
Re: The ambiguity of 0^0 on N
Posted:
Sep 20, 2013 11:08 AM


On Friday, September 20, 2013 8:09:22 AM UTC4, 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/G1zxy5ZcXMJ > > > > 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 > > selfapplication. 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 nonstandard. > > > > 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/whyhasntmereologysuceededasanalternativetosettheory/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 settheoretic notion A^B = {f  f:B>A} ? This is an interesting analogy, but I was looking for a proof ultimately based on some Peanolike 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

