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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 ]
 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: The ambiguity of 0^0 on N
Posted: Sep 20, 2013 4:37 PM

Dan Christensen wrote:

>
> Do you mean Percy's 0^0=1 since, by definition, x^0=1 for all x in N?
> Easily dismissed. He is basically assuming what we are trying to
> prove since we already have x^0=1 for x=/=0 in N.
>
> Or do you mean Michael's analogy to natural number exponentiation
> based on the set-theoretic notion |A|^|B| = |{f | f:B->A}| ? This is
> an interesting analogy, but I was looking for a proof ultimately
> based on some Peano-like axioms for the natural numbers. If that's
> not possible, just say so. You could assume addition and
> multiplication on N, and whatever arithmetic theorems you may require
> -- just not x^0=1 for all x in N. How about it?

Suppose you wish to formally prove something about a two-place function
symbol f in a theory T. How will you proceed? It may be that you
already know some theorems about f and the required result follows from
them. But suppose that a proof /ab initio/ is required. Two
possibilities occur to me:
i) f is an undefined (often I write "primitive") symbol of T, in which
case the proof begins with T's axioms;
ii) f is defined in terms of T's undefined symbols, in which case the
proof begins with f's definition.
What other possibilities are there?
So, if T is your theory and ^ is undefined, what axioms govern it? If ^
is defined, what is its definition?
When I have proved that 0^0 = 1 I have proceeded from the definition of
^ (on the natural numbers) that I know. What else can I do? Nothing
without you answering: if ^ is undefined, what axioms govern it? If ^
is defined, what is its definition? Note that even with answers to
those question (or the one that applies) the question what is 0^0? may
be unanswerable if the theory is weak or the definition is incomplete.
If something (say 0^0 = 1) is not provable in T one may be able to show
as much by exhibiting a model of T's axioms in which the formula in
question is false. Have you tried to find such a model?

--
Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies

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