Re: how to list all of the real numbers

In article <1188193676.421234.264380@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> wrote:


Are you asking me? Others here would probably be familiar with the
sketch of the argument that any theory containing all elements that ZF
contains would find that the domain of discourse of ZF, its
"universe", is the Russell set and thus contains itself, so, there is
a universe in ZF, its own. (ZF is inconsistent.)

ZF may have a "universe", but there is no way in ZF to establish that
that universe is a set.

NBG has a universe wish is specifically not a set.


Pedagogically it is acknowledged that the universe is not deemed a set
in regular set theories (those with the axiom of regularity). While
that may be so, the set-theoretical universe is generally introduced
for fundamental considerations of set theory, "naive" set theory.

So don't be naif.

That is to say, in classrooms around the world the universe and sets'
complements within the universe (domain of discourse) are considered
as sets, casually and usually without complication. (If you find that
offensive please do.)

Until students newly being introduced to sets have some idea about what
sets can do, it is pedagogically foolish to rub their noses in what sets
do not do.

And naive set theories in which everything is constrained to some
relatively small set as a universe, as in, say, Euclidean plane
geometry, is quite valid, both pedagogically and mathematically.
.

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