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Topic: Why Mathematics contradicts set theory
Replies: 41   Last Post: Jan 21, 2014 11:29 PM

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 Michael F. Stemper Posts: 125 Registered: 9/5/13
Re: Why Wolkenmuekenheim fails to understand set theory
Posted: Jan 21, 2014 7:13 PM

On 01/21/2014 05:56 PM, Virgil wrote:
> mueckenh@rz.fh-augsburg.de wrote:

>> On Tuesday, 21 January 2014 05:41:54 UTC+1, Virgil wrote:

>>> WM must be using a highly non-standard definition of countability if he
>>> claims that sets which cannot be listed can be counted.
>>> In standard mathematics "countable" and "listable" are equivalent.

>
>> A set that is not uncountable but cannot be listed is not a set in standard
>> set theory.

>
> In standard mathematics and standard set theory, countability and
> listability are equivalent properties, and any set that has one has
> both, and any set that lacks either lacks both.
> An the presence or absence of either of these in no way limits what
> constitutes a set,
>

>> For instance the set of all real numbers that can only be defined
>> as limits.

>
> Unless one can unambiguously determine of every real whether it an be
> defined OTHER than by a limit, that property is far too ambiguous to
> define a set at all.

Of course, if you define the reals as "equivalence classes of Cauchy sequences
of rationals", then all reals are defined by limits. By definition.

--
Michael F. Stemper
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