
Re: Matheology § 233
Posted:
Mar 30, 2013 5:19 PM


On Mar 30, 1:17 pm, fom <fomJ...@nyms.net> wrote: > On 3/30/2013 8:38 AM, WM wrote: > > > On 29 Mrz., 19:34, Virgil <vir...@ligriv.com> wrote: > > >>> So we have established the fact that an irrational number has no node > >>> of its own. > > >> No number in any infinite binary tree has any node "of its own", as > >> every node has two child nodes belonging to necessarily different > >> numbers. > > > That is correct, but only establishes the fact that no actually > > infinite path can be distinguished from all rational paths as should > > be possible in a Cantorlist  but is not. > > WM failed the science lesson again today. > > The Cantor argument is an argument scheme. > > It presupposes a standard, classical use of > of the quantifier "all". > > WM has never defined his nonstandard uses > for the word "all". It has no agreed upon > usage. It is meaningless by WM's own standards > of meaning through pragmatic agreements between > language users. > > By definition, all paths in the complete infinite > binary tree are infinite whether or not they > become eventually constant. > > Any purported countable listing of all the paths > of that tree will result in a successful > defeat of the claim by a Cantor argument.
A breadthfirst traversal of the paths, as the level of the tree goes to infinity, is the same as depthfirst of the nodes, for the infinite tree, for each path, and each node, in reading out that paths as nodes, as paths. The depth first traversal goes through nodes in a transfinite sequence, as it were, that is the same as the readout of the nodes of the paths of the breadthfirst traversal of paths, as modeled from the finite, in exhaustion, in the infinite.
And, it is very similar to n/d, with n, d e N and d > oo, n>d (or, "EF"), in this manner: unfilled nested intervals and the antidiagonal result don't follow from the premises.
And besides that rays through countable ordinal points are dense in the paths, of the infinite balanced binary tree (and ordinals are well ordered). This is similar to the notion of that the rationals are dense in the reals: that for other sets dense in the reals (countable or not): the rationals are dense in those, and stronger: for the ordinal points as wellordered, each is identified with a distinct path, here regardless of what its elements as nodes are, except: first, and last.
Then transitively that 2^w <> P(N) and P(N) </> N that N </> 2^w, due the settheoretic Cantor's theorem of set and powerset (where even infinite sets are _axiomatized_ to be wellfounded and the universe doesn't exist), well, with ubiquitous ordinals that S of Cantor's theorem is {} (not containing any elements of the set) for n > n+ 1 , again there are systems with mapping the continuum of the naturals through the sweep of the continuum of the segment, that it is 11, onto, and furthermore constant monotone: and where it is not onto if not constant monotone or sweep.
So: draw a line. Cantor's theorem for reals is simply read as that the drawing of the point is through the stroke, not each mark, for each mark besides beginning or end as of the others: of the drawing, of the points, of the line: all of them.
The drawing, the stroke, the sweep of the line, is through all, of infinitely many: only at once.
There are proofs that irrationals exist without the uncountable. It would be of tremendous interest to many to find application solely due transfinite cardinals. To accommodate the results of real analysis matching geometry: countably additivity of no greater than infinitesimal reals, to accommodate the existence of a universe: non sets in set theory, it's easier to find nonapplication, and eschewal, of transfinite cardinals: for results.
Then, with regards to the universal quantifier: sometimes: transfer carries. That is where: "for each" _is_ "for all". This with regards to "the" universal quantifier, here over sets.
So: make a science lesson today.
Regards,
Ross Finlayson

