, it does not follow that those axioms can tell you everything about all the models of those axioms. We have axioms for mammals, and for universes of sets, but those axioms can't tell us whether a particular mammal has a trunk, or whether a particular universe of sets satisfies the continuum hypothesis.

PS---I think (but am not sure) that some of your confusion might arise from thinking of ZFC as a set of axioms for sets, whereas it is better to think of them as axioms for universes of sets.