On 12/26/2012 5:29 AM, Zuhair wrote: > On Dec 26, 12:21 pm, Virgil <vir...@ligriv.com> wrote: >> >> For any finite set of such strings, finite initial segments suffice to >> distinguish all of them from each oterhbut for at least some infinite >> set, no finite set of finite initial segments suffices. >> > Yes but a countable set of them suffices! no? >
The diagonal argument generates a distinct logical type given a particular assumption.
A completion of an incomplete dense topological space populates the namespace of the new logical type.
The completed infinity of eventually constant infinite strings is required to separate the completed infinity of arbitrary strings.
Again, THE COMPLETED INFINITY OF EVENTUALLY CONSTANT INFINITE STRINGS is required to separate the completed infinity of arbitrary strings. These are not the finite strings. The finite strings collect the infinite strings into the open sets of the topology.
This is what is meant by a dense set. Think of the fact that there is a rational between every two irrationals and an irrational between every two rationals. The order relation of the rationals grounds the order relation of the reals, and hence gives an identity criterion for the reals in terms of antisymmetry. But, this is not an existence criterion.
A completed infinity is admitted as a base assumption of the original argument.
A completed infinity is required to substantiate any named particular of the new logical type.
There is no countable-to-uncountable magic until one claims that there is a correspondence -- only type distinctions.
These constructions trace back to the incommensurables of Grecian geometry. The counter-intuitiveness arises from geometric intuitions associated with invariance under rigid motion and the measurement of area.
The reason I mention the measurement of area is because of an example from a statistics book I once read. Area is a non-linear function of linear measurements. As such, the random errors of linear measurements can produce systematic error in the subsequent calculation of area. This does not happen with linear functions.
This example speaks to the general treatment of differentials in calculus. Modern differentials use the little-oh concept to deal with small linear errors in the neighborhood of a definite value.
delta(y)=f'(x)h + o(h) where o(h)->0 as h->0
Whereas it once seemed that the calculus depended on the structure of the real numbers, that dependency is less apparent now. At the time Cantor and Dedekind did their investigations, infinitesimals were still actively being researched. Continuity seemed to depend upon the geometric completeness of the real line, but there had been a strong desire on the part of late nineteenth century mathematicians to get away from geometric reasoning. As various algebraic systems different from naturals, rationals, or reals had been shown to have arithmetic relations, the notion of "number" had been expanded. Thus Cantor and Dedekind had little inhibition against introducing logical types having the same relations as the numbers with which everyone had been using and then calling them "numbers."
But, if you need something intuitive to ground what Cantor and Dedekind were doing, look to the fundamental theorem of algebra which asserts that the field of complex numbers is algebraically closed. The proof depends upon the fact that every positive real number has a real positive square root and that every polynomial of odd degree over the real numbers has a root in the real numbers. These assertions depend in the first instance on the existence of irrational numbers and in the second instance on the intermediate value theorem (meaning no "gaps" like with the rational numbers).
So, in order for (x^2-2)=0 to have a solution, there has to be something that is not a rational number but behaves like a number.
All known proofs of the fundamental theorem of algebra depend on real analysis (which is not the same as calculus or approximation theory).