> 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:
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:
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.