John,

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Allow me please, one more remark:

I allow you an infinity of remarks. But not one more :)

my ID for an axiom is a "ground-rule" derived to facilitate theacceptance of a theory.

`Hmm... That is not the standard idea. An axiom is simply an`

`hypothesis. Like the hypothesis that there is a moon, or that 0 + x =`

`x, etc. It is what we accept to proceed.`

I suspect the axioms were invented AFTER the theoreticalconsiderations to make them acceptable.

`That is true, but they are useful to communicate ideas and beliefs to`

`others. When formalized, the axioms and theorems don't depend on the`

`many interpretations that they can have. In applied science, we cannot`

`use such axiom, and so must do some semi-axiomatization, with implicit`

`hypotheses, like the existence of the domain of application, like when`

`we send persons or robots to the moon.`

They are called axioms because we cannot justify their acceptability.

Yes.

I am not ready to defend this.

`Without (semi)-axioms, we remain unclear and non refutable, so we`

`can't so easily progress.`

Bruno

JMOn Wed, Feb 27, 2013 at 2:40 PM, meekerdb <meeke...@verizon.net>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 show that ifwe don't assume them, or equivalent (basically anything TuringUniversal), then we cannot derive them.I'm not sure how you mean that?I meant that you cannot build a theory, simpler than arithmetic inappearance, from which you can derive the existence of the numbers.All theories which want talk about the numbers have to be turinguniversal.So I meant this in the concrete sense that if you write youraxioms, and want talk about numbers, you need to postulate them, orequivalent. You can derive the numbers from the equational theory:Kxy = x Sxyz = xz(yz) + few equality rules,But that theory is already Turing universal, and assume as much thenumber than elementary arithmetic.We know that we experience individual objects and so we can countthem by putting them in one-to-one relation with fingers ornotches 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 it logically.But "justify logically" seems like a bizarre concept to me. We justmake up rules of logic so that inferences from some axioms, which wealso make up, preserve 'true'. To say that it is "justifiedlogically" seems to mean no more than "we have avoided inconsistencyinsofar as we know." Sure it's important that our model of theworld not have inconsistencies (at least if our rules of inferenceinclude ex contradictione sequitur quodlibet) but mere consistencydoesn't justify anything.Brent --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. --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.

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