Re: How many real numbers are there?

Mike Deeth wrote:
Jonathan Hoyle wrote:

<most of rant snipped>

is countable. Let r be a real number. By our above observations, r is
just some ZF set, and hence r in S_i for some i, hence S in T.

(snip)

You appear to be confusing the theory (ZFC) with your model (R). You
realize that ZFC does have both countable and uncountable models, don't
you?

Jonathan Hoyle


Nowhere in the constructions of the real numbers do we make any
assumptions that we are in some exotic set theory with supernatural
powers. This is not X-Men mathematics! The set T which I contructed
in my original post satisfies all the axioms of ZF and therefore the
contructions of the reals work fine therein. For those who naysay
because of choice, it would be trivial to alter my definition to
include that as well. Please stop picking on strawmen and admit the
deep flaw in the Cantor Cabal.

You have not proved that every set (or even every real number) is in some S_i. In fact, you yourself construct a set not in any S_i, namely T. Therefore you have not found any flaw in set theory.

Your mistake is thinking all real numbers are constructible.

--
Eric Schmidt

--
Posted via a free Usenet account from http://www.teranews.com

.

Relevant Pages

  • Re: Platonism
    ... I said that they'd retreat to ZFC when pressed. ... To most mathematicians, there is something very ... The set of reals is not hard to construct in ZFC, ... that provide intuition, and better intuition is that which avoids ...
    (comp.theory)
  • Re: Platonism
    ... I said that they'd retreat to ZFC when pressed. ... To most mathematicians, there is something very ... The set of reals is not hard to construct in ZFC, ... that provide intuition, and better intuition is that which avoids ...
    (sci.math)
  • Re: Constructions of Real Numbers
    ... (how one verifies the field properties of the reals when ... issues related to various constructions of the real numbers, ... Southeast Asian Bulletin of Mathematics 1, ... Arnold Knopfmacher and John Knopfmacher, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >> Anders Goeransson says... ... First note that all the axioms of ZFC holds in L. ... Now M contains all reals and is an ... of the naturals. ...
    (sci.logic)
  • Re: computation and cardinality
    ... so William is just right and you are just wrong. ... There is a model of ZFC in which the 0-ary functor 0 ... INDEPENDENT terms of some LOGICAL piece of language (possibly ... or didn't get all the wide world of reals into THAT LARGER ...
    (sci.logic)