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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 8:31 PM

On 9/19/2013 4:34 PM, Dan Christensen wrote:
> On Thursday, September 19, 2013 5:04:44 PM UTC-4, Peter Percival wrote:
>>
>>>> The first principle is the definition of x^y, but when I prove 0^0 = 1
>>
>>>>
>>
>>>> from the definition in two lines you object. What you seem to object to
>>
>>>>
>>
>>>> is that it is just two lines.
>>
>>>>
>>
>>>
>>
>>> So, why don't we just include Goldbach's Conjecture in the axioms of number theory and move on?
>>
>>
>>
>> Because we don't yet know that Goldbach's conjecture is true. But a
>>
>> more significant point so far as the discussion of 0^0 is concerned is
>>
>> this: Goldbach's conjectute isn't a definition.
>>

>
> 0^0=1 can be seen as just another conjecture.
>

WRONG!

Whatever "number theory" is, its current representation
arises from the "foundational crisis" introduced by
non-Euclidean geometry. The "arithmetization of mathematics"
had been a response to the uncertainty introduced by
this development.

The attempt to give mathematics a more secure logical
foundation had been ongoing through this period, possibly
beginning with Bolzano. In one direction, it led to
Cantor and Frege. In another it led to Whitehead and
Russell through De Morgan.

When Hilbert had been confronted with the problem of
indemonstrable infinities with respect to his formalistic
axiomatics, he directed his attention to the arithmetization
of proofs (defeated by Goedel and Gentzen).

The prominent role of arithmetic in modern foundations
is a response to historical developments.

The moral of this little story is that the continuity
of polynomials in real analysis had been one part of the
"insecure" mathematics which one would expect to be "grounded"
by foundational theories.

One can, perhaps, formulate useful systems using different
conventions. But, you are arguing for the (metaphysically)
"correct" interpretation of a term relative to an established
theory.

Under such conditions, you may not suddenly invoke the
arbitrariness surmised from the phrase "just another".

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