Re: how to list all of the real numbers

In article <1188260188.635917.246960@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> wrote:

On Aug 27, 12:03 am, Virgil <vir...@xxxxxxxxxxx> wrote:

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.

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.)

Since ZF contains all the elements that ZF contains but does not have a
universe which is a set of any kind in ZF, Ross trivially proves
himself dead wrong.

(In the dually-self-intraconsistent null axiom theory the universal
set is a set, the ur-element, void's complement. Set theories where
it's not a set are not pure set theories.)

There is no ur-axiom that requires any set theory to conform to Ross'
"dually-self-intraconsistent null axiom theory" nor that any set need
have a universal compliment. Ross is off in some sort of unreal world
again.

(It is not actually novel to suggest the physical
universe contains itself, in a sense.)

But it is quite novel to suggest that the physical universe is a set
theory.
.

Relevant Pages

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