# Could the incompleteness theorems have been proved by Tarski if Gödel had not proved them first?

As I know, Tarski had proved his undefinability theorem by $1936$, $5$ years after Gödel's incompleteness theorems had been discovered. I wonder whether his original proof was built upon the work of Gödel or not.

Could the incompleteness theorems have been proved by Tarski by $1936$ if Gödel had not already discovered them?

• Such question can be asked about any theorem, and on my opinion this is out of scope of this site as an "alternative history". – Alexandre Eremenko Apr 8 '16 at 19:03
• It mainly asks about if we can logically prove undefinability theorem first (using techniques that are different from Godel's) and the deduce Incompleteness from it. – Fawzy Hegab Apr 8 '16 at 19:13
• For the English translation of Tarski's 1936 paper, see here, and specifically page 247. – Mauro ALLEGRANZA Apr 8 '16 at 19:24
• My sense also is that the title question is too speculative, I hope it will be changed, and "can we prove undefinability theorem first (using techniques that are different from Godel's) and then deduce Incompleteness from it?" is a question for Math SE. The first two questions in the body however are perfectly legitimate. – Conifold Apr 8 '16 at 23:18
• @Conifold I will ask such a question there soon. it's such a great question. – Fawzy Hegab Apr 12 '16 at 16:44

In principle, yes.

See:

Tarski made clear his indebtedness to Gödel's methods. [...] On the other hand Tarski strongly emphasized the fact that his results were obtained independently.

Tarski did not claim any priority for Gödel's own results. In footnote 2 to the paper Some observations on the concept of omega-consistency and omega-completeness (1933) he wrote:

Already, in the year 1927 (...) I also communicated the example of a consistent and yet not omega-consistent system which I give in the present article in a slightly altered form. Naturally it is not hereby claimed that I already knew then the results later obtained by Gödel or had even foreseen them. On the contrary, I had personally felt that the publication of the work of Gödel was a most exciting scientific event.

Though Tarski saw the similarity of the system $P$ used by Gödel in the paper of 1931 and his own system used in 1933 he adds [...] the remark that (...) the abstract character of the methods used by Gödel renders the validity of his results independent to a high degree of the specific peculiarities of the science investigated.

It is known that Tarski's theorem on undefinability of truth implies the existence of undecidable sentences, hence Gödel's first incompleteness theorem. So one can ask whether Tarski was close to this theorem.

• So, can we say that the method Tarski used to prove his undefinability theorems are different than that of Godel's arithemetization? Where can I find such a proof? I think it's in his 1936 paper titled "the concept of truth in formalized systems", Will I find his own proof there? and is there a modern presentation of his original proof? Thanks! – Fawzy Hegab Apr 8 '16 at 18:58
• @MathsLover - see Tarski's undefinability theorem with references. – Mauro ALLEGRANZA Apr 8 '16 at 19:15
• So, far as I know, Tarski's Undefinability Theorem does not imply Gödel's First Incompleteness Theorem exactly, but a weak form of it (it does imply the existence of true but unprovable sentences but not gives any clue as to how to construct it). – user459 Oct 8 '17 at 13:14