This question is based off of a post I saw on the Physics StackExchange: http://physics.stackexchange.com/questions/20370/does-the-banach-tarski-paradox-contradict-our-understanding-of-nature
While the question refers to physics, it also refers to mathematics. From the third answer on the page:
Even within pure mathematics, the mechanism of logical deduction is always a finite computation. If you are given a well defined collection of axioms, or axiom schemas all of whose axioms can be listed by a computer program (this includes every reasonable mathematical theory), you can write a computer program to deduce all the consequences of these axioms. Godel's completeness theorem states that every deduction will be reached by the rules of first order logic, and that when there is an undecidable statement, one which cannot be proved or disproved by the axioms, there is always a model of the axioms where the statement is true, and a model where the statement is false.
This means that when you are given a set theory, which talks about infinite non-denumerable collections, you can understand that the theory is really talking about its countable models, and this gives a countable computational interpretation to every theorem. You can then ignore the jibber-jabber about the theory talking about some enormous sets, and consider the theory as talking about its countable models.
(You might also need some of the context from the post, it's fairly long)
Now, while a countable model of (say) ZFC is possible, what I'm wondering about is does the "computational" nature of logic render the idea of "enormous" sets as just "jibber-jabber" that you don't need? Or perhaps, maybe a better way of asking the question is: how come that many (I believe) mathematicians would imagine there to really be such a vast, uncountable set when they think of, say, the Reals, as opposed to it "really" being countable inside a countable model of ZFC? What's the strongest justification that can be given for this view and also for the view (if possible) mentioned in the quoted message?
I suppose this also ties into a similar question I have been wondering about: namely, that is, given the very fact that there is not one single model of ZFC, how can we trust that familiar objects like the Reals are "what we think they are"? Namely, am I right in believing there exist models of ZFC where the model of the natural numbers is (when viewed from "outside") a "non-standard model of arithmetic", and therefore the naturals and the Reals contain nonstandard numbers: numbers that from "outside" look to be "infinitely big" and "infinitely small"? If so, then how do we reconcile this with the fact our intuition about the reals says no such things should exist -- though of course since we cannot "see" them from inside the model, it doesn't affect the math done within ZFC?