Re: Uncountable sets in CZF?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 09/08/04 

Date: 7 Sep 2004 18:24:33 -0700

kramsay@aol.com (KRamsay) wrote in message news:<20040907121638.22164.00000614@mb-m05.aol.com>...
> In article <3c6b9c1e.0409030813.584832fe@posting.google.com>,
> raf@tiki-lounge.com (Ross A. Finlayson) writes:
> |Keith presented a statement that he could map a proper subset of the
> |naturals bijectively to the reals. What's the deal with that?
>
> Do you think you're quoting me at all accurately here? You do agree
> that you should attempt to, don't you?
>
> Keith Ramsay

Yes, I think you claimed there was a surjection from some proper
subset of N onto R, and through Cantor-Schroeder-Bernstein as there is
a trivial surjection from R onto any subset of N there is a bijection.

You say specifically that "it doesn't follow that there is a
bijection." Yet, it necessarily does, until you present some disproof
or negation of the Cantor-Schroeder-Bernstein theorem in that context.

I don't base my arguments (that the reals and naturals are equivalent)
upon what you said, I haven't seen your explanation of a surjection
from some proper subset of the naturals to the reals, and I have my
own explanations for why the naturals can biject with some proper
subset of the reals.

Apparently, so do you. I think that's good, and progress.

Regards,

Ross F.

Relevant Pages

  • Re: Uncountable sets in CZF?
    ... It doesn't follow that there's a bijection. ... > naturals to the reals. ... What that means is that one of the reasons that people call the reals ... That implies it is not a mathematical fact and to promote the other ...
    (sci.math)
  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >>> A proper subset IS smaller than the superset, ... >> When TO claimed that the sets of rational and reals are of the same ... > that they are certainly NOT an equivalent set to the naturals. ...
    (sci.math)
  • Re: Cardinality of Real Numbers
    ... Cantor's first assumes the existance of a bijection between the ... >> natural numbers and the reals. ... From this, a contradiction is reached ... >naturals to the reals, and shows that there is some real not in the ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... What is "a function which is a bijection"? ... exists another function (algorithm), f^-1, with the property for all b ... well-defined (exists for all naturals n, m) and S is a bijection - ... of reals, S". ...
    (comp.theory)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... What is "a function which is a bijection"? ... exists another function (algorithm), f^-1, with the property for all b ... well-defined (exists for all naturals n, m) and S is a bijection - ... of reals, S". ...
    (sci.math)