In 2004 Ben Green and Terence Tao ostensibly proved a result which is now called the Green-Tao theorem. It asserts that there are arbitrarily long arithmetical sequences of prime numbers.
That is, given a natural number n, there is a sequence of prime numbers of the form p+mk, k=0,1,…,n-1 where p and m are natural numbers. For example 5, 11, 17, 23, 29 is a sequence of 5 primes in arithmetical progression with difference m=6, while 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a sequence of 10 primes in arithmetical progression, with difference m=210.
Up to now the longest sequences of primes in such an arithmetical sequence that I know about was found in 2010 by Benoãt Perichon: it is
43,142,746,595,714,191 + (23,681,770)( 223,092,870)k , for k = 0 to 25.
The proof of Green and Tao is clearly a tour-de-force of modern analysis and number theory. It relies on a result called Szemeredi’s theorem along with other results and techniques from analytical number theory, combinatorics, harmonic analysis and ergodic theory. Measure theory naturally plays an important role.
Both Green and Tao are brilliant mathematicians, and Terence Tao is a Fields Medal winner. Terence is also originally Australian, and spent half a year at UNSW some time ago, where I had the pleasure of having some interesting chats over coffee with him.
Is the Green-Tao theorem true? This is actually quite an interesting question. The official proof was published in Annals of Math. 167 (2008), 481-547, and has been intensively studied by dozens of experts. No serious problems with the argument has been found, and it is now acknowledged that the result is firmly established. By the experts.
But is the Green-Tao theorem true? That depends not only on whether the arguments hang together logically when viewed from the top down, but also crucially on whether the underlying assumptions that underpin the theories in which those arguments take place are correct. It is here that one must accept that problems might arise.
So I am not suggesting that any particular argument of the Green-Tao paper is faulty. But there is the more unpleasant possibility that the whole edifice of modern analysis on which it depends is logically compromised, and that this theorem is but one of hundreds in the modern literature that actually don’t work logically if one descends right down to the fundamental level.
Let me state my position, which is rather a minority view. I don’t believe in real numbers. The current definitions of real numbers are logically invalid in my opinion. I am of the view that the arithmetic of such real numbers also has not been established properly.
I do not believe that the concept of a set has been established, and so consequently for me any discussion involving infinite sets is logically compromised. I do not accept that there is a completed set of natural numbers. I find fault with analysts’ invocation of limits, as often this brazenly assumes that one is able to perform an infinite number of operations, which I deny. I don’t believe that transcendental functions currently have a correct formulation, and I reject modern topology’s reliance on infinite sets of infinite sets to establish continuity. I believe that analysts are not being upfront in their approach to Wittgenstein’s distinction between choice and algorithms when they discuss infinite processes.
Consequently I find most of the theorems of Measure Theory meaningless. The usual arguments that fill the analysis journals are to me but possible precursors to a more rigorous analysis that may or may not be established some time in the future.
Clearly I have big problems.
But as a logical consequence of my position, I cannot accept the argument of the Green-Tao theorem, because I do not share the belief in the underlying assumptions that modern experts in analysis have.
But there is another reason why I do not accept the Green-Tao theorem, that does not depend on a critical analysis of their proof. I do not accept the Green-Tao theorem because I am sure that it is not true. I do not believe that there are arbitrarily long arithmetical progressions of prime numbers.
Let me be more specific. Consider the number z=10^10^10^10^10^10^10^10^10^10+23 that appeared in my debate last year with James Franklin called Infinity: does it exist?
My Claim: There is no arithmetical sequence of primes of length z.
This claim is to be distinguished from the argument that such a progression exists, but it would be just too hard for us to find it. My position is not based on what our computers can or cannot do. Rather, I assert that there is no such progression of prime numbers. Never was, never will be.
I do not have a proof of this claim, but I have a very good argument for it. I am more than 99.99% sure that this argument is correct. For me, the Green-Tao argument, powerful and impressive though it is, would be better rephrased in a more limited and precise way.
I do not doubt that with some considerable additional work, they, or others, might be able to reframe the statement and argument to be independent of all infinite considerations, real number musings, and dubious measure theoretic arguments. Then some true bounds on the extent and validity of the result might be established. That would be a lot of effort, but it might then be logically correct– from the ground up.