Python cryptography, Rust, and Gentoo

There are two parts.

The first is that any sufficiently powerful[1] system of arithmetic is incomplete. In this sense it means that there are statements one can make in the system for which no proof exists (of either its truth or falsity).

The second is that in such a system, the consistency of the system itself is one such statement.

It makes no such claim as to which statement is required.

> That's why we can't prove logic correct, because we have nothing else to throw into the mix.

Sure, but *that* system is also not provably correct. So what have you gained? You (claim to have) jumped one rung up a countably infinite ladder among a countably infinite selection of such ladders. Yay? :)

> So I have no qualms about throwing that infinity stuff into the proof, because otherwise you can't class zero as a number, because it behaves completely differently to all the other numbers.

Zero is a number. It works just fine. Division has a singularity at its value, but all kinds of functions have singularities. Do we need something else for tan(π/2)? Why not extend to the complex numbers with sqrt(-1) while we're at it? Quaternions? Octonions? Sedenions? Each of these is a separate algebra, an extension of algebra over the reals. We don't use them in general because we don't need the additional power they offer in day-to-day uses.

> Yes, because my logic (as per Godel) MUST be either incomplete, or inconsistent. Without that rule, it's inconsistent. With that rule it's incomplete. Pick one ... I've gone for consistency.

You're using the wrong definition of "consistency". It isn't consistent as in "all values must be able to take places of all other values in all expressions".[2] It is consistent as in "there are no contradictions between provable statements" which is *way* more important in (useful) mathematics.

[1] Peano arithmetic is sufficiently powerful. Arithmetic with just the natural numbers, addition, and multiplication, I believe, is not.
[2] You're still "inconsistent" in this sense about the square root of negative numbers for example. Why not toss those in? Why stop at trying to make division "consistent" in this sense when you're leaving out the trigonometric functions, square root, and the other infinite singularity-containing or domain-limited functions alone?