On 31 May 2012, at 18:13, meekerdb wrote:

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On 5/29/2012 11:39 AM, Bruno Marchal wrote: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, whichgives sense to any non stooping program notion.Comp is ontologically finitist. As long as you don't claim thatthere is a biggest prime number, there should be no problem withthe comp hyp. Infinities can be put in the epistemology, or atthe meta-level: they are mind tool, souls attractor etc.BrunoBut diagonalization arguments assume realized infinities.Set theoretical diagonalizations, à-la Cantor, assume realizedinfinities (like analysis, by the way). I don't use them, if onlyto explain diagonalization.Computer science or "arithmetical" diagonalization does not assumerealized infinities, only potential. Kleene second theorem isconstructive. Gödel's diagonalization is constructive: for eacheffective theory, it provides the undecidable sentences.But they do depend on infinity (i.e. the axiom of succession).

`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.`

`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.`

The intensional diagonalization, leading to reproduction, self-generation and self-reference are all constructive concepts.Can you explain "intensional diagonalization"?

`It is when you build an expression involving a formal diagonalization,`

`like writing a program which refer to itself.`

`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.

`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).`

Bruno

The theory of everything is really just logic andAx ~(0 = s(x)) (For all number x the successor of x is differentfrom zero).AxAy ~(x = y) -> ~(s(x) = s(y)) (different numbers havedifferent successors)Ax x + 0 = xAxAy x + s(y) = s(x + y) ( meaning x + (y +1) = (x + y) +1) =laws of additionAx x *0 = 0 AxAy x*s(y) = x*y + x laws of multiplicationThe observer is the same + the induction axioms. To define it inthe theory above is of course a very long subtle and tediousexercise.Bruno http://iridia.ulb.ac.be/~marchal/ --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-l...@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

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