Refuting Gödel's 1931 Incompleteness Theorem in one sentence

[PeteOlcott]

I already said that is only decides one single instance of the "impossibly decidable" generic halting problem proof pattern.

And I already told you that your algorithm can't decide its own undecidability if:
1. Your language is Turing-complete
2. You allow me to craft the input.

And you are provably incorrect in your assumptions.

[Univalence]

And you are provably incorrect in your assumptions.

We have established empirical standards for deciding this. There is no need to stoop down to formalism.

I await your algorithm.

[PeteOlcott]

And you are provably incorrect in your assumptions.

We have established objective standards for deciding this.

I await your algorithm.

Great how about in the mean time we go back to my basis for refuting Gödel and Tarski?

If the notion of True(x) is defined as provable from axioms and axioms are stipulated
to be finite strings having the semantic property of Boolean true then every expression
of language that not a theorem or an axiom is not true.

One guy in another forum said this:
"I agree with you that if your definitions are accepted then we can eliminate incompleteness."

Thus indicating that I finally (after 22 years) met the threshold of sufficient clarity.

http://liarparadox.org/index.php/2019/0 ... cal-logic/

[Univalence]

If the notion of True(x) is defined as provable from axioms and axioms are stipulated
to be finite strings having the semantic property of Boolean true then every expression
of language that not a theorem or an axiom is not true.

http://liarparadox.org/index.php/2019/0 ... cal-logic/

Because your definition is too strict and produces false positives.

Until you actually solve the halting problem you have incompleteness to worry about.

True statements which are not deducible axiomatically are not probably true, but they can be verifiably true.
This is the P vs NP problem.

[PeteOlcott]

True statements which are not deducible axiomatically are not probably true, but are verifiably true.

Within my specification (and the way that conceptual truth actually works) this is impossible.

[Univalence]

Within my specification (and the way that conceptual truth actually works) this is impossible.

I don't care about your artificial specifications.

There is more to truth than formalism.

Prove that you love your family.

[PeteOlcott]

Within my specification (and the way that conceptual truth actually works) this is impossible.

I don't care about your artificial specifications.

There is more to truth than formalism.

Prove to me that you love your family.

Conceptual truth is nothing more than an interconnected set of semantic tautologies.

https://en.wikipedia.org/wiki/Soundness
A system with syntactic entailment ⊢ and semantic entailment ⊨ is sound if for any sequence
A1, A2, ..., An of sentences in its language, if A1, A2, ..., An ⊢ C, then A1, A2, ..., An ⊨ C.
In other words, a system is sound when all of its theorems are tautologies.

To the extent that your test case involves subjective value judgement it is not
an example of truth. Whether or not I have more than zero love for my family
(rather than a subjectively assessed threshold of sufficient quantity of love) is proven
by any loving act that meets the axiomatically specified properties of loving acts.

[Univalence]

To the extent that your test case involves subjective value judgement it is not
an example of truth.

OK, but by your own critrerion of provability "Pete Olcott loves his family" is not true.

Whether or not I have more than zero love for my family
(rather than a subjectively assessed threshold of sufficient quantity of love).

Except I stated the question so that you can DECIDE where to draw the line on a sufficient quantity for "truth"

[PeteOlcott]

To the extent that your test case involves subjective value judgement it is not
an example of truth.

OK, but by your own critrerion of provability "Pete Olcott loves his family" is not true.

Whether or not I have more than zero love for my family
(rather than a subjectively assessed threshold of sufficient quantity of love).

Except I stated the question so that you can DECIDE where to draw the line on a sufficient quantity for "truth"

That I have more than zero love for my family can be proven by a single instance of a loving act.

That I have sufficient love for my family is a matter of opinion and thus outside of the scope of facts and truth.

it is like trying to mathematically prove what the "best" flavor of ice cream is, subjective thus not objective thus not truth.

[Univalence]

That I have more than zero love for my family can be proven by a single instance of a loving act.

Insufficient. Love also implies the absence of hateful acts. Prove to me that you stopped beating your wife.

You are no longer in the domain of deduction.

[commonsense]

Just an observation.

If P.O.’s refutation of conventional patterns of proof refutes anything, it only shows that there are some patterns of proof that will not refute decideability.

No matter how many counterexamples are provided, this does not mean that there is not at least one example where decideability cannot be determined.

If there is at least one instance where decideability cannot be determined, then Gödel’s idea of incompleteness has not been refuted.

P.O.’s refutation does not prove that conventional patterns fail to demonstrate indecideability; it only shows that there are some conventional patterns of proof that do not show that there are no expressions that cannot be decided.

This is trivial and does not advance any undertaking to refute incompleteness.

[Univalence]

If P.O.’s refutation of conventional patterns of proof refutes anything, it only shows that there are some patterns of proof that will not refute decideability.

Precisely. The general case of the halting problem is conceptual. Individual proofs are mere instances of the intuition. There are very many ways to arrive at the result.

No matter how many counterexamples are provided, this does not mean that there is not at least one example where decideability cannot be determined.

The counter-examples themselves will have non-halting states. Because they will be Turing machines.

[commonsense]

No matter how many counterexamples are provided, this does not mean that there is not at least one example where decideability cannot be determined.

The counter-examples themselves will have non-halting states. Because they will be Turing machines.

Exactly. I saw this in your earlier posts.

[PeteOlcott]

Just an observation.

If P.O.’s refutation of conventional patterns of proof refutes anything, it only shows that there are some patterns of proof that will not refute decideability.

No matter how many counterexamples are provided, this does not mean that there is not at least one example where decideability cannot be determined.

If there is at least one instance where decideability cannot be determined, then Gödel’s idea of incompleteness has not been refuted.

P.O.’s refutation does not prove that conventional patterns fail to demonstrate indecideability; it only shows that there are some conventional patterns of proof that do not show that there are no expressions that cannot be decided.

This is trivial and does not advance any undertaking to refute incompleteness.

When true is defined as deductively sound conclusions:
(1) True is ALWAYS defined.
(2) Everything else is defined as untrue.

Here is the one sentence version:
If the notion of True(x) is defined as provable from axioms and axioms are
stipulated to be finite strings having the semantic property of Boolean true
then every expression of language that not a theorem or an axiom is not true.

Within the SOUND deductive inference model undecidability is categorically impossible.
Every expression of language that would otherwise be classified as undecidable
is instead decided to be untrue.

[Univalence]

Every expression of language that would otherwise be classified as undecidable
is instead decided to be untrue.

Does untrue mean the same as false?