Stanford Encyclopedia of Philosophy link
I had a hard time accepting Godel’s papers,
“Some metamathematical results on completeness and
consistency“,
“On formally undecidable propositions of
Principia Mathematica and related systems I“,
and
“On completeness and consistency“
Kurt Godel
(1930b, 1931, and 1931a)
I studied these papers in the collection From Frege to Godel, edited by Jean van Heijenoort. Eventually, I accepted the results. But, in doing so, I became more disturbed by another implication. This was a result of footnote 17 on page 599 of that book. It said it was assumed that there were denumerably many signs at our disposal for each type of variables. At first, I did not understand this correctly – he is only speaking of variables here. That is OK. He assumes the signs “0” (zero) and “f” (successor) can be used to represent all individuals. But it is known that that the number of real numbers is not denumerable. The total number of statements that can be constructed by Godel’s system is denumerable. But the total number of mathematical facts is not denumerable. E.g. there is a non-denumerable number of facts x=x, where x is a real number. Some (actually most) such x’s cannot be represented. At least, there is no known method of representing them. This is known as “Church‘s Thesis” (actually one consequence of it). The book I that I learned this from was Computability and Logic, Third edition, by George S. Boolos and Richard C. Jeffrey. There is now a newer edition. It is also in The Oxford Dictionary of Philosophy. (page 62). Thus there are a non-denumerable number of mathematical facts that cannot be represented (let alone proven) in Principia Mathematica, or any known system. It can be proven that they cannot be represented in any system that has been checked. All systems checked have equivalent power (or less power). It cannot be proven that there is not a totally different sort of system in which they could be represented. That is why it is a thesis not a theorem.
There is more on this at Church-Turing Thesis as is also sometimes called.
Note: I am not talking just about irrational numbers, such as pi. Pi is computable, I think. At least the nth digit of it is a computable function – so it can be described with a finite sized expression. A program to compute the digits of pi could never finish. Usually, algorithms are required to halt with an answer, so pi, in this sense, could be considered uncomputable, but finitely describable. But there are real numbers (I obviously cannot specify any) that are not describable in a finite sized expression. All sentences are finite. There are only a denumerable number of finite sentences, and there are a non-denumerable number of real numbers. So there are real numbers for which no subexpression of any sentence corresponds. Such numbers would be irrational, but not all irrational numbers would be such.
I have refined this at my Alonzo Church page. I have there a lengthy quotation explaining this better.
There are multiple ways to think about these facts. One is that there “really” are real numbers, and that we just cannot specifically talk about some of them. This is not far fetched, as there are many things unknown to us humans. But it is apparently, at least, in conflict with the philosophy of mathematics known as logicism.
Another way to think about this is that the objects of mathematics are “ideal” objects which do not “really” exist. They are very useful for calculating, say in physics. Also the various systems of logic can be interesting to study, but have no metaphysical import. This would be akin to Hans Vaihinger‘s The Philosophy of As If, to which I, mainly, agree.
There are also other philosophies of mathematics to study, such as Intuitionism, but I am not familiar enough with them to comment on them here.
I also have pages for Church, Tarski and Turing.