Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?

On Wed, 15 Aug 2007 21:10:58 -0000, Scott <ToaTerra@xxxxxxxxx> said:
On Aug 15, 10:52 am, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Tue, 14 Aug 2007 20:49:08 -0000, Scott <ToaTe...@xxxxxxxxx> said:

...
Wow, you actually have a good point here. Yes, there does not seem to
be a language about describing formal systems.

You're quite wrong about that. Formal systems theory can be done
entirely in the language of set theory. Like most mathematics, in
most (English) texts, it is actually done somewhat informally in
"mathematical English", i.e., English supplemented with mathematical
constructs. But that is purely for pedagogical reasons.

I was not aware of this.

Indeed.

Can you show me how set theory handles a formal system capable of
expressing and manipulating "{ x : x notin x }", both in terms of
symbols and its contents?

Well, provided you're talking about a coherent formal system, it would
do the obvious things: it would "handle" the symbols by defining the
syntax of your system and it would "handle" its content by defining its
semantics. Here's a simple example: von Neumann-Bernays-Gödel (VNBG)
set theory without the axiom of infinity. The language of this theory,
as with any formal language, is easily defined recursively in ZF set
theory. The theory itself can be shown in ZF to have a model in which
(i) set variables range over V_omega, the set of all finite sets, and
(ii) class variables range over, and class terms are interpreted in,
V_omega+1, the set of all sets of finite sets. It is easily provable in
ZF that, in this model, "{x : x notin x}" denotes V_omega.

If you'd study a little mathematical logic and set theory you'd be able
to figure all of this elementary stuff out for yourself.

.

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