On Tuesday, June 11, 2013 10:14:46 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > On Tuesday, 11 June 2013 18:50:54 UTC+2, Zeit Geist wrote: > > > However, in set theory, if we have an axiom of infinity then we can define a function ( with finite number of symbols ) to do the task. This works very well enumerating the set of rationals, or to find the diagonal of a countably infinite list of real numbers. > > > > Why should it work less well for the aleph_0 steps of well-ordering the rations by magnitude? >
An enumeration of the rationals is a well ordering. However, no such enumeration will be with respect to magnitude.
> > Every step is well-defined such that we, you and me, when starting with the same enumaration of |Q, will up to step n (for every n) get the same result including as many rationals as we like. And if we apply the axiom we get that for all steps. >
I think your algorithm fails at the limit stage.
BTW, is this the other thread?
> > > > Mathematics is an ideal theory that doesn't not necessarily correspond directly with reality. > > > > Matheology is in direct contradiction with reality, science, and sanity. > > > > > Despite the lack of correspondence, Science finds Mathematics useful in its endeavors. > > > > Yes, but never matheology which is even less scientific as astrology. > > > > > Even the Infinite! Supertasks are not physically possible. You can't increase the efficiency of a machine, or operation, an infinite number of times ( in reality ). Supertasks are a nice heuristic theory used to explain the Infinite to people who lack the Mathematical intuition of the Infinite. > > > > Every application of the axiom of infinity generates a super task. Enumerating all rationals is but one example that is so simple that most people do not realize its character as a super task. >
A "supertask", it when you introduce the concept of infinitely divisible finite time. I guess, I just don't like them.
Why add the extra construct? Just take the natural numbers with a point at infinity appended. Everything works out just fine.