Followup to: Standard and Nonstandard Numbers

So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was forever and truly impossible, even unto a superintelligence, no matter how clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers.

"Right."

How could you, in your finiteness, possibly know that?

"Have you heard of Godel's Incompleteness Theorem?"

Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are true within that system, which can't be proven within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list <3, 7, 1, 4> could be uniquely encoded as:

2³ * 3⁷ * 5¹ * 7⁴

And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list <3, 7, 1, 4> back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ:

¬Φ = 2² * 3^#Φ

Φ ∧ Ψ = 2³ * 3^#Φ * 5^#Ψ

It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.'

Godel then put in some further clever work to invent statements which referred to themselves, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula.

And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-ord...

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