On 12 Jun., 06:38, Owen Jacobson <angrybald...@gmail.com> wrote: > On 2009-06-11 16:49:49 -0400, WM <mueck...@rz.fh-augsburg.de> said: > > > On 11 Jun., 22:31, Virgil <virg...@nowhere.com> wrote: > >> In article > >> <1348b29b-3f39-40a7-be0e-5bd24a4e3...@s21g2000vbb.googlegroups.com>, > > >> WM <mueck...@rz.fh-augsburg.de> wrote: > > >>> I need no axioms. > > >> Then there is absolutely nothing that you take to be true? > > >> There are absolutely no truths from which you can deduce other things? > > > There are truths, like 1 + 1 = 2, or, for linear sets, ExAy <==> AyEx > > These "truths" are the axioms of the theory you're using, or are > implied by them.
I do not deny that. But I deny that axioms can be chosen. > > > Incorrect are such lies as the axiom of infinity or the axiom of > > choice. > > The mathematics community mostly does not pursue a universally true set > of axioms.
But mathematics does.
> Instead, mathematics takes sets of axioms and attempts to > demonstrate that they lead to interesting results.
Here is an interesting quote by Weyl (1946): "We accept the hierarchy of types; but we assume only one category of primary objects, the numbers; and one basic binary relation between numbers, namely "x is followed by y." All other relations of the various types are explicitly constructed, the quantifiers (Ex) and (Ax) being applied only to numbers and not to arguments of higher type. No axioms are postulated."
Does he no longer belong to the math. community?
> The results derived > from a given theory (a set of axioms) are only valid within that > theory.
Mathematics (if done correctly) is true without any theory.
> There are informal "families" of theories, within which the > informal definitions of various terms are portable (for example, there > is a family of "set theories," each of which has a loosely similar > definition of what a "set" is), but informal similarity does not mean > that proofs from theories in the same "family" can necessarily be used > together.
All transfinite set theories are nonsense, formal or informal. > > ZF set theory is no more "true" or "false"
It is false, no doubt. A man who believes in or even attempts to prove something by means of "higher cardinals" is a fool. > > So it is with the idea that all sets are finite. The ZF axiom of > infinity is only the contradiction of the axiom "there are no infinite > sets," and the resulting theory is both consistent and useful.
That is wrong. Not a single useful bit has ever been obtained from finished infinity.
> There > are also consistent, useful theories that do not prove or require the > existence infinite sets.
This is a prerequisite of a consisten theory like the convergence of a sequence to zero, shall the due series converge. > > If you would only name the theory you're using, or sketch out its > fundamental truths (that is, its axioms), you'd probably find people a > lot more receptive to your ideas.
I am doing simply correct mathematics, at least I am striving for that goal.