In 1977, Henry Pogorzelski published what some believed was a claimed proof of Goldbach's Conjecture in Crelle's Journal (292, 1977, 1-12). His argument has not been accepted as a proof of Goldbach's Conjecture, but as far as I know it has not been shown that his argument is incorrect.
Pogorzelski's argument is said to depend on the "Consistency Hypothesis," the "Extended Wittgenstein Thesis," and "Church's Thesis." Pogorzelski has a Ph.D. in mathematics (his advisor was Raymond Smullyan).
Daniel Shanks says in Solved and Unsolved Problems in Number Theory (fourth edition, 1993) that: "It seems unlikely that (most) number-theorists will accept this as a proof [of Goldbach's Conjecture] but perhaps we should wait for the dust to settle before we attempt a final assessment." (page 222)
Did Pogorzelski claim to present a proof of Goldbach's Conjecture? If so, and this claimed proof has not been disproven after 33 years, I am curious why this would be the case, given that Shanks considers it important enough to mention in his book.
Best Answer
In the 1970's Pogorzelski published a sequence of four papers in Crelle concerning the Goldbach Conjecture (and various generalizations and abstractions):
The last paper is the one referred to in the question. (Edit: as I have been writing this, the question has been edited to make this reference explicit, which is good.)
I think that describing Pogorzelski's last paper as a purported proof of the Goldbach Conjecture is a mischaracterization. Rather what he shows is an implication: three statements which are not known to be true imply Goldbach. (I don't pretend to understand these three statements. The only one that I recognize at all is Church's Thesis, but although I think I know what that means, it does not denote to me a precise mathematical conjecture, so I am for sure out of my depth here.)
So far as I can see, Pogorzelski himself never claimed that his 1977 paper is a proof of Goldbach. Indeed, he worked for many years thereafter on the problem and published nearly two thousand pages of further work. Specifically, in 1982 he published the first of a proposed seven volume series, Foundations of a semiological theory of numbers, whose ultimate goal is to prove Goldbach by showing that a disproof is impossible in a certain formal system. Volume 1 is 608 pages. Volume 2 (743 pages) appeared in 1985. Volume 3 (522) pages appeared in 1988. MathSciNet does not list any further volumes.
To summarize, his programme for proving Goldbach seems to be as yet unfinished (and, of course, may well be unfinishable), but none of the reviews I read -- some of which are written by leading mathematicians -- raised any mathematical objections to the work that has been published.