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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 UTC5, 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



