Re: Attempts to Refute Cantor's Uncountability Proof?

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?

.

Relevant Pages

  • Re: Attempts to Refute Cantors Uncountability Proof?
    ... Consider Goedel, vis-a-vis incompleteness, and the physicists' notion ... no "Theory of Everything" in ZF or other regular set theories. ... arithmetic is complete where Peano arithmetic is incomplete. ... other is inconsistent or incomplete, ...
    (sci.math)
  • Re: Godel misuses ZF
    ... Goedel proves in his paper (in PRA) that if ZF is ... omega-consistent it is incomplete ...
    (sci.logic)
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