Re: how to list all of the real numbers

On Aug 27, 7:56 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1188260188.635917.246...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@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.


A discussion point is that the collection of sets that comprise ZF, in
any supertheory containing more sets than ZF, would be the Russell set
and irregular.

It seems you admit that the theory ZF is a collection of all the sets
of ZF.

Consider that with regards to statements about existential quantifiers
being statements of elementhood, elementness, in ZF as a set, i.e.,
"exists x" means "x is an element of ZF."

Then, when there are considerations of sets of elements, any given
particular subdomain of discourse would be a subset of ZF, the set.

Modelled (technically) by the cumulative hierarchy, ZF is thus a
collection with "element" and "subset", defined by its elements: a
set.

Then, it is the Russell set, and contains itself.




(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.


Yeah, there isn't. Your insult doesn't seem a reasonable statement.



(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.

No, it's not.

EF is a function. Half of the integers are even.

Ross

--
Finlayson Consulting

.

Relevant Pages

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