The succinct way of defining the topology (on NxN) is the product topology of the discrete topology(/ies). Maybe that’s the discreteness you’re feeling?
Axiom of Choice is not regarded as a big deal by most set theorists, but it’s interesting when it comes up. The diagonalisation proof that there are undetermined games uses choice to well-order the set of strategies, so it’s actually the other way around: without choice it is consistent (assuming consistency of some other stuff) that all games in this formulation are determined. This is called the Axiom of Determinacy.
The axioms in question are power set and replacement: to prove full Borel determinacy you need to apply the power set axiom infinitely many times (using the replacement axiom). These two axioms are what gives the ZFC axioms their power, really.
Set theorists nowadays I don’t think debate about axioms per se. Set theory nowadays is at once somewhat pluralistic and somewhat settled (paradoxically). I’ll explain a little: set theorists are basically agreed that the ZFC axioms are natural, intuitively “true” (many set theorists would not put scare-quotes there, but I would), powerful enough to do all ordinary mathematics and more, and probably consistent. They also generally agree that many large-cardinal axioms are natural and probably consistent, though there is a wide variation in whether people think they are “true”; there is not nearly so much intuition that such huge objects could exist. This is different than our intuition behind the axiom of infinity, because that axiom is actually needed to do some ordinary mathematics (though you can do without it for a lot!)
The Projective Hierarchy continues the stratification of the Borel Hierarchy even further. If you assume infinitely many woodin cardinals, then you can prove Projective Determinacy. I have heard Tony Martin being quoted as saying that “if Projective Determinacy were found to be inconsistent” (and hence infinitely many Woodin cardinals is inconsistent) “then I’d be having serious doubts about [the axiom of] Replacement.” This gives you a flavour of how people think about the relationship between these concepts.
The succinct way of defining the topology (on NxN) is the product topology of the discrete topology(/ies). Maybe that’s the discreteness you’re feeling?
Axiom of Choice is not regarded as a big deal by most set theorists, but it’s interesting when it comes up. The diagonalisation proof that there are undetermined games uses choice to well-order the set of strategies, so it’s actually the other way around: without choice it is consistent (assuming consistency of some other stuff) that all games in this formulation are determined. This is called the Axiom of Determinacy.
The axioms in question are power set and replacement: to prove full Borel determinacy you need to apply the power set axiom infinitely many times (using the replacement axiom). These two axioms are what gives the ZFC axioms their power, really.
Set theorists nowadays I don’t think debate about axioms per se. Set theory nowadays is at once somewhat pluralistic and somewhat settled (paradoxically). I’ll explain a little: set theorists are basically agreed that the ZFC axioms are natural, intuitively “true” (many set theorists would not put scare-quotes there, but I would), powerful enough to do all ordinary mathematics and more, and probably consistent. They also generally agree that many large-cardinal axioms are natural and probably consistent, though there is a wide variation in whether people think they are “true”; there is not nearly so much intuition that such huge objects could exist. This is different than our intuition behind the axiom of infinity, because that axiom is actually needed to do some ordinary mathematics (though you can do without it for a lot!)
The Projective Hierarchy continues the stratification of the Borel Hierarchy even further. If you assume infinitely many woodin cardinals, then you can prove Projective Determinacy. I have heard Tony Martin being quoted as saying that “if Projective Determinacy were found to be inconsistent” (and hence infinitely many Woodin cardinals is inconsistent) “then I’d be having serious doubts about [the axiom of] Replacement.” This gives you a flavour of how people think about the relationship between these concepts.