It's just a magnificent toy, this ZF construction. And those others set theories as well. No wonder some people here don't want it to be broken. With passion, may I add?
Let us take the aleph-zero dimensional R. Countably infinite dimensional Euclidean space, in other words. Then take a point T and all those points which are finitely far away from this point T. By the standard metrics of sqrt(dx1^2+dx2^2+...).
This space is separable into continuum many such subspaces. Where in every such subspace every two points are close. Close means only a finite distance. And every two spaces are far from each other. Where far means an infinite distance between any two points from two different subspaces.
You probably are familiar with this also.
No wonder some people here don't want it to be broken. With passion, may I add?
Well, whomever is going to show ZF to be contraddictory is sure to receive great glory in the mathematical community, and is going to be probably considered the next Godel.
If only ZF wasn't already been shown coherent with the notion of an inaccessible cardinal. Which has shown to be coherent with the notion of a measurable cardinal. Which has shown to be coherent with the notion of a compact cardinal... and so on. ZF's coherence is a hairy, hairy subject.
...Then take a point
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