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I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics.

In this question I assume some fixed formal system F expressive enough for the theorems to go through.

The essence of the theorems seems to be that there are number-theoretic facts that are true but unprovable.

Now, let's talk about possible foundations of mathematics. Arithmetic of natural numbers is an integral part of mathematics, so the foundation should include arithmetic by definition. Also, the foundation has to have some ontology to it. For example, the ontology of ZFC includes pure sets only and nothing else.

Here's where things get interesting: every object of ZFC is a set, including numbers, so the incompleteness spreads not only to numbers, but to sets as well, giving rise to all of the undecidable problems seemingly completely unrelated to arithmetic.

Now, can we come up with a foundation which has the following ontology: numbers exist, and some other fundamental objects exist as well having no formal connection to said numbers?

Now, these numbers in this foundation never leave the "quarantine zone", with their unprovability problems never spreading like wildfire across the whole foundation, leaving the other fundamental objects "healthy".

Is such a thing feasible? Is the situation with ZFC inevitable for every conceivable foundation of mathematics?

Is my understanding flawed?

  • It is not really about numbers, it is about the mixture of infinite and discrete content. Remember the Real and Complex Closed Fields are decidable, and those are numbers. – jobermark Oct 1 '16 at 1:23
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It is a natural idea, but unfortunately the answer is no, it is not feasible. The root of incompleteness is not numbers, but the possibility of (implicit) self-reference, arithmetic is just the simplest structure that already realizes that possibility. In fact, one does not even need the Peano arithmetic, but a much weaker Robinson arithmetic without even induction for the proof to go through. In the end what matters is not whether the theory has numbers, or sets, or something else, or how the pieces are connected to or isolated from each other, but only the expressive power of the theory. As long as it can imitate the minimal expressive power of arithmetic the incompleteness sets in, whether we connect numbers to other objects, or whether we even have numbers at all, makes no difference. The incompleteness does not spread from numbers, it is inherent in anything that can simulate numbers. If the other objects can not they remain "healthy", but then their theory is weaker than arithmetic so we can drop them altogether for we are essentially committing to reducing all mathematics to arithmetic (this is not to say that some useful complete fragments weaker than arithmetic do not exist, elementary Boolean algebra and elementary plane geometry are examples).

One can go a surprisingly long way with that actually, this is called predicative mathematics. As nominalists like Field showed, while it is weaker than classical mathematics, it is enough for all the purposes of classical physics at least. In predicative mathematics the incompleteness is essentially reduced to that of arithmetic only. Wittgenstein was willing to go even further, and reduce mathematics to primitive recursive arithmetic, which is finitist, see Was Wittgenstein anticipating Gödel? But if we really want to beat incompleteness without trivializing mathematics, structural manipulations won't help, we have to give up one of Gödel's other premises: either that mathematics is recursively axiomatizable (axioms are recognizable as such), or that it is consistent (or both). Again, Wittgenstein was willing to give up consistency and confine contradictions using what later developed into non-classical ("dialetheic") logic. Development of these ideas led to modern inconsistent mathematics, which produces complete inconsistent arithmetics that can prove non-triviality of their consistent parts, see Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole? and In which text/paper was the concept of dialetheism first introduced as a serious position?

It is unlikely that predicativism, finitism or dialetheism will become mainstream positions among mathematicians however. They are viewed as too restrictive and/or artificial to support the existing mathematical practice, which does not really need either completeness or foundations.

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To compliment Conifold's answer, here's another way to look at it: Statements about number theory always end up being statements in number theory as well. Take any number theoretical theorem and replace the symbols with numbers using a suitable encoding, and you end up with an equation.

Because of this, any system rich enough to encompass the arithmetic of natural numbers cannot avoid self-reference. And as Conifold points out self-reference makes the paradox at the heart of Godel's theorem inevitable.

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Gödel's Incompleteness Theorem is proven for any self-referential system which is omega-consistent. However, there have been some clever workarounds. Dan Willard's work, in particular, explored a clever workaround if one does not assume multiplication is a total function (i.e. there exist numbers which cannot be multiplied together). He was able to develop such systems where he could prove all of the truths of arithmetic, but the system was ever so slightly too weak to admit the diagonalization lemma essential to Gödel's proof. Such systems could prove their own consistency, but could not prove the lemma needed for Gödel's proof to work.

  • When you say "all of the truths of arithmetic" you mean excluding the 'missing' multiplications, right? Anyway, link? – Eliran May 1 '16 at 20:51
  • @EliranH True, though there's some fascinating details that show up when you start down that line. In particular, you can get into systems where you can't prove multiplication is total, but you also cannot prove that there must exist two numbers, a and b, such that there in no c such that a*b=c. His website has many papers in PDF form: cs.albany.edu/FacultyStaff/profiles/willard.htm – Cort Ammon May 1 '16 at 22:31
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Yes, by joining the logicist school. Gödel sentence presents no threat to logicism because a self-referential sentence is meaningless to logicists.

The meaning of Gödel sentence G cannot be determined until each of G's constituents is determined; one of G's constituents is G itself, thus a vicious circle ensue. To formalists, mathematics is ink blobs that have no meanings, thus Gödel sentence presents a valid argument. To logicists, meaning is fundamental; since Gödel sentence has no meanings, Gödel sentence G has no impact on logicism.

Any branch of mathematics that has practical application must concern itself with meanings.

Assigning number to magnitude is called measurement; measurement is not fundamental to mathematics.

  • Goedel's number is not a sentence, so this is not a way out. It only represents a sentence, and the function around it only represent arguments. But it is set up in such a way that if the whole system succeeded, then the simulation inside it would also succeed. And none of G's constituents is G itself, one of G's constituents is a number that is assigned to represent G in the simulation. I can assign some token to represent myself in a game and playing out the whole game can still be instructive. If the game tells me something about myself, it has been meaningful despite being abstract. – jobermark Oct 1 '16 at 15:51
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    If you are dealing in insults rather than arguments, you don't belong here. You are addressing none of my objections. Spewing bile is not the point of comments on this exchange. – jobermark Oct 1 '16 at 16:37
  • Gödel can make waves among formalists because nonsense is what formalism is all about. Between a formalist and a logicist, there is no argument: one talks nonsense; the other doesn't - this is exactly what happened when Russell met Gödel. I will follow Russell's example and leave you guys alone. Best wishes. – George Chen Oct 1 '16 at 16:49
  • Russel fully acknowledged Goedel's work, even if he gave it an odd take. See the answers here: philosophy.stackexchange.com/questions/3951/… You continue to spout falseness. – jobermark Oct 1 '16 at 17:18
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    @user170039 Or you can weaken other assumptions. I am not on the opposite side here. I just think that more respect is owed on all sides than the answer affords. Mr Chen is being a bigot and excusing abuse on that account. – jobermark May 3 '18 at 15:40

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