On 28 Feb 2013, at 19:09, Craig Weinberg wrote:

## Advertising

On Thursday, February 28, 2013 1:03:40 PM UTC-5, Brent wrote: On 2/28/2013 4:46 AM, Bruno Marchal wrote:On 28 Feb 2013, at 05:04, meekerdb wrote:On 2/27/2013 7:17 PM, Bruno Marchal wrote:On 27 Feb 2013, at 20:40, meekerdb wrote:On 2/27/2013 2:59 AM, Bruno Marchal wrote:On 26 Feb 2013, at 21:40, meekerdb wrote:On 2/26/2013 1:24 AM, Bruno Marchal wrote:How did number arise? We don't know that, but we can showthat if we don't assume them, or equivalent (basicallyanything Turing Universal), then we cannot derive them.I'm not sure how you mean that?I meant that you cannot build a theory, simpler than arithmeticin appearance, from which you can derive the existence of thenumbers. All theories which want talk about the numbers have tobe turing universal.So I meant this in the concrete sense that if you write youraxioms, and want talk about numbers, you need to postulatethem, or equivalent. You can derive the numbers from theequational theory:Kxy = x Sxyz = xz(yz) + few equality rules,But that theory is already Turing universal, and assume as muchthe number than elementary arithmetic.We know that we experience individual objects and so we cancount them by putting them in one-to-one relation with fingersor notches or marks. So what are you calling an "assumption"in this?A theory is supposed to abstract from the experiences. Theexperiences motivates the theory, but does not justify itlogically.But "justify logically" seems like a bizarre concept to me. Wejust make up rules of logic so that inferences from some axioms,which we also make up, preserve 'true'.We have intuition and/or evidences for accepting the axiom. Herethe question is really: do you accept the axiom of PeanoArithmetic (for example). And with comp we don't need more. Thenwe prove theorems, which can be quite non trivial.To say that it is "justified logically" seems to mean no morethan "we have avoided inconsistency insofar as we know."Yes. We can't hope for more. But in case of arithmetic, we dohave intuition. Well, in physics too.Sure it's important that our model of the world not haveinconsistencies (at least if our rules of inference include excontradictione sequitur quodlibet) but mere consistency doesn'tjustify anything.Consistency justifies our existence. I would say. We are ourselfhypothetical with comp. We are divine hypotheses, somehow. Ithink you ask too much for a justification.You are assuming that justification comes from logic; and indeedit is too much to expect from such a weak source. I look for such"justification" as can be found from experience, which you demotedto mere "motivation".Where did I say "motivation"?"The experiences motivates the theory, but does not justify itlogically."I use the term "intuition", and I demote nothing, as it correspondto to the first person (the hero of comp, the inner God, the thirdhypostase, Bp & p; S4Grz1, etc.).But justification for me invokes "proof", formal or informal.Logical proof is relative to axioms. So the justification can be nostronger than the axioms.And axioms are no stronger than the capacity to make sense of them.

`That's correct, and that's why when we assume comp we need at the`

`start a Turing universal theory, capable of representing the partial`

`computable functions, incmuding universal functions. With comp the`

`universal machine, which compute the universal function (alxays in the`

`Church Turing sense) are capable of making sense of the axioms, and`

`prove a lot of things about them. Such machines can prove their own`

`limitation, and bet on what is beyond them.`

Bruno

I don't think that what you ask is possible, even if I am prettysure that x + 0 = x, x + s(y) = s(x + y), etc.I'm not at all sure that there is successor for every x.Then you adopt ultrafinitism, and indeed comp does not make sensewith such hypothesis, and UDA1-7 suggests that ultrafinitism mightsave physicalism, but step 8 put a doubt on this.The axiom that all natural numbers have a successor is used inbasically all scientific paper though.It is assumed, but I'm not sure it is used in an essential way. Irecognize it difficult to do mathematics without it, but still itmay be only a convenience.You need it, or equivalent, to define "machine", "formal systems","programs", "Church's thesis", "string theory", "eigenvector","trigonometry", etc.My intuition doesn't reach to infinity. It seems like anhypothesis of convenience.I propose a theory, that's all. You don't need to believe ininfinity, unlike in set theory (yet also used by many). You needjust to believe (assume) that 0 ≠ s(x), and that x ≠ y entailss(x) ≠ s(y). Notion like provability and computability are basedon this.I think you need to accept that every number has a successor inorder to prove things like Godel's theorems.BrentBruno http://iridia.ulb.ac.be/~marchal/ No virus found in this message. Checked by AVG - www.avg.comVersion: 2013.0.2899 / Virus Database: 2641/6133 - Release Date:02/25/13--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To unsubscribe from this group and stop receiving emails from it,send an email to everything-li...@googlegroups.com.To post to this group, send email to everyth...@googlegroups.com.Visit this group at http://groups.google.com/group/everything-list?hl=en.For more options, visit https://groups.google.com/groups/opt_out.--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To unsubscribe from this group and stop receiving emails from it,send an email to everything-list+unsubscr...@googlegroups.com.To post to this group, send email to everything-list@googlegroups.com.Visit this group at http://groups.google.com/group/everything-list?hl=en.For more options, visit https://groups.google.com/groups/opt_out.

http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.