The first sentence does not make sense, the truth predicate applies to sentences, not proofs. And are you asking us what the mistake might be in the Tarski's proof? That would be Math Overflow material.

@Conifold I fixed the mistake. True consequences necessarily follow from true premises and valid inference, thus always selecting only true sentences from the set of all sentences. Does this also always select all true sentences?

"Definable" means definable in the object language, "conventional formal proofs of mathematical logic" is not what one can express in typical languages of formulas for which the proofs are written. And if the truth predicate can be defined within a language then (under some richness assumptions) such a language is inconsistent. Natural languages are examples.

@Conifold An object language defines a predicate True(x) as ⊢x. If one plugs the Liar Paradox in for x (as Tarski did) we get ¬True(x).

"⊢x" is not a formula of object language, "⊢" is a meta-symbol. And this is not the right site for this topic, Math SE or Overflow are.

@Conifold here is the object language: researchgate.net/publication/…

It makes no difference what your meta-language for the purposes of Tarski's theorem.

@Conifold It is both the metalanguage and the object language. Any proof that this does not work seems to fail.

"It" is not the language Tarski used, so his theorem is not obligated to apply. And "any proof that this does not work seems to fail" is very strangely phrased. There supposed to be a proof that it does work (whatever "work" means here), and if you think there is one submit it for peer review, instead of wasting time on a Q&A site.

@Conifold Tarski's proof is a proof by contradiction, and that contradiction cannot be formed within my definitions. Tarski: "the sentence x which is undecidable in the original theory becomes a decidable sentence in the enriched theory." and decidable within the single language having my definitions.

@Conifold ""It" is not the language Tarski used, so his theorem is not obligated to apply" If Tarski proved that True(x) is not definable in mathematics, and I defined it in mathematics that would prove him wrong. Most people are unaware that his proof applies to arithmetic AND ABOVE, not just arithmetic. It applies to all of math.

That it applies to arithmetic "and above" is written in Wikipedia, so it can't be that unknown. But if you think this means "applies to all of math" you are very naive about how theorems work. "Definable in mathematics" is not exactly reassuring either. Tarski proved a technical result about a class of formal languages, not "all of math", and so far you displayed little awareness of what the technical conditions are.

@Conifold The only thing that I really need to know is that whenever an otherwise undecidable sentence is evaluated by my true(x) predicate, it correctly decides this sentence.

@Conifold If you could only imagine the situation where the consequence of the subset of conventional formal proofs having true premsies were somehow selected, then you would comprehend that this would necessarily provide the functional result of a true(x) predicate for whatever formal system that this is implemented within. You seem stuck in notions that this is not the way we usually do things instead of thinking the hypothetical possibility that I propose ALL THE WAY THROUGH.

@Conifold I refer you to this exchange on math.stackexchange. I'm not convinced this discussion will be of any use, unfortunately.

@DonThousand and you can easily verify that the referenced question was a perfectly valid math question with a perfectly valid math answer therefore I was cheated. I have two people on SE that agreed that I am correct about how my definitions do eliminate incompleteness.

On a tangent, if you can help me state this: philosophy.stackexchange.com/q/63515/33787 paradox more clearly, then possibly I can help with this Question...

@christo183 I made my ideas progressively simpler until they were finally acknowledged and understood as correct. I had to get them down to the junior high school level before college graduates could understand them.

@PL_OLCOTT I'm really blown away by the lack of self-awareness you have in presenting your issue. Even if you've solved the Riemann hypothesis, it's your pejorative to present your ideas in the structures built to present ideas, not for others to dance to your tunes.

@PL_OLCOTT Try presenting this to professors as you've claimed you would, and let us know what they think. I'm pretty sure they'll have the same reaction we had.

@DonThousand I have one famous professor that is published in the field that has reviewed my work. No comment from him yet. As soon as I can make sure that my words are understood they have been accepted. Nearly all of the "errors" that anyone has ever pointed out are: "we simply don't believe you".