Is mathematics mostly chaos or mostly order?

> Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.

As someone who also has never fully made his peace with the diagonality argument, but just chosen to accept it as true, as a given, this kind of bumps up against an interesting implication of different cardinalities of infinity.

To precisely define an arbitrary real you'd need some kind of finite string that uniquely identifies that real number. Finite strings can be mapped, 1 to 1, to natural numbers. Therefore there can't be a finite string for any real number that uniquely identifies it. Otherwise we'd have a mapping between natural numbers and real numbers.

In fact, the set of uniquely identifiable real numbers is a countable subset of real numbers. [1]

Somehow, this realization has helped me make peace with the uncountability of real numbers.

[1] Sorry if use words like "unique", "identify", "define" in not quite the right way. I hope the meaning I'm going for comes across.

A way to make peace with the Reals is to understand them as "potential numbers". Every where you look, there is Real number. Everyone logical agrees about that.

But what about where you don't look? Either you take the orthodox axiomatic view that Real numbers are there too, or you take the constuctivist or finitist (or perhaps quantum mechanical?) view that nothing is there until you look, because the act of looking is the same as the act of creation.

I wouldn’t call it quantum mechanical. The “looking” in math is not like the measurement of an observable/operator in quantum mechanics. When you consider a thing in math, there’s no alternative thing that you could have considered instead which would correspond to a different operator that doesn’t commute with the first one.

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.

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.