Re: Well Ordering the Reals

Randy Poe said:
>
> Tony Orlow wrote:
> > Randy Poe said:
> > > So if you declare something like "all sequences contain their limits",
> > > then you can prove it pretty easily. Since that will give a result
> > > which is both provably true and provably false, it might cause
> > > other people problems. But you've shown in the past you have
> > > no problem accepting such things.
> > Of course I do, and this is exactly what I was saying about axiomatic systems.
> > You can prove anything just by declaring it as an axiom, but that's just a
> > waste of time.
>
> And leads, as here, to self-contradiction.
>
> > Axioms must be justified logically based on more fundamental
> > facts.
>
> Nonsense.
>
> > But, you're not making much sense here.
>
> No? Let me try again: we have a set of self-consistent axioms. You
> can't prove anything to be both true and false within that system.
>
> You have several different new axioms YOU want to add, any one
> of which will introduce contradictions in the system that weren't there
> before.
>
> Without your additions, we can prove certain things true and other
> things false, but nothing is both true and false.
>
> With your additions, everything is provably true and false.
>
> So there's no incentive for your additions. As you say, a waste of
> time.
>
> - Randy
>
>
Actually, most of the standard axioms would get scrapped regarding anything
transfinite. I don't think this foundation can be properly fixed without
removing the rot, or we'll never get rid of the bugs or the cracks in the
walls. Of course, I understand you're attached to the sill and the post, having
become part of the infrastructure, and I apologize, but promise they will be
recycled appropriately.
--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

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