Re: how to list all of the real numbers

On Aug 26, 9:32 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1188182979.830990.26...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

On Aug 26, 4:15 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1188166938.280884.122...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

In a physical universe where physical objects are mathematical
objects, and functions between them physical objects,

You mean in a nonsense universe, as that is not the case in any known
universe.

No, I don't, and no.

Called "Cantor's paradox", that a universe as set would be its own
powerset,

In ZF and NBG there is no way to make the "universe" a set, so that no
set need be its own power set.

has basically that to resolve the antinomy either a) the
universe doesn't exist, or b) Cantor's theorem doesn't hold.

In ZF and NBG, the universe does not exist as a set. In what set theory
does Ross claim it is a set?

I prefer

to believe in existence, and many or most are accustomed to assuming a
set-theoretic universe in their set theory over which to quantify,
with, for example, properties that for a set that its complement is a
set.

In ZF and NBG, there is no "set theoretic inverse" nor "compliment".

Their union is a set.

Since "they" do not both exist as sets, their "union" need not be a set
either.


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.)

(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.)

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.
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.)

Cantor himself felt that the universe was to be an element of his
Mengenlehre, the "domain principle", and if I'm correct in my
understanding of the historical situation he kept a universe in his
theory despite the "Cantor paradox", then it was removed some few
years later in the axiomatization for reasons largely to do with that
the excision of the universe from the theory was considered a
resolution of the "Russell paradox."

So, considering some fundamental physical particles as basically point
particles, the functions between them represent the interactions
between them, as do those between theirs and etcetera ad infinitum,
so, the physical universe contains infinitely many of those types of
things, and, the functions between them are again as well part of the
same thing, thus, the physical universe is an example of self-same
set: and powerset. (It is not actually novel to suggest the physical
universe contains itself, in a sense.)

Returning to the subject of "how to list all of the real numbers", I
encourage you to consider further the natural/unit equivalency
function, in regards to the discussion as above. Is it a function?
(Axiomatic support is readily supplied.) As a function, is it a
cumulative distribution function, besides being dense in the unit
interval and having a range that is naturally well-ordered, by the
natural integers in the real numbers natural total linear ordering?

It's quite a simple thing.

Ross

--
Finlayson Consulting

.

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