On 28 Feb 2013, at 19:03, meekerdb 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 areyou 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."

`Indeed. It is the inductive inference part of the learning process. It`

`is very important. All theories, including "brains" comes from this.`

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.

Sure.

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.

`OK. I don't see the problem with this. Convenience is a fuzzy notion.`

`A brain too is convenient. Universes can be convenient. I am not sure`

`to see your point.`

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.

`That follows from the axiom above. It is easy to prove this, and`

`Gödel's incompleteness, *in* the theory, if the theory has the`

`induction axioms. If the theory has no induction axioms, you can still`

`prove this at the meta-level, *about* the numbers in the theory.`

`You have the terms 0, s(0), s(s(0)), etc. When you say that you are`

`not sure all numbers have a successor, you are telling me that there`

`is a number x = to some s(s(s(...(0)...) for which has no successor.`

`this can be true for some weird notion of numbers, but non sensical in`

`all models of Robinson Arithmetic, Peano Arithmetic, of the arithmetic`

`in all model of set theories like ZF, etc. I am not sure you why you`

`want things like that, unless you are searching for some ultrafinitist`

`theory, which is your right, and is a mean to escape the UDA1-7`

`conclusion. But still not the step 8 (on which I will come back soon`

`on the FOAR list of Russell Standish).`

Bruno 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.