fom
Posts:
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Registered:
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Re: The ambiguity of 0^0 on N
Posted:
Sep 19, 2013 8:13 PM


On 9/19/2013 3:36 PM, Dan Christensen wrote: > On Thursday, September 19, 2013 12:50:02 PM UTC4, Rotwang wrote: >> On 18/09/2013 15:53, Dan Christensen wrote: >> >>> For the natural numbers, exponents greater than 1 are naturally defined for x in N as follows: >> >>> >> >>> x^2=xx >> >>> x^3=xxx >> >>> x^4=xxxx >> >>> x^5=xxxxx >> >>> >> >>> and so on. >> >>> >> >>> More formally, exponentiation can be defined as a binary function on the set of natural number N such that: >> >>> >> >>> (1) Ax in N: x^2=x*x >> >> >> >> Let's see your proof of that identity. Sorry, simply defining it as such >> >> won't do in this context. > > It is not a theorem that requires a proof. It is a definition, or if you like, an axiom. Unlike your 0^0=1, it is justified by the natural development of exponentiation that I give above. >
As noted elsewhere, if the base element is impredicatively defined by the formula
x^x * x = x
and asserted to be unique, then your arguments are sensible and 0 is not the base element of number theory
That is what may be discerned from the natural development of exponentiation given above.
In the modern presentation of number theory, 0 is included as the additive identity to support the recursive definition of multiplication in terms of addition.
Skolem is credited with having introduced this conception. His purpose had been to formulate a quantifierfree foundation for arithmetic, and thus avoid the problems of classbased construction associated with the use of quantifiers.
Your views of "number theory" are influenced by a modern axiom set that obfuscates the contrary programs of research from which it arises.
You may have number theory without 0. If you insist on 0, then you must admit either classbased justifications or recursive definition of operations.
You are welcome to introduce new ideas...
But it will take more than
x^2 = xx
x^3 = xxx
x^4 = xxxx
and so on.
Mathematics is funny that way.
It is hard.
Honest toil makes it even harder.

