I have recently become suspicious of the real numbers. For nearly 3 decades I accepted their axiomatic existence as a complete, ordered archimedian field. The Dedekind-cut, and Cauchy sequence, and "infinite decimal" constructions all made sense to me.(adsbygoogle = window.adsbygoogle || []).push({});

And then I started reading about models of ZFC. And I became concerned. Perhaps the power set axiom really didn't say what I thought it did, at least not for the power set of a countably infinite set. If the collection of subsets that were members of a model of ZFC weren't *all* the possible subsets (perhaps I should use a different word than "subset" here, I'm not sure) of a given infinite set, then perhaps Cantor's proof only showed that a surjection from N to 2^{N}wasn't a function in our model.

The lack of an actual model for ZFC started to concern me, too. I feel...uncertain...as to what is allowed, and firm ground, and what is mere conjecture. I never worried overmuch about what structure might be large enough to contain all of ZFC, or whether or not a Grothendieck universe actually existed. I'm a simple person at heart, willing to leave some questions unanswered.

But this doubt....what does the Skolem paradox mean? What are these c.t.m. "extensions" M[G]? Levy collapse? How exactly does "forcing" work? Why are "generalized ultrafilters" so mysterious? I want to understand....and I'm a bit hesitant, too, at the same time. And, you just can't "do" topology without running into some of these questions.

Someone help me out, here...what's going on?

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# Uncountable sets and forcing

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