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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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 fom Posts: 1,968 Registered: 12/4/12
Re: The ambiguity of 0^0 on N
Posted: Sep 20, 2013 8:09 AM

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:

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:

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.

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