Timm Lampert quotes from Wittgenstein's "notorious paragraph" (§8 of Remarks on the Foundations of Mathematics, Appendix 3) in http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate
I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says ‘P is not provable in Russell’s system’. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that is not provable. Thus it can only be true, but unprovable.”
Just as we ask, “‘Provable’ in what system?”, so we must also ask, “‘true’ in what system?” ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system. – Now what does your “suppose it is false” mean? In the Russell sense it means ‘suppose the opposite is proved in Russell’s system’; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by ‘this interpretation’ I understand the translation into this English sentence. – If you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up.[…]
When we examine these things using the mathematical formalist approach:
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.
When analytical truth in language is defined as tautologies between finite strings then:
(1) There is no possible way that truth can diverge from provability.
(2) Truth is always definable.
(3) Incompleteness gaps cannot possibly exist.
Tractatus Logico-Philosophicus, by Ludwig Wittgenstein (1921)
seems to have this same view on tautology.
This is how my view unifies with the view of Wittgenstein:
Analytical truth expressed in language is merely finite strings that have been defined to have the semantic property of Boolean true (axioms) and relational connections to these finite strings (Theorems). Any expression of language that is not defined to have the semantic property of Boolean true (axiom) or has a relational connection to these finite strings (theorem) is necessarily untrue.
The following can be understood from the mathematical formalist perspective of [tautological] relations between finite strings:
Formalizing Wittgenstein's highlighted words:
P ↔ (RS ⊬ P)
∀x (True(RS, x) ↔ (RS ⊢ x))
∀x (False(RS, x) ↔ (RS ⊢ ¬x))
Examples of Provability relations:
(PA ⊢ "2 + 3 = 5") // Provability relation exists----------AKA True
(PA ⊬ "2 + 3 = 7") // Unprovability relation exists------AKA ¬True
(PA ⊢ ¬"2 + 3 = 7") // Refutability relation exists-------AKA False
Within the truth predicates derived from Wittgenstein's specification if Unprovability is not proved within Russell's System then Unprovability does not count as true within Russell's System.
This view (based on Wittgenstein) seems to be irrefutable
Only statements inside the scope of provability within Russell's System derive truth values from Russell's System. The Truth of the statements of a Theory is only relative to that Theory. The formalization of (the Wittgenstein specification of) P relative to RS shows that P is self-contradictory relative to RS thus not true in RS even if it is true elsewhere.
These are Two Different Questions:
(1) Is P true in RS?---NO
It cannot be proved in RS that P cannot be proved in RS because P is self-contradictory in RS thus ill-formed in RS and therefore untrue in RS.
(2) Is P True outside of RS?---YES
It can be proved outside of RS that P cannot be proved in RS.
When the above two questions are conflated together as one question [Is P true?] it really seems that Gödel Incompleteness and Tarski Undefinability have been proven. When this one question is divided so that it is relative to a formal system then the Gödel and Tarski conclusions cease to be sustained.
Actual truth in the world is simply relations that have been defined between finite strings. The relation [is a] has been defined between "cat" and "animal". You can pile layers of extraneous complexity on top of that to confuse the issue, yet the self-evident truism remains, conceptual truth in language is simply defined relations between finite strings. Provability is nothing more than determining whether or not one of these defined relations exists.
This aspect will be continued in a new question:
Are truth and provability inherently inseparable because every means of ascertaining conceptual truth can be construed as a formal proof?
The key issue is not a math error by either Tarksi or Gödel. If Truth and provability are truly inseparable then Tarksi and Gödel are wrong.
Wittgenstein proves that Gödel's G is merely self-contradictory when Truth and provability are inseparable. When truth and provability mutually define each other then Tarski Undefinability ceases to exist.
The actual refutation Tarksi and Gödel requires a proof that Truth and provability be proved inherently inseparable and cannot be defined coherently otherwise.