Re: infinity

In article <MPG.1da4a2efd403d42898a3aa@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> David R Tribble said:
> > Tony Orlow wrote:
> > >> Are you distinguishing between strings of length N and strings
> > >> of length 2^N? Do you not have, in your theory, the same
> > >> unending set of infinite bit strings?
> > >
> >
> > David R Tribble said:
> > >> ... as the infinite members of *N? Yes. There are (surprise!)
> > >> 2^aleph_0 (2^N) of them in *N. But P(*N) has 2^2^aleph_0
> > >> (2^2^N) members.
> > >
> >
> > Tony Orlow wrote:
> > > Yes, you can do a little limited math with aleph_0, although it's
> > > not quite correct. Your N is the size of the set of finite
> > > naturals, yes? And *N includes the infinites? There are
> > > infinitely more infinite naturals than finite ones. 2^N is
> > > infinitely smaller than N*.
> >
> > Actually, card(*N) = 2^aleph_0. So *N has exactly 2^N members in
> > your terminology. That's why I can define a bijection between *N
> > and the reals in [0,1] or between *N and P(N).


> Prove that *N has 2^N members. It doesn't

Claims without valid proofs from TO should be ignored as the
foolishness they are.


> That assumes an infinite
> number of natural finites.

Dedkind infiniteness of the set of finite naturals is an immediate
consequence of the successor function on the set of finite naturals.

And that is the only infiniteness needed.


> The number of infinite naturals is
> infinitely larger than the number of finite naturals

Except that the set of finite naturals is infinite enough for all
standard mathematics, and TO's version of "infinite naturals" have no
existence outside TOmatics. Even Robinson's "infinite naturals" behave
quite differeently.


, standard theory
> notwithstanding. This is just another example of where your axioms
> shoot from the hip and hit their own feet.
> >
> >
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... and therefore never becomes infinite as long as the elements ... >> finite naturals, you say the size is larger than all finite naturals, so it ... > Did those infinite members of N disappear when you removed the 1? ...
    (sci.math)
  • Re: infinity
    ... Tony Orlow writes: ... > Daryl McCullough said: ... the set is unbounded but not infinite. ... >> What is the size of the set of all finite naturals? ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... or the value of the string I am discussing. ... There is no point at which any finite natural has an infinite ... They ARE the finite naturals. ... At which point has my abstraction differed from what you're ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Randy Poe said: ... Do any of your naturals achieve any infinite values? ... or the value of the string I am discussing. ... They ARE the finite naturals. ...
    (sci.math)
  • Re: infinity
    ... there is no exact size of the set. ... >>> not matter whether N is finite or infinite, ... >> resulting set suddenly only has half the number of members. ... Thus, for the set of finite naturals, which does have a smallest, to ...
    (sci.math)
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