philosopher wrote: ↑Mon May 20, 2019 8:53 pm
PeteOlcott wrote: ↑Mon May 20, 2019 8:46 pm
philosopher wrote: ↑Mon May 20, 2019 8:40 pm
Sorry, but this has already been attempted by Paul Finsler, Ernst Zermelo and Ludwig Wittgenstein in the early and mid-20th century, without success.
I have faith in that they - all of them great mathematicians - have constructed better systems than any 21st century internet discussion members.
I'm also sure if there is ever found anything that even hints at refuting Gödel's Incompleteness Theorem, then it will be all over the media and the foundation of modern mathematics (which relies heavily on Gödels Incompleteness Theorem) will be shaken and teared apart.
You can't prove a solid theorem false in just one sentence. It is ridiculous to think you can do so.
Since I have proven two theorems false with this one sentence:
If the notion of true is defined a provable from axioms and axioms
are defined to be finite strings having the semantic property of
Boolean true then any expression of language that is not provable
is not true.
Your mere disbelief to the contrary provides no rebuttal at all.
In short plain English your sentence says:
"You can prove any axiom to be true or false."
Not really not at all.
Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015):28
1.4 An Axiom System for the Propositional Calculus page 27-28
A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
If Γ is the empty set ∅, then ∅ ⊢ C if and only if C is a theorem. It is customary
to omit the sign “∅” and simply write ⊢ C. Thus, ⊢ C is another way of asserting that
C is a theorem.
It is exactly: ⊢ C with the additional stipulation that Axioms are actually true,
not this conventional true-ish:
http://mathworld.wolfram.com/Axiom.html