This re-instated what passes for faith in humanity

[Thecla]

I wish I'd known that before I had the hubris to add 2 + 2 in expectation of  meaningful result.

Fair enough. (Especially since my previous statement was wrong, and one can add cardinal numbers after all. My mistake. :) )

With more seriousness, I doubt I can master the particular field of abstract mathematics that you speak of fast enough to respond to your statements in a meaningful manner; and when I speak I find that I lose you in what appears to be "philosophical pyrotechnics."

If you're going to talk about "infinity" then I'm afraid you have no alternative other than to understand something about how it is defined in mathematics; otherwise what you say is liable to degenerate into meaninglessness, and what seems counter-intuitive or contradictory may just turn out to be an unexpected property of infinite numbers.

My difficulty is that it would seem that you claim a class of numbers neither finite nor infinite in its properties: not infinite in that it is bounded, yet not finite in that it has no upper limit. This strike me as contradictory, as what is bounded is bounded by limits, and what is unbounded is unlimited:

For example, if I interpret you correctly, you don't like the fact that for every natural number there is a natural number which is larger than it (the natural numbers have 'no upper limit') yet there is a number that is larger than every natural number (the natural numbers are 'bounded'). That isn't a contradiction, however; that's how it works.

Of course, any number that is larger than every natural number cannot itself be a natural number -- it must be an infinite number. This means one has to define, as Cantor did, what one means by an infinite number and show that this statement is true. Much of our intuition goes out the window when dealing with infinite numbers, which is why it's essential to base everything on precise definitions and logical deductions from them.

It would seem that to compare one of these infinite sets to another one introduces language that denotes difference between magnitudes of infinity; yet if infinity is unbounded, unlimited, and un-numbered, then how can one assign magnitude to it?

Exactly what Cantor did was to show that in a precise way there are different 'magnitudes of infinity' (in fact, that there are infinitely many different cardinal numbers or `magnitudes of infinity') and that, for example, the real numbers are 'more infinite' than the integers or the rational numbers.

But perhaps it isn't magnitude, but correspondence that is introduced? Correspondence between what? Correspondence between the magnitude of the quantity. What is the magnitude of the quantity? Infinite. Magnitude cannot be assigned to infinity, but perhaps it isn't magnitude, but correspondence that is introduced? Correspondence between what? ...

You start with sets (say sets of numbers to be definite, but it doesn't matter) and define what it means for two sets to `have the same cardinality': namely, two sets have the same cardinality (`magnitude') if they are in one-to-one correspondence with each other.

Then you can say a set has cardinality 1 if it is in one-to-one correspondence with the set {1} with one element; cardinality 2 if it is in one-to-one correspondence with the set {1,2} with two elements,...; cardinality aleph_0 if it is in one-to-one correspondence with the set of natural numbers; cardinality c if it is in one-to-one correspondence with the set of all real numbers;... and so on.