I am currently studying ZF set theory in terms of first-order logic and I am having trouble understanding the motivation behind this axiomatic formulation of set theory.
ZF set theory is a first-order language with the proper axioms of extensionality, foundation, specification, union, pairing, replacement, power set, well- ordering and infinity.
But why?
Is ZF set theory just a model for universal set theory in the same manner that Lagrangian Classical Mechanics is just a model for the universe. I.e neither are perfect?
Otherwise, what is the justification in choosing these axioms, what 'logic' are we basing our choices on?
For when we extend the theory to include the axiom of choice, we have added a new axiom.
What is the justification of adding the axiom of choice, and not other statements which we cannot prove (have been proved to be unprovable in set theory); such as the continuum hypotheses, provided they are consistent with the current axioms?
Why can't we add every statement that has been proved to be unprovable within the system, why do we only add the axiom of choice?
In this light, what is the scope of ZF(C) set theory?
Best Answer
If we add to ZFC every statement that is proved to be unprovable in ZFC, we would obtain an inconsistent system, because there are sentences that are both not-provable and not-disprovable in ZFC.
Take the particular case of the continuum hypothesis, CH. Both CH and its negation are unprovable in ZFC. So we can only add one of them to ZFC at a time. But then we have to decide which one. Of course, we can assume either CH or its negation on a temporary basis, but if we wanted to add one "once and for all" we would have to decide whether there is a consensus to assume CH, or a consensus to assume the negation. Because there is not a widely accepted argument for either of these options, we don't add either axiom to ZFC on a permanent basis.
The axioms that are in ZFC are all consistent with a particular understanding of sets as forming a cumulative hierarchy. Indeed, ZFC as a theory is set up for studying the cumulative hierarchy, rather than any sets that might exist outside of it. But our understanding of the cumulative hierarchy is not complete, so that we have no idea whether CH (for example) should hold.
There has been a lot written on whether new axioms should be added as "basic axioms" in mathematics, extending ZFC. For example, see