3

I'm currently reading a book on the famous Gödel incompleteness theorems which, at least as I originally understood it, purport to prove that mathematics itself cannot be axiomatized (that is, there exists no axiomatic system upon which all of math can be constructed). Upon further inspection, however, it seems that the truth is a bit more murky. Wikipedia describes the theorems as "that [which] demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic"... a bit vague.

What do Gödel's incompleteness theorems actually tell us, and how do they go about proving it, roughly?

No doubt I will have a deeper understanding after spending more time with this, however I can't help but want to anticipate the conclusion now. I'm looking for an answer in simple terms (I'm familiar with fundamental mathematical proof techniques, logic, e.g. at the undergraduate level). Is it true that math itself somehow cannot be axiomatized?

10GeV
  • 1,371

1 Answers1

3

In simple terms the way I understand it is this. If you pick any consistent set/system $A$ of axioms (which obeys certain conditions), and thus build a (math) theory, there will always be statements $S$ in this theory which you can formulate and which are true, but you cannot prove just by using your set $A$ of axioms.

This is Goedel's 1st incompleteness theorem.

That's why it's called incompleteness theorem. Because any consistent system of axioms is not complete i.e. cannot prove all the statements which can be formulated.

There's also a 2nd incompleteness theorem by Goedel which states that no set/system of axioms can prove its own consistency.

Note that both formulations given here are rough and loose.

See also:

How Goedel's incompleteness theorems work?

Book:

Ernest Nagel, James Newman, "Gödel's Proof", 1958

peter.petrov
  • 12,568
  • 1
    Interesting. This may be outside the scope of the question (perhaps it deserves its own), but is there an example of this, somewhere? – 10GeV Feb 02 '21 at 08:20
  • @KeithMadison I read a very good blog post somewhere a few months ago. It had details in layman terms and also a reference to a good book. I will try to find this link for you. – peter.petrov Feb 02 '21 at 08:22
  • I am not aware of an example. Probably it's not easy to construct one just like that. E.g. say we pick math's group theory and its set of axioms. I am not aware of a sample statement which can be formulated and is true, but one cannot prove (just with the axioms of group theory). – peter.petrov Feb 02 '21 at 08:33
  • 1
    @peter.petrov - Pay attention, whenever you say "consistent set/system of axioms", you should say "consistent and recursively enumerable set/system of axioms". Otherwise it is easy to build a consistent and complete set of axioms: take for instance all the statements that are true for the structure $\mathbb{N}$ with addition and multiplication. – Taroccoesbrocco Feb 02 '21 at 08:34
  • 1
    This can't be correct as stated. Pick a specific model of your theory and make every statement satisfied by the model an axiom of your theory. This theory violates your statement of the first incompleteness theorem. What's missing is that the set of axioms must be decidable, or at least recursively enumerable. – Robert Shore Feb 02 '21 at 08:35
  • @RobertShore OK, note taken. But the OP wanted something in layman terms. I am not claiming that I am stating Goedel's theorems in their formally precise and complete form. I thought this was needless to say. – peter.petrov Feb 02 '21 at 08:37
  • 1
    @peter.petrov There's a difference between imprecise and incorrect. Noting that you need some loose conditions on the set of axioms without stating those conditions would be imprecise. Your actual statement, which lacks any sort of qualification, is just incorrect. – Robert Shore Feb 02 '21 at 08:37
  • @RobertShore OK... I added a small note there. – peter.petrov Feb 02 '21 at 08:40