> On 14 Dez., 05:54, Owen Jacobson <angrybald...@gmail.com> wrote: > >> The key feature of this systems is that the strings we manipulate under >> it *have no intrinsic meaning* - > > Mathematics has an intrinsic meaning. > And if your strings have lead you to the result that there are more > natural numbers than any natural number can indicate, i.e., there are > m natural numbers with m > n for every natural n, then your strings > have lead you astray. > > And please, do me the favour and do not ask me where the > error lies. I am not interested in a system that, according to its > result, has been proven in conradiction with mathematics.
You've missed my point.
You've made it abundently - no, obesely - clear that you don't like some of the theorems that are entailed by ZFC's axioms under a particular (and particularly common) assignment of intuitive meanings to the symbols involved, and that's fine in as far as any aesthetic judgement is fine. However, the tools I'm describing - a deeply formalistic and mechanical view of mathematical logic, at the level where a machine could carry it out - are incredibly powerful and flexible and you're absolutely free to use to build the mathematical universe you prefer.
I think you'd be better off using your time to build up the axioms and working out the so-entailed theorems describing what you think is "real" math: for the sake of clarity, let's call it WMT, for WM's Theory. It's on you to describe the axioms of WMT, upon which every theorem of WMT is entailed, in formal language so that subsequent proofs can be unambiguously verified. It's also on you to kickstart your own research program that explores the logical space grounded in your axiom set the way Metamath has explored first-order logic's and ZFC's.
If it turns out that WMT appears to be formally consistent, and if it entails something recognizable as finite arithmetic, and also entails the non-existence of infinite sets, and if it turns out that WMT's axioms are formally independent* of the axioms of any other established theory, then you might be onto something interesting to other mathematicians and to other posters in this newsgroup. Don't count on a massive amount of buy-in from others right away: you've done a fantastic job of building yourself a reputation as an argumentative bag of vinegar, and it'll take a lot of work before you establish yourself as a competent mathematician, but dedication will win out eventually.
Alternately, you can keep going around the same five examples with the same five people for the rest of your literate life. It's your choice.
Good luck!
-o
* That is, collectively neither entailed by nor entailing the axioms of some other theory.
