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Topic: The ambiguity of 0^0 on N
Replies: 106   Last Post: Sep 29, 2013 10:06 AM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
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 UTC-4, 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

Skolem is credited with having introduced this
conception. His purpose had been to formulate
a quantifier-free foundation for arithmetic, and
thus avoid the problems of class-based 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 class-based
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.

Date Subject Author
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Virgil
9/18/13 Dan Christensen
9/18/13 Rotwang
9/18/13 Rock Brentwood
9/18/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/19/13 Virgil
9/19/13 Virgil
9/19/13 Rotwang
9/18/13 Virgil
9/18/13 fom
9/18/13 Rotwang
9/28/13 Shmuel (Seymour J.) Metz
9/29/13 Marshall
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Michael F. Stemper
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/19/13 fom
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Helmut Richter
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 fom
9/19/13 JT
9/19/13 JT
9/19/13 Michael F. Stemper
9/19/13 JT
9/19/13 JT
9/19/13 JT
9/19/13 Helmut Richter
9/28/13 Shmuel (Seymour J.) Metz
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Karl-Olav Nyberg
9/19/13 fom
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/25/13 Rotwang
9/26/13 Dan Christensen
9/27/13 Brian Q. Hutchings
9/19/13 fom
9/18/13 Rock Brentwood
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 Peter Percival
9/20/13 Peter Percival
9/20/13 Dan Christensen
9/20/13 Virgil
9/20/13 Peter Percival
9/20/13 fom
9/20/13 Michael F. Stemper
9/20/13 LudovicoVan
9/21/13 Michael F. Stemper
9/21/13 LudovicoVan
9/21/13 Richard Tobin
9/20/13 Peter Percival
9/20/13 Peter Percival
9/21/13 Dan Christensen
9/19/13 Karl-Olav Nyberg