Re: Well Ordering the Reals


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

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Relevant Pages

  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >> Randy Poe said: ... we have a set of self-consistent axioms. ... everything is provably true and false. ...
    (sci.math)
  • Re: Peanos second axiom.
    ... primitives in PA. ... that contain other axioms beside these. ... To me it is not a waste of time. ... by the way what does Sheesh ...
    (sci.logic)