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

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Dan Christensen

Posts: 2,883
Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 20, 2013 11:08 AM
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On Friday, September 20, 2013 8:09:22 AM UTC-4, fom wrote:
> On 9/20/2013 12:39 AM, Dan Christensen wrote:
>
>
>

> > fom, how about some honest toil to prove 0^0=1? Choose any axioms or definitions you like.
>
>
>
> I doubt you have the stomach for my version of "honest toil". Nor
>
> do you have any comprehension of "logical priority" as a principle
>
> of analytical reduction. A system analyzed by such means must be
>
> presented synthetically.
>
>
>
> Thus, before we get to any discussion of natural numbers, we must
>
> introduce the logical constants of our language:
>
>
>
> https://groups.google.com/forum/#!original/sci.logic/V41XKFS9zSU/4vcFbzXcyRMJ
>
>
>
> One cannot even think of formulating definitions in a logical
>
> language without these constants.
>
>
>
> Next, we must formulate some axioms justifying the interpretation
>
> of the logical constants for propositional logic as functions:
>
>
>
> https://groups.google.com/forum/#!original/sci.math/zuLUred6O3U/G1zx-y5ZcXMJ
>
>
>
> To understand the second link, let me suggest you begin reading
>
> Frege. You will eventually find the comparison he makes between
>
> his system of logic and a system of parts without a whole. This
>
> comparison is in reference to the compositional structure of
>
> nested formulas characterizing his advance in logic. The equational
>
> axioms in the second link characterize the functional relations
>
> of logical connectivity without invoking truth table representations.
>
> Such representations presuppose propositional attitudes and
>
> would otherwise commit the system to free Boolean algebras.
>
>
>
> In Feferman you will find a discussion of the paradoxes which
>
> attributes the problem to a combination of negation and
>
> self-application. The systems above had been discerned by
>
> considering the structure of (mathematical) logic without
>
> a unary negation. The axioms of the second link serve as a
>
> prerequisite to formulating a representation of the connectivity
>
> of propositional logic as an applicative structure.
>
>
>
> You can find the definition of an applicative structure in
>
> "The Lambda Calculus" by Barendregt. Its origins are attributed
>
> to Shoenfinkel, and, the idea had been developed by Curry for
>
> his combinatory logic.
>
>
>
> As to the content of the first link, the ideas are simple.
>
>
>
> When Frege retracted his logicism, he suggested geometry as
>
> the basis for mathematical foundation. This is in contrast
>
> to Hilbert or Brouwer who remained committed to arithmetization
>
> in one form or another. It is also in contrast to Russell
>
> who remained committed to logicism.
>
>
>
> In Wittgenstein, you will find the idea that a name is like
>
> a geometric point. Wittgenstein considered the sign of equality
>
> as a relation of identity eliminable. The use of a geometry
>
> to present the names of the logical constants follows this
>
> notion. It is also compatible with statements in both Kant
>
> and Strawson attributing the numerical identity of objects
>
> to geometric intuitions.
>
>
>
> I recognize that my views on the foundations of mathematics
>
> are essentially incomprehensible to others. But, unlike you,
>
> I have done (and continue to do) the work needed to place
>
> those ideas in the context of the subject.
>
>
>
> Eventually, my constructions would lead to development
>
> of a formal theory interpretable as a theory of classes.
>
> It is non-standard.
>
>
>
> The role of the empty set in that formulation arises from
>
> considerations different from Frege's use of an empty class
>
> to ground his arithmetic. You can find the role it plays
>
> explained here:
>
>
>
> http://mathoverflow.net/questions/58495/why-hasnt-mereology-suceeded-as-an-alternative-to-set-theory/127222#127222
>
>
>
> The sense of this comes from lattice theory where atomistic
>
> geometric lattices are characterized by having a bottom.
>
>
>
> Again, the common theme here is that individuation -- that
>
> is, numerical identity -- is fundamentally a geometric
>
> notion in so far as one is speaking of mathematical foundations.
>
>
>
> Within that theory, the Fregean considerations
>
> come into play and the empty set grounds the natural
>
> numbers. Those natural numbers correspond with the finite
>
> von Neumann ordinals, and, the requested proof is one given
>
> to you repeatedly by others.


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?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com


Date Subject Author
9/18/13
Read The ambiguity of 0^0 on N
Dan Christensen
9/18/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/18/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/18/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/18/13
Read Re: The ambiguity of 0^0 on N
Virgil
9/18/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/18/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/18/13
Read Re: The ambiguity of 0^0 on N
Rock Brentwood
9/18/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/20/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Virgil
9/19/13
Read Re: The ambiguity of 0^0 on N
Virgil
9/19/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/18/13
Read Re: The ambiguity of 0^0 on N
Virgil
9/18/13
Read Re: The ambiguity of 0^0 on N
fom
9/18/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/28/13
Read Re: The ambiguity of 0^0 on N
Shmuel (Seymour J.) Metz
9/29/13
Read Re: The ambiguity of 0^0 on N
Marshall
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Michael F. Stemper
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/20/13
Read Re: The ambiguity of 0^0 on N
fom
9/20/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/20/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Helmut Richter
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
JT
9/19/13
Read Re: The ambiguity of 0^0 on N
JT
9/19/13
Read Re: The ambiguity of 0^0 on N
Michael F. Stemper
9/19/13
Read Re: The ambiguity of 0^0 on N
JT
9/19/13
Read Re: The ambiguity of 0^0 on N
JT
9/19/13
Read Re: The ambiguity of 0^0 on N
JT
9/19/13
Read Re: The ambiguity of 0^0 on N
Helmut Richter
9/28/13
Read Re: The ambiguity of 0^0 on N
Shmuel (Seymour J.) Metz
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/19/13
Read Re: The ambiguity of 0^0 on N
Karl-Olav Nyberg
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/19/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/25/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/26/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/27/13
Read Re: The ambiguity of 0^0 on N
Brian Q. Hutchings
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/18/13
Read Re: The ambiguity of 0^0 on N
Rock Brentwood
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Rotwang
9/19/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
fom
9/20/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/20/13
Read Re: The ambiguity of 0^0 on N
fom
9/20/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/20/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/20/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/20/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/20/13
Read Re: The ambiguity of 0^0 on N
Virgil
9/20/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/20/13
Read Re: The ambiguity of 0^0 on N
fom
9/20/13
Read Re: The ambiguity of 0^0 on N
Michael F. Stemper
9/20/13
Read Re: The ambiguity of 0^0 on N
LudovicoVan
9/21/13
Read Re: The ambiguity of 0^0 on N
Michael F. Stemper
9/21/13
Read Re: The ambiguity of 0^0 on N
LudovicoVan
9/21/13
Read Re: The ambiguity of 0^0 on N
Richard Tobin
9/20/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/20/13
Read Re: The ambiguity of 0^0 on N
Peter Percival
9/21/13
Read Re: The ambiguity of 0^0 on N
Dan Christensen
9/19/13
Read Re: The ambiguity of 0^0 on N
Karl-Olav Nyberg

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