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.

  • $\begingroup$ If Pete Clark's answer is correct (and you seem to believe that it is, since you have accepted it), it would be nice to change the title of this question to more accurately reflect the actual nature of Pogorzelski's work, e.g. "Has Pogorzelski's program to proof Goldbach reached fruition", or something of this nature. $\endgroup$ – Emerton Sep 10 '10 at 18:31
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    $\begingroup$ I have changed the title and rewritten the question. I do not want to change it too much, however, or the comments will not make much sense. $\endgroup$ – Marko Amnell Sep 10 '10 at 19:01
  • $\begingroup$ Have you some link to the paper? $\endgroup$ – Sergei Tropanets Sep 14 '10 at 12:48
  • $\begingroup$ deepdyve.com/lp/de-gruyter/goldbach-conjecture-frA6e4f0DS If you do not have an account, click the button "Rent for free." $\endgroup$ – Marko Amnell Sep 14 '10 at 15:56

In the 1970's Pogorzelski published a sequence of four papers in Crelle concerning the Goldbach Conjecture (and various generalizations and abstractions):

MR0347566 (50 #69) Pogorzelski, H. A. On the Goldbach conjecture and the consistency of general recursive arithmetic. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. J. Reine Angew. Math. 268/269 (1974), 1--16.

MR0505402 (58 #21554) Pogorzelski, H. A. Dirichlet theorems and prime number hypotheses of a conditional Goldbach theorem. J. Reine Angew. Math. 286/287 (1976), 33--45.

MR0434999 (55 #7961) Pogorzelski, H. A. Semisemiological structure of the prime numbers and conditional Goldbach theorems. J. Reine Angew. Math. 290 (1977), 77--92.

MR0538046 (58 #27414) Pogorzelski, H. A. Goldbach conjecture. J. Reine Angew. Math. 292 (1977), 1--12.

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.

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    $\begingroup$ Thanks. That is very informative, but it raises the further question of why people such as Daniel Shanks believed that Pogorzelski had claimed to have a proof of Goldbach's Conjecture. $\endgroup$ – Marko Amnell Sep 10 '10 at 19:05
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    $\begingroup$ You're welcome. I agree that this is a natural followup question, to which I certainly do not know the answer. $\endgroup$ – Pete L. Clark Sep 10 '10 at 19:15

For what it's worth, the 12/12/2012 article at http://bostonglobe.com/news/science/2012/12/11/math-even-mathematicians-don-understand/e41V2lixnAVyB63X4NTPaO/story.html indicates that Pogorzelski believes that he has established a legitimate demonstration of the Goldbach conjecture (which he claims he obtained in the 1990s). The article quotes from his manuscript's preface that his proof was submitted for publication in 2002, and he received no response.


"The procedure by which proofs become accepted as essentially correct is by publication in a journal after peer review": that's false. Many published proofs contain substantial mistakes. Nobody should accept a result only because it has been published.

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    $\begingroup$ If you want to stretch a point (and quote out of context) it is neither necessary nor sufficient. Conventionally, though, it is necessary, the point I was trying to make. $\endgroup$ – Charles Matthews Sep 10 '10 at 17:03

The "why not" is about the onus of proof. The word "false" does not properly qualify a proof, but a proposition. The procedure by which proofs become accepted as essentially correct is by publication in a journal after peer review (you don't say where Pogorzelski published). All in all, a proof not accepted after 30 years is in most cases better forgotten, though I suppose not in absolutely every case.

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    $\begingroup$ Pogorzelski published his purported proof in Crelle's Journal (292, 1977, 1-12) and has a Ph.D. in mathematics (his advisor was Raymond Smullyan). If his 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. $\endgroup$ – Marko Amnell Sep 10 '10 at 15:55
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    $\begingroup$ This should be included in the question. Without this, it feels slightly hard to distinguish this from the likely hundreds (thousands?) of other claimed proofs of Goldbach's conjecture which have not been definitively refuted (primarily for lack of interest in doing so). $\endgroup$ – Cam McLeman Sep 10 '10 at 16:02
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    $\begingroup$ I replaced the word "false" with "incorrect" when describing the purported proof. $\endgroup$ – Marko Amnell Sep 10 '10 at 16:03
  • $\begingroup$ @Cam: Good point. I added the information to the question. $\endgroup$ – Marko Amnell Sep 10 '10 at 16:08
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    $\begingroup$ OK, it seems it was some kind of conditional proof. $\endgroup$ – Charles Matthews Sep 10 '10 at 16:46

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