Continuum Hypothesis(CH) is independent of ZFC Axioms, which means there exist models of ZFC where CH is true, and models where CH is false.

Can I say something similar for groups? Something like the following:

The statement $\forall a\forall b[a,b\in G\Rightarrow ab=ba]$ (i.e. any two elements commute) is independent of the axioms of a group because there are examples of groups where this is true (Abelian groups), and groups where this is false.

By "axioms of a group", I mean things we have in the definition of a group (associativity, the existence of an identity, etc). A group is a set, so set theory axioms are included here as well.

Can I say $\forall a\forall b[ab=ba]$ is independent of group axioms, and can I refer to an example of a group (e.g. $\mathbb Z_n$) a model of group axioms?

This might be a strange question, but it is important. The meaning of "independence of CH" and the concept of Model is very difficult for a learner to understand; the fact that a group can be either Abelian or not can be easily understood. So are they really the same thing according to my interpretation above? If they are not exactly the same, then how are they different?

PS: there are a lot of cross-overs between set theory and Algebra. For example, boolean algebra. I do find some very general discussion on the topic, for example here, but none of them go into such details.


Yes. That is correct. The axioms of a group is a theory, and groups are exactly the models of this theory.

And the "commutativity axiom" (or whatever you'd want to call $\forall a\forall b(ab=ba)$, maybe the abelian axiom) is independent of the theory of groups because there are groups which are commutative and groups which are not.

Similarly, $\exists x(x\cdot x=1+1)$ is independent from the theory of fields, because in some fields there is such $x$ (e.g. $\Bbb C$) and in others there isn't (e.g. $\Bbb Q$).

The main difference between basic algebraic theories (and structures) and set theories (and their models) is that set theory is significantly more complex, and its "natural interpretation" deals with "infinite objects" whereas groups, etc., are not generally concerned with the structure of "elements of the group".

There is a conceptual jump there. But on the paper, yes, CH and being an abelian group share this similarity that you mention.

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    $\begingroup$ +1 Let me add that one difference between group theory and set theory is that the axioms for groups were, from the start, intended to apply to many examples with obvious differences between them. The idea was to axiomatize the similarities that people had noticed between those many examples. The axioms of set theory (like those of Euclidean geometry and Peano arithmetic) were intended to describe one "structure", not to unify many topics. As a result, it is expected that some interesting statements are independent of the group axioms; for set theory it's surprising. $\endgroup$ – Andreas Blass Jun 5 '19 at 16:19

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