On 29 May 2012, at 19:27, meekerdb wrote:

On 5/29/2012 12:27 AM, Bruno Marchal wrote:

I doubt infinities.

I can doubt actual infinities. Not potential infinities, which gives sense to any non stooping program notion.

Comp is ontologically finitist. As long as you don't claim that there is a biggest prime number, there should be no problem with the comp hyp. Infinities can be put in the epistemology, or at the meta-level: they are mind tool, souls attractor etc.


But diagonalization arguments assume realized infinities.

Set theoretical diagonalizations, à-la Cantor, assume realized infinities (like analysis, by the way). I don't use them, if only to explain diagonalization.

Computer science or "arithmetical" diagonalization does not assume realized infinities, only potential. Kleene second theorem is constructive. Gödel's diagonalization is constructive: for each effective theory, it provides the undecidable sentences.

The intensional diagonalization, leading to reproduction, self- generation and self-reference are all constructive concepts.

The theory of everything is really just logic and

Ax ~(0 = s(x)) (For all number x the successor of x is different from zero). AxAy ~(x = y) -> ~(s(x) = s(y)) (different numbers have different successors)

Ax x + 0 = x
AxAy x + s(y) = s(x + y) ( meaning x + (y +1) = (x + y) +1) = laws of addition

Ax   x *0 = 0
AxAy x*s(y) = x*y + x    laws of multiplication

The observer is the same + the induction axioms. To define it in the theory above is of course a very long subtle and tedious exercise.



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