Scott Mayers wrote: ↑
Wed May 15, 2019 9:32 pm
PeteOlcott wrote: ↑
Wed May 15, 2019 9:01 pm
Scott Mayers wrote: ↑
Wed May 15, 2019 8:45 pm
Okay. I give up on you on this quest. It is like one asserting they can prove the Pythagorean Theorem as false. I'm not sure what you're expecting but while the theorem speaks of incompleteness, it is precisely only about limitations upon using a binary-only system of logic based on 'true/false' factors. I already agree that there are ways around any difficulties of the limitations. But the limitations in reality, apart from those incorrectly interpreting the theorems as making logic itself out to be faulty, are superficial with respect to the whole.
You only have to take a few minutes and trace through this simple reasoning to confirm that I am obviously correct.
Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015) Pages 27-28
The notion of complete and consistent formal systems is exhaustively elaborated as conventional formal proofs to theorem consequences where axioms are stipulated to be finite strings with the semantic property of Boolean true.
// LHS := RHS the LHS is defined as an alias for the RHS
∀x True(x) := ⊢x
∀x False(x) := ⊢¬x
Because valid deduction from true premises necessarily derives a true consequent we know that the above predicate pair consistently decides every deductively sound argument.
You don't make sense here at all. I don't know if you are being real and it appears that you MAY be just playing people here. I notice that if you discover someone like myself use some logical argument elsewhere, you opt to select a more obscure one you hope the readers couldn't follow but make it seem that you know what you are talking about.
Let's see, YOU upturn Godel's intensely argued theorem in only a mere few minutes of reading? And though you fail constantly, you keep up the same tactic.
WARNING TO OTHERS: DO NOT USE THIS GUY'S LINKS.
There is something wrong here and it CAN be a "honeypot" being used to DROP something on your computer! If this guy cannot argue here and ONLY here without an insistence on his links, there is something amiss. Pete, you keep begging you have powerful proof of something but cannot competently express this to me.....someone who DOES have a lot of background on this and STILL can't understand you.
Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
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”...
I only have to use these links to avoid violating copyright law.
If you really do understand formal proofs of mathematical logic
then you don't need to go to these linked textbook pages.
It you can't understand what I mean by formal proof by the above paragraph alone
then you simply lack too much of the required prerequisite knowledge.
The textbook pages take someone all the way through the whole process
of learning everything about formal proofs of mathematical logic.