# Re: was Relativity of Existence

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On 31 May 2012, at 18:13, meekerdb wrote:```
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```On 5/29/2012 11:39 AM, Bruno Marchal wrote:
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On 29 May 2012, at 19:27, meekerdb wrote:

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```On 5/29/2012 12:27 AM, Bruno Marchal wrote:
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```I doubt infinities.
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I can doubt actual infinities. Not potential infinities, which gives sense to any non stooping program notion.
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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.
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Bruno
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But diagonalization arguments assume realized infinities.
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Set theoretical diagonalizations, à-la Cantor, assume realized infinities (like analysis, by the way). I don't use them, if only to explain diagonalization.
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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.
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But they do depend on infinity (i.e. the axiom of succession).
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The axiom of succession is not an axiom of infinity. It just says that the numbers have each unique and different (from other's) successors. All the standard numbers are finite.
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In ZF there is an axiom of infinity, for you cannot prove infinity from below. Unless you have more powerful axiom like the scheme of reflexion.
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The intensional diagonalization, leading to reproduction, self- generation and self-reference are all constructive concepts.
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Can you explain "intensional diagonalization"?
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It is when you build an expression involving a formal diagonalization, like writing a program which refer to itself.
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For example like with the self-duplicating expression Dx = 'xx', so that DD generates 'DD', i.e. its description. That is what Gödel did to prove the existence of a sentence referring to itself, and notably asserting that she is not provable. And that's what Kleene did to prove the existence of a number e, such that phi_e (x) = T(e, x), or generalization. In that case the program e compute T on itself with parameter x.
```Intensional diagonalization concerns codes.

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Extensional diagonalization concerns set of functions, like the Cantor one, showing that N^N is not enumerable, or Kleene one showing that comp-N^N is not recursively enumerable. (comp-N^N = the partial computable functions from N to N).
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Bruno

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The theory of everything is really just logic and

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

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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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Bruno

http://iridia.ulb.ac.be/~marchal/

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