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Topic: Constructing the Addition, Multiplication and Exponentiation
Functions from Peano's Axioms

Replies: 93   Last Post: Jan 13, 2014 4:06 PM

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 Roland Franzius Posts: 371 Registered: 12/7/04
Re: Constructing the Addition, Multiplication and Exponentiation
Functions from Peano's Axioms

Posted: Jan 10, 2014 3:18 AM

Am 10.01.2014 06:39, schrieb Dan Christensen:
> On Thursday, January 9, 2014 6:40:04 PM UTC-5, Martin Shobe wrote:
>

>>>> You,
>>
>>>>
>>
>>>> however, have failed to justify the final step in your "proof"
>>>> that it

>>
>>>>
>>
>>>> should be left undefined.
>
>
> The onus is on you. You claim that 0^0=1 but seem unable to derive
> this result or in any way justify it based on the usual definitions
> of natural numbers, addition and multiplication (i.e. on Peano's
> axioms.)
>
> I have shown that infinitely many binary functions on N satisfy any
> reasonable requirement for exponentiation, each such function
> differing only in the value assigned to 0^0. (See formal proofs in
> "Oh, the ambiguity!" at my math blog.)

Exponentiation with nonnegative integer exponents generally count equal
factors in associative and commutative products of any kind.

Generally, the absence of a factor x can be factored out or indicated as
x^0. There exist other notations to denote missing or discarding named
factors.

This algebraical uniformization trick canonically extends to the integer
factor 0 in products of integers.

In the same way it extends to the notation of absent factor X of the
empty space {} in multilinear and tensorial products of spaces.

It is easy to show that that adjunction of the rule
nothing^0 = multiplicative identity
to the rule
anything^0 = multiplicative identity
together with the standard rule derving from the identity axiom
identity^anything=identity
is an independent axiom that does not interfere with any material axiom
but only uniforms theorems by the reduction of the number of exceptions
on general formulas.

--

Roland Franzius

Date Subject Author
1/7/14 Dan Christensen
1/7/14 Robin Chapman
1/7/14 Dan Christensen
1/7/14 Robin Chapman
1/7/14 Dan Christensen
1/8/14 Robin Chapman
1/8/14 Dan Christensen
1/8/14 Robin Chapman
1/8/14 Dan Christensen
1/8/14 Robin Chapman
1/8/14 Dan Christensen
1/8/14 Robin Chapman
1/8/14 Dan Christensen
1/8/14 Martin Shobe
1/8/14 Dan Christensen
1/8/14 Martin Shobe
1/8/14 Dan Christensen
1/8/14 Martin Shobe
1/9/14 Dan Christensen
1/9/14 Martin Shobe
1/9/14 Dan Christensen
1/9/14 Martin Shobe
1/9/14 Dan Christensen
1/9/14 Robin Chapman
1/9/14 Martin Shobe
1/9/14 Dan Christensen
1/9/14 Peter Percival
1/9/14 Robin Chapman
1/9/14 Dan Christensen
1/9/14 Martin Shobe
1/9/14 Dan Christensen
1/9/14 Martin Shobe
1/9/14 Dan Christensen
1/10/14 Dan Christensen
1/10/14 Roland Franzius
1/10/14 Martin Shobe
1/10/14 Dan Christensen
1/10/14 Martin Shobe
1/10/14 Martin Shobe
1/10/14 Dan Christensen
1/10/14 Martin Shobe
1/10/14 Dan Christensen
1/10/14 Martin Shobe
1/10/14 Dan Christensen
1/10/14 Martin Shobe
1/10/14 Dan Christensen
1/11/14 Martin Shobe
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1/11/14 Peter Percival
1/11/14 Dan Christensen
1/11/14 Peter Percival
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1/12/14 Peter Percival
1/12/14 Dan Christensen
1/11/14 Martin Shobe
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1/12/14 Dan Christensen
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1/12/14 Dan Christensen
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1/12/14 Dan Christensen
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1/8/14 Robin Chapman
1/8/14 Dan Christensen
1/8/14 Robin Chapman
1/8/14 Peter Percival
1/8/14 Dan Christensen
1/7/14 Roland Franzius
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