Expand on that please, so I can make sure I understand what you are writing here.

This is a brief reply - I don't have time for more detail now, I am intending to write this up, but it's about 50th on a very long "To Do" list.

People have this concept that infinity is kind of, well, "out there", as far as you can go. You can't go any further, it's all there is. People talk about parallel lines "meeting at infinity" and "there is no last point on the line" sort of thing.

And that's what their intuition is telling them. There's nothing beyond infinity.

Then when we talk about the cardinals we tend to reinforce that by showing that the odds, evens, squares and primes all have the same cardinality. That's a bit weird for people, but they're getting the idea that infinity is odd, but they can cope with odd.

Then we show that N^2 and Q are both countable, can be put in one-one correspondence with N. They're getting cool with that too. We continue to reinforce the idea that there's infinity, and every time we do something we get the same infinity.

But that's now what they expect. Infinity is kind of as far as you can go. We're matching the set-theoretic cardinals to their geometrical intuition of "out there."

No wonder they get confused, upset, and occasionally angry when we then introduce 2^N as being "bigger."

I'm finding that talking about geometry and the concept of infinity, then talking about sets and the concept of intinity as a different thing, despite the same name is helping people to create different models in their minds. These different models then help them not use the same intuition, and they don't get confused in the same way.

Then, as the final modification, I don't start by showing that loads and loads of things are all countable. I talk about "same size" as being 1-1 matching, and discuss the idea that there could be something "bigger than N."

I do that first, and then go hunting, pointing out from the beginning that historically people found this difficult.

Now they get excited when I show them that 2^N is uncountable. They've been primed to want to find it, and their intuition about "infinity" isn't challenged.

That's an incomplete summary - I hope it helps. Ask for more, or email me if it's not clear.

As a closet constructivist I feel compelled to point out that 2^N certainly has a more complicated internal structure, but "bigger" is entirely a question of interpretation.

There are a countable number of possible constructions of something in 2^N, and therefore in the traditional interpretation of mathematics all but a countable number of members of 2^N are unconstructable. However a constructivist looks at this and asks what sense it makes to talk about the real "existence" of mathematical objects that cannot be represented in any way, shape or form, even in principle? To a constructivist these don't really exist at all, so there really can't be more of them.

Incidentally the classic picture most mathematicians have of the infinite cardinals being nicely sorted by size depends on the axiom of choice. Specifically Friedrich Hartogs proved in 1915 that if all infinite sets are comparable in size, then the axiom of choice holds. The reverse statement is also true, and is easy to prove from the well-ordering principle. So if you want to be a classical mathematician but are heretic enough to doubt the axiom of choice, then again the simple picture people have of infinite cardinals breaks down.

In this way, infinity of any form doesn't actually exist.

Add to the mix the axiom of infinity and it then does exist, but we're close to the constructivist's world.

However, having said that, there's a lot that I do that relies, underneath, on having the more usual model. Allowing the declaration of the existence of all elements of 2^S for any set S is just too useful not to explore. That's the world I usually inhabit.

However, it's not clear that getting into those discussions will, in any way, help people to understand the more "regular" viewpoints and reasoning. Encouraging people's intuition can be good, but sometimes it simply prevents them from reasoning clearly.

And perhaps that's the most easily defensible aspect of the usual idea (within mathematics) of infinity. Dealing with it forces you not to take things for granted, and to reason carefully about properties. Discovering that 1+aleph_0 = aleph_0 = aleph_0+1 from first principles is useful. Then finding that 1+omega != omega+1 is useful.

Mathematical logic and reasoning is so useful, we need places to exercise it. This is one, there are others.

But personally, I like the idea that there are different sizes of infinity, and the resulting implications such as the Banach-Tarski theorem, the existence of uncomputables, and the tension between Zorn's Lemma and the well-ordering principle.

Although mostly it doesn't really matter.

From that point of view, it would appear that your intuitions are not helping you. More, under some models of geometry it makes a lot of sense to say that parallel lines really do meet at infinity. It makes some theorems a lot easier to state, and easier to prove.

So you may actually agree with a lot of what I'm saying, but your arguments to support that claim appear, at least on the surface, to be wrong. It may be that you have some understanding of these "paradoxes," but your other statements suggest that your inderstanding is not that of current or classical mathematics.