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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 ]
 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 20, 2013 11:08 AM

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

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