Re: Attempts to Refute Cantor's Uncountability Proof?


David R Tribble wrote:
Ross A. Finlayson wrote:
There's only one theory with no axioms.


Jonathan Hoyle wrote:
Correct. It is the one with no theorems.


Ross A. Finlayson wrote:
No that's not what it is.

Consider Goedel, vis-a-vis incompleteness, and the physicists' notion
of a "Theory of Everything". Apparently, those people never heard of
Goedel, or didn't agree that his results about incompleteness hold in
their case, because they talk about a "Theory of Everything."

You're confused. The physicists' TOE delas with unifying gravity and
quantum physics, and while it uses a lot of complex math, it has
nothing to do with set theory.

There is
no "Theory of Everything" in ZF or other regular set theories. There
is no universe in ZF. Quantify over sets, in ZF: it's not a set. So,
it's a non-sets theory.

If you mean there can be no "set of all sets", yes, that is well known.

There's only one theory with no axioms. It has all the theorems. Any
other is inconsistent or incomplete, just ask Goedel. Your regular set
theory is incomplete, via Goedel, and inconsistent, via universal
quantiification, not to mention paradoxes in them, generally paradoxes
of unrestricted comprehension or the Liar, or about
symmetry/antisymmetry. Incomplete means inconsistent, of a universal
theory.

There's only one theory with no axioms, the null axiom theory.

I refute.

It still sounds like gibberish.
Could you show us a theorem in this theory with no axioms?

I think that the axioms of ZF besides regularity are theorems. While
that is so, some interpretations of what the objects are lead to
differences among what you'd expect primitive objects to be. There are
some more theorems than that, that may provide a physical basis.

About a "Theory of Everything", superstrings are mathematical
infinitesimals. Technicolor is a notion that there are particles
comprising quarks and particles comprising those etc ad infinitum, yet
while that is so something along the lines of the atom is a nilpotent
infinitesimal, in a sense. The particle/wave duality is not so far
removed from, say, various vacuous statements about null vis-a-vis the
universe. Colder than absolute zero is hotter than the sun, i.e.,
infinity = negative one. That might seem rather unobvious, or not.
Points are polydimensional, for example real numbers in or on the real
number line.

The universe is everything. It's all-encompassing. Were there a
parallel "universe", it would be part of the universe also. There's
only one universe, by definition, and conveniently in expression, the
universe contains itself, it's irregular, not well-founded, as it were,
in the views of some cosmologists. The universe is infinite.

There is no universe in ZF. Infinite sets are equivalent because
they're infinite. Basically infinite sets aren't regular.

So, there is no universe in ZF. Where there are only sets in set
theory, yet no sets, there is nothing. It's only truth is in vacuously
being the null axiom theory.

I borrow the book mentioned about universal sets in set theory, I'll
look at it. That is to say: there is a "sets of all sets" considered
as parts of set theory. It is agreed that there is no set in ZF.

Infinite sets are equivalent.

Makes sense to me.

Ross

.

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