Clasification set theory axioms

Question

The axioms for real numbers can be classified in field axioms, order axioms and completness axioms. There is also a classification for axioms of geometry. I am wondering if there is a classification for set theory axioms. Thank you for your attention.

Answer 1

To start my answer, I believe that an answer to this question is necessarily subjective. So please don't put too much value on it.

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I think that the axioms for the real numbers are classified in these three groups, because these subdivisions are proper research fields on their own. Taking only the field axioms, gives you field theory. Taking only the order axioms, gives you the theory of linear orders. Therefore the classification makes immediately clear that the real numbers are a field, linearly ordered, and complete. Consequently, results from field theory or order theory can be readily applied to the real numbers.

To my knowledge, no such classification really makes sense for the axioms of set theory. For example, with $\mathsf{ZFC}$, there are many possible weaker theories and stronger theories, and many of those are extensively studied. However, these are still only of real interest to Set Theory, and are generally not seen as individual research fields on their own.

In a sense, the study of Set Theory is very intimately related with exactly what the effect of the axioms of set theory are. In Set Theory, a huge part is the study of the logical relations between the axioms themselves. On the other hand, with the real numbers, we're usually only interested the consequences of the axioms, instead of their individual power or the interplay between the axioms.

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Nevertheless, commonly used axioms in set theory can certainly be classified if one looks at their general purpose. To make some attempt at classifying some types of axioms in set theory:

• there are many variants of the axiom of choice,
• there is also a family of axioms that deal with the determinacy of infinite games (some of the stronger of these axioms contradict the axiom of choice),
• non-well-founded set theories study axioms that negate the axiom of regularity, such as the anti-foundation axioms.
• many axioms exist that stipulate the existence of infinite cardinals with certain nice properties, which are called large cardinal axioms
• certain axioms are added to $\mathsf{ZFC}$ to control arguments involving the method forcing, such as (variants of) Martin's axiom or the proper forcing axiom.

Answer 2

Vsotvep has already given a good answer concerning (mainly) axioms that go beyond the standard ZFC axioms; I'll concentrate instead on ZFC. I regard the ZFC axioms as being of three broad sorts.

First, there are the axioms of extensionality and regularity. These tell us what sorts of things sets are, as opposed to lists, multi-sets, ill-founded sets, etc.

Second, there are axioms asserting the existence of sets with specified elements. These include (with some redundancy) axioms of null set, pairing, union, separation, replacement, and power set. I'd also include here the axiom of infinity, because, even though its usual formulation ("there is an inductive set") doesn't fully specify the desired elements, it's equivalent to "there is a smallest inductive set", which specifies the desired elements as the natural numbers.

Finally, there's the axiom of choice, which asserts the existence of sets without fully specifying what elements they should contain.