[zamadatix]

This is what I mean in that it only appears more grounded if you already understand why a countable set has a different type of infinity than an uncountable set in the first place and what type of universe that implies. Otherwise you're left wondering what type of universe is needed and why it is that type of universe can account for some infinities but not others. The latter part is just the answer to the original question of what the difference between a countable and uncountable set is again so if you can answer that you didn't have the question to start with!

[aithrowawaycomm]

I think you are getting away from the actual original question, which is why (intuitively) the rationals are dense in the reals despite being a different form of infinity. The confusion wasn't about different forms of infinity, it was really about the topology of R with respect to Q - why is Q "big enough" yet Z "too small" despite the sets having the same cardinality? And that is intimately related to any fixed real number having a computable/rational approximation up to any accuracy, yet most real numbers not actually being computable.