Re: Extrapolating linear ratios

On Dec 19, 5:08 pm, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
Cantor's Theorem is a theorem OF SET theory!
In that context, EVERYTHING is a set!

With exception of the set of everything. So not everything is a set.

If you want the universal then just migrate UP one from set theory
to CLASS theory. THERE, EVERYthing is a class, INCLUDING the
universal class.

INCLUDING NUMBERS!
Since numbers ARE sets,

There is not the least definition or axiom "what" a set is.

EVERY axiom of set theory is telling what a set is.
But under this paradigm, ANYthing can "be" a set.
Your left big toe could be the empty set if you wanted it to.
The theory DOESN'T CARE.

There is no hint in the axioms of set theory that numbers exists at all,

Of course not, but people who USE set theory to reason about or with
numbers (and EVERYbody uses SOMEthing) HAVE DEFINED some sets
TO BE numbers. And I'm sorry you didn't know this, but you CAN'T
ARGUE
with a DEFINITION.

In particular, set theory DOES not only HINT but PROVE that the empty
set exists.
So it is hardly odd to let THAT set represent zero.

let alone cardinal numbers.

***.
Obviously, the empty set can easily represent the cardinal zero.
After that, we just DEFINE cardinals as initial ordinals.
In order for that to make sense, you have to know what an ordinal is.
Zero is an ordinal.
The successor of any ordinal is an ordinal.
So 0={ } and 1={0} and 2={0,1} and in general, the finite ordinal n =
{0,1,2,...,n-1}.
All these finite ordinals ARE ALSO cardinals, i.e., ARE NATURAL
NUMBERS.
By *DEFINITION*.
Now, of course, these are not definitions of THE mystical zero, or
one, or two,
or the holy trinity. But if you think those things exist then THESE
ARE SET
THEORY'S NAMES for them. There are many different systems of
NUMERATION
in the world and these are Set Theory's NUMERALS for these numbers.
You should, after all, also bear in mind that 0,1,2, etc. are numerals
rather than
numbers and the the Chinese and the Hindus (not to mention the Romans
and the Greeks) DON'T spell them this way.

"there exists a set" IS
sufficient to conclude that that set has a cardinal number,
especially if that set *IS* a cardinal number!

Who does endorse that rubbish?
What axiom identifies a set with a
cardinal number?

IT's NOT an AXIOM! It's a DEFINITION!
The word "cardinal number" DOES NOT APPEAR in the axioms of
set theory! We just PICK some sets and CALL them cardinal numbers!
JUST CUZ WE *FEEL* like it.
And everybody who, UNlike you, ACTUALLY KNOWS SOME MATH,
LETS us do this because ALL THE THEOREMS YOU THOUGHT you knew
regarding cardinal numbers as presented IN ANY OTHER fashion
GO THROUGH under this encoding and this axiomatization!

Of course, you may have thought you had some other theorem in mind,
e.g., For Every Infinite set, if there is a predicate that is true of
every
finite non-empty subset of it, then that predicate must be true of the
infinite
set as well, but "WM's theorem", unlike Cantor's, CAN be refuted by
logic
BECAUSE IT IS EASY TO DERIVE *CONTRADICTIONS* from it.
Which you so far HAVE NOT with all your ranting about how "There can't
be
more" intervals. THERE ARE NOT more; there are THE SAME (cardinal)
NUMBER.
There can, however, be ORDINALLY more. Of course, how two sets can
have the
same cardinality yet different "ordinality" is something that you,
being too stupid
to understand, would simply(and wrongly)dismiss as "a contradiction".


Under this paradigm,

Who has stated that paradigm?

Zeremelo, Fraenkel, Cantor, Cohen, ad nauseam.

It's time for changing the pardigm.

It's a FREE country, DUMBASS!
ANYbody can use ANY paradigm they want!
But you CAN'T go around telling people that they CAN'T use a paradigm
they want
to use! THAT is what is NOT allowed! If you want to do THAT then THE
ONLY
way you can do it is BY DERIVING A CONTRADICTION from THEIR paradigm!
Or, in a pinch, by solving some important class of problems more
elegantly
than theirs. But in your case, we're not holding our breath, since
you just
LIE all the time and then claim to have found contradictions where
none exist.


EVERYTHING, INCLUDING CARDINAL NUMBERS,
is REPRESENTED or ENCODED as a set.

Nonsense! Everything is not a set.
Or does it contain its power set?

DIP***: "Everything" refers to each of MANY DIFFERENT things.
No, NO individual thing contains its own powerset.
You are intentionally mis-using every AS "all" here AGAIN.

 In particular, every set
representing a cardinal number simply IS its own cardinality.

Including the cardinal number of its own power set.

Oh, BULL***. No set is EQUAL to its powerset, so obviously
no cardinal is also the cardinality of its own powerset -- that is
always,
AS CANTOR'S THEOREM PROVES, a BIGGER cardinal. It is specifically
2^(the original smaller cardinal).
.

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