What is the role of forcing in understanding uncountable sets in set theory?

[Deveno]

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.

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?

 

[Preno]

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

What's going on is that first-order logic is simply incapable of dealing properly with infinities. First-order set theory is really an ersatz theory of infinity, suitable for many purposes but ultimately flawed. (This is my opinion, not the received view, mind you.)

 

[Deveno]

What's going on is that first-order logic is simply incapable of dealing properly with infinities. First-order set theory is really an ersatz theory of infinity, suitable for many purposes but ultimately flawed. (This is my opinion, not the received view, mind you.)

that's the impression that i get, too. one web-site described it as: first-order logical theories are extremely non-categorical. but in what I've been reading (or trying to read...it's hard going) the tactic of "forcing" to create non-standard models equi-consistent with ZF(C), significantly alters what the power set axiom "means".

ok, with respect to the real numbers: it seems that it is possible (given "a" model of ZF(C)) to insist that any element of your "universe" be countable, which at face value, means that we don't get to count the whole of the real numbers (in the usual sense) as a "set". but i find it hard to imagine what a maximal countable subset of the reals might even look like, and which numbers we "don't get".

some have responded that this is a compelling reason to use second-order logical systems, which eliminate some of these ambiguities. but i feel like I'm "not getting" some vital pieces of the puzzle. like these generalized filters based on dense sets in some poset in the universe. even if these constructions are more curiosities then actual useful enities, i'd still like to have a better grasp of what is intended.