[scrubs]

I'm will give this more consideration; thank you for the comment.

For now I just want to add you hit a bit closer into the slight of hand in Cantors argument (for me) which is alluring but hard to surmount in the last 10% of the argument.

The natural numbers are constructible, finite. They are finite to write down. It requires a finite amount of code (tape) to output one etc. The 1:1 mapping business gets the concept of infinity onto the table but without engaging a completed infinity. So far, it's solid followable etc ... now the next 5% you toss real numbers in rhs ... then produce another real off the diagonal for 5% more ... and |Z| /= |R|.

Here real numbers live under the shadow or reflect the light of nats, which is misleading. The reals are not well defined objects.

Now, the realist (the mathematician) will argue: the point of Cantor's argument is not to construct reals as part of the solution to |Z| /= |R|. The point is only to establish there's no bijection. In truth I agree: the focus is on the mapping not getting dragged into the mud of construction.

However, I remain unclear if too much got swept under the rug that (practical minded) argument. I will have to re-read Chatin/Kolmogorov ... so I need 4 semesters now. This is my spooky action at a distance problem.

[dawnofdusk]

In the diagonalization you don't need to assume the existence of any real numbers. Just on the left hand side you write down, formally, any sort of numbers that have decimal expansions that may be infinite. Rational numbers have infinite decimal expansions too, it's just that they will eventually repeat, but at this stage it's not necessary to think about what the properties of these infinite decimal expansions actually mean. Then the diagonalization argument shows that this set of numbers with infinite decimal expansions are uncountable and also contain the rationals. This still doesn't define the real numbers yet: to do so one needs to think about the Euclidean metric on the rationals and how to complete it.