Re: Countable union of countable sets

From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 02/07/05 

Date: 6 Feb 2005 21:25:43 -0500

In article <slrnd080kj.p80.tim-via-n.i.net@soprano.little-possums.net>,
Timothy Little <tim-via-n.i.net@little-possums.net> wrote:
>I've seen stated that proving that a union of countably many countable
>sets is countable is not possible in ZF alone. What is wrong with the
>following argument?

>Definition: set X is countable iff there exists surjective f:N->X.

>If we have a set S of countably many sets then, by definition, there
>exists surjective f:N->S. Each X in S is countable, so there exists
>g_X:N->X for each. Let U = Union S.

There is a big difference between there exists, for each
X, a function h:N->X, and there exists a function g such
that for each X, g_X:N->X. 

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

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