Date: Sep 19, 2013 10:19 PM
Author: Dan Christensen
Subject: Re: The ambiguity of 0^0 on N
On Thursday, September 19, 2013 7:36:04 PM UTC-4, fom wrote:

> On 9/19/2013 1:26 PM, Dan Christensen wrote:

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> >>

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> >> Why apart from? Why are you leaving it out?

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> >

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> > We can't divide by 0. Unless you want to assign a value to 0/0 as well.

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> >

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>

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> And, what about division is "number theoretic"?

>

As you can see in my proof, I am actually using the right-cancelability property of natural number multiplication:

x*y = z*y & y=/=0 => x=z

"Dividing" both sides by y, to cancel off the factor of y, OK?

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com

>

>

> What axiom of number theory ensures closure under division?

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>

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> You cannot distinguish an additive group from a multiplicative

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> group except in relation to a ring structure. Otherwise

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> the difference is mere purport.

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>

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> Relative to ring theory, one may speak of division as some

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> essential numeric operation. In fact, that is what characterizes

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> "division rings".

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>

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> http://en.wikipedia.org/wiki/Division_ring

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>

>

> So, you reject "set theoretic" considerations, but you invoke

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> "ring theoretic" considerations (or, since I am not going to

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> research obscure notions of "division", some other

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> "???-theoretic" considerations which are not "number-theoretic"

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> considerations).

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>

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> You cannot invoke "number theory" as a restriction and then

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> use arguments outside of "number theory" in your defense.

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>

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> In "number theory" you cannot divide by 2 unless you wish

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> to assign natural numbers to

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>

>

> 1/2, 3/2, 5/2, ..., (2n+1)/2, ...

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>

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> n in {0, 1, 2, ...}

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>

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> The standard definitions based upon the established operations

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> of the language of the theory suffice to prove the fundamental

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> theorem. If you wish to begin from the exponentiation associated

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> with the fundamental theorem then you need to present a different

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> theory for consideration as "number theory".