divided by
p. Likewise for
a·b. So by definition, if
a,b are in
Fp, then for
p bigger than max of
a+
b,a·b, a+
b and
a·b coincide in Z and in
Fp.", so now you are saying that this is the matter of definition.
My first remark is technical. The problem deals only with rings and has nothing to do with division. So, it is not necessary to consider the field Fp and only primes. The problem is whether Z is the limit of rings Rp=(0,1,…p-1) (with operations modulo p) when p→∞.
Again, in mathematics, mathematical statements should be formulated unambiguously such that different interpretations should be excluded. For this reason the words “for any” and “there exist” are often used in mathematical statements. However, saying about a and b you are not using those words, and one can only guess what you mean. Consider you “definition” “So by definition, if a,b are in Fp, then for p bigger than max of a+b,a·b, a+b and a·b coincide in Z and in Fp.” literally. For example, if a=0 and b=0 then for p > 0, a+b and a·b coincide in Z and in Fp. Or if a<10 and b<10 then for p > 100, a+b and a·b coincide in Z and in Fp.”. So your “definition” is indeed obvious.
I guess that probably you meant something like this: for any p0 there exists a set S such that for any a+bϵS,a·b ϵS, a+b and a·b coincide in Z and in Fp for any p >= p0, and card(S)→∞ when p0→∞.
However, even if my guess is correct, this still cannot be a correct definition that Rp→Z when p→∞. The definition should be such that not only for two elements from S their sum and product coincide in Z and in Rp but that it is possible to find a number n such that for any m<=n the result of any m operations of multiplication, summation or subtraction of elements from S should be the same in Z and in Rp, and that n→∞ when p→∞.
The exact formulation of the definition is given in my paper, and I prove that with this definition indeed Rp→Z when p→∞. As I said, the definition should be to some extent analogous to the definition that the sequence (an) →∞ when n→∞: for any M>0 there exists n0 such that an>=M for any n>= n0.
I asked several mathematicians to give me a reference where this is proved but nobody gave such a reference. The response of some of them was analogous to yours: this is obvious. Then I asked that if this the case then why in mathematical textbooks this is not even mentioned and standard math starts from Z from the beginning, but again no response. As I wrote, they don’t care that standard math has foundational problems (as follows e.g. from Gödel’s incompleteness theorems and other considerations). But when I asked Prof. Zelmanov (who is the Fields Medal laureate) he did not say that this is obvious and advised me to look at Terence Tao’s blog where ultraproducts are considered. In my paper I thank Prof. Zelmanov for his advice and refer to the blog.
Technically indeed it is possible to prove that Rp→Z follows from the results on ultraproducts although in ultraproducts they consider only fields and their goal is to use finite fields for proving some features of fields of characteristic zero. Nevertheless, this is not a direct proof, and the construction is rather sophisticated.
In summary, I think that, with the probability 99.99 %, in the literature there is no direct proof that Rp→Z when p→∞ and so my proof is new. Let me note that my paper contains not only this result: I explain that this result is the first step in proving that finite math is more fundamental than standard one: the latter is a special degenerated case of the former in the formal limit p→∞.
However, it seems obvious that you even did not try to carefully read my proof and other results of the paper. You noticed that I prove that Rp→Z, immediately (within minutes) decided that this is obvious (as you say, even without the machinery of ultraproducts) and immediately wrote a rejection. I am amazed that the attitude to my paper at such a prestigious journal was on such a level.
For me it is not a great tragedy that my paper will not be published in NDJFL. I have no doubt that the results are fundamental, they will be acknowledged sooner or later and published elsewhere. However, I treat such an attitude to me as disgraceful from the professional point of view. Such an attitude in fact means that you treat me as unprofessional who submitted to NDJFL a junk which does not deserve consideration.
Of course you have a right to have such an opinion. However, if you think that your attitude was a mistake I would be grateful if you tell me this and will be fully satisfied. I understand that we are only people, everybody makes mistakes, you are very busy handling such a journal, you have to look at many papers and probably some of them are indeed junk, so probably mistakes in your work are inevitable. However, decent people acknowledge that they make mistakes when this becomes obvious.
В этом ответе вначале популярно объясняю, что его объяснения не имеют смысла. Пишу, что был очень удивлен, что в таком престижном журнале моя статья была рассмотрена на таком уровне.
В конце пишу, что для меня главное – не то, что статья не будет опубликована в его журнале, а то, что отношение к статье было позорным. Как будто я полностью не профессионален и послал в журнал мусор на который не стоит тратить время. Пишу, что все мы
