On 2009-06-11 16:49:49 -0400, WM <mueckenh@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.
> 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. Instead, mathematics takes sets of axioms and attempts to demonstrate that they lead to interesting results. The results derived from a given theory (a set of axioms) are only valid within that 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.
ZF set theory is no more "true" or "false" than it is "blue" or "grape-flavoured" ? it is only a set of axioms, from which results that are, most of the time, consistent with intuition can be derived. It also implies some results that are less intuitive, like the existence of more than one size of infinite set. There are other theories which imply that there are no infinite sets at all. Both are useful; neither is more right than the other, and the existence of infinite sets in ZF is not "inconsistent" (in the technical sense of the word) with the non-existence of infinite sets in another theory.
The pursuit of a single "true" set of axioms is at best an exercise in philosophy, not mathematics, and at worst a grave misunderstanding of how math works. Fortunately, you are not alone in this search; there are others posting to sci.math who believe that there is a "true mathematics."
>> That makes for a very simple mathematics in which nothing need be true. >> >> Consider axioms like in the American Declaration of Independence: >> >> "We hold these truths to be self evident..." >> >> Unless WM can list the things he "holds to be self evident", he does not >> have anything to build on. > > There is no necessity to list them in mathematics. Every sober mind > knows them.
If that were true, this thread would've ended years ago.
You really do have to get your head around the idea that the people you're talking to aren't idiots, don't have any personal animosity towards you or your ideas, aren't yanking your chain (much), and quite honestly don't share your intuition. The repeated requests by some posters that you break your claims down into very small, easily digested pieces are not attempts to waste your time, but rather (mostly) honest attempts to understand the rules of the intellectual framework you're using to make your pronouncements.
Mathematicians, by long and painful experience, generally distrust "intuitive" truths. The parallel postulate is intuitively true, but its contradiction is not only mathematically consistent but extremely useful: geometries without the parallel postulate are used to model the way light and matter interact. Similarly, it was once intuitively true that all numbers could be expressed as the ratio between two whole numbers, but the contradiction of that axiom is (again) both mathematically consistent and useful.
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. There are also consistent, useful theories that do not prove or require the existence infinite sets.
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.
