Why do you believe that? And do you also believe that ZF is inconsistent?
Yes. It's not the Choice axiom which is problematic, but the infinity itself. So it doesn't mater if ZF or ZFC.
Why do I believe this? It's known for some time now, that you can't have an uniform probability distribution over the set of all naturals. That would be an express road to paradoxes.
The problem is, that even if you have a probability distribution where P(0)=0.5, P(1)=0.25, P(2)=0.125 and so on ... you can then invite a super-task of swapping two random naturals (using this distribution) at the time 0. Then the next swapping at 0.5. Then the next s...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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