Godel’s incompleteness theorem was proved specifically for an axiom system that includes addition, multiplication, numbers, and more than that too, that those all have to have particular properties. And mathematicians generally believe them to be consistent, At any rate, there are many other complete, decidable and consistent axiom systems. Let’s unpack this in more detail.

Kurt Gödel who proved the remarkable incompleteness theorems. Turing and Church proved related theorems at around the same time.

There are many such systems of axioms but one of the easiest to state is the system of Peano axioms. It consists of some niggly axioms you have to add to define the operations of equality and adding 1. Then you also say that there’s a first number 0 (this is an axiom system for the non negative numbers only).

Then it has one very special axiom, the “induction axiom”. If a property is true of 0, and if you can prove that whenever it is true of a number n, it is also true of n+1, then this property must be true of all numbers.

This is the axiom that cause all the trouble. It lets you define addition, multiplication, and indeed what it means to take one number to the power of another one, e.g. 2^3, 5*7, etc etc. You can go on and prove many complicated theorems. For instance, you can prove that every number has a unique prime factorization. E.g. 30 = 2*3*5 where 2, 3 and 5 are all prime (are not divisible by any other prime numbers). And this is the only way you can express 30 as a product of primes. That’s the same for all numbers, they can only be expressed as a product of primes in one way.

Those were the main properties of numbers that Godel used to prove his theorem, together with frequent use of proof by induction. Once you have that much deductive power, then yes, you can’t prove that the resulting system is consistent.

So - that’s a big chunk of mathematics. We use numbers almost everywhere. But there are many areas of maths that don’t need numbers, or don’t need numbers with all that apparatus of deduction rules to prove things about them.

One excellent example here is geometry using rulers and compasses. Euclidean geometry. Euclid worked out the basic axiomatization, though he left out some things that seemed obvious to him, so obvious he didn’t realize they needed an axiom. One of the rules he left out is that if a line enters a three sided triangle, crossing one of its sides, it has to exit the triangle by crossing one of the other sides or the opposite vertex. It was so obvious that even with his careful logical mind, he didn’t realize he had it as an assumption.

Anyway - if you have a proper axiomatization of geometry - well there’s no mention of numbers there. And it turns out, not only that Godel’s theorem doesn’t apply, but if you are careful in how you set out your axioms, you can also prove that the resulting theory is consistent, decidable and complete. You can use Tarski’s axiomatization of geometry to prove this.

What’s more, even addition and multiplication are not enough for Godel’s theorem. They have to have very special properties, with quite a powerful deduction system. One nice system that you can prove to be consistent is the theory of real closed fields.

This is a theory with addition, multiplication, the numbers 1 and 0, fractions, so you can use it to express any ratio like 2/3, 4/5 etc. Also it’s “closed” meaning that given any sequence of numbers in it, then the limiting point of that sequence is also in it. In particular it’s going to include every infinite decimal, such as PI as the limiting point of 3, 3.1, 3.14, 3.141, 3.1415, …

It also has to have the <= relation which also has to work just as it does on the real numbers - given any pair of numbers, one is always smaller or equal to the other. But as well as that it has to have a total ordering. Given any pair of numbers, either they are identical or one of them is smaller than the other, and that if a<=b and b<=a then a = b, Also if a<=b and b<=c then a<=c. In short, it’s a total order.

Well - that may seem very similar to what you get with Peano’s axioms - after all we now have perfectly good numbers we can use for addition and multiplication. But - it turns out we just don’t have the same deductive power we have for Peano’s axioms. We couldn’t, for instance, use these rules to define exponentiation, primality, unique factorization etc because it doesn’t have an induction rule for the numbers. So we can’t prove Godel’s theorem for this theory either.

Well it turns out, if we add two more axioms, an axiom that it includes the square root of any number, and an axiom that it includes at least one solution of any polynomial equation of odd degree (linear, cubic, quintic etc) - then we can prove that the resulting theory is complete, consistent and decidable - that given any postulate you can state within the theory, then it is either true or false, and what’s more, there’s a procedure you can follow that is guaranteed to find the right answer.

Now - this process for finding the right answer to any question in the language is not very practical (nor is it practical for Tarski’s geometry either). The procedure is immensely complex and might take pretty much for ever on human timescales. Nevertheless, in the sense of Godel’s theorem, it’s decidable, consistent and complete.

So - now we’ve seen some rather powerful theories that are decidable, consistent and complete. So now let’s look at the opposite. Something that seems like a very weak theory, yet, it’s enough to prove Godel’s theorem, so it can’t be shown to be a consistent theory.

This is Robinson arithmetic

The axioms are, where “successor” here means the result of adding 1:

0 is a number and is not the successor of an number

If the successor of x equals the successor of y, then x = y.

Every number except 0 has a predecessor.

Adding 0 to a number has no effect: x+0 = x

Multiplying any number by 0 gives 0: x * 0 = 0.

Then, we have a couple of rules to define addition and multiplication. So, if S is the successor operation, then for every x and y:

x + Sy = S(x+y)

x . Sy = (x.y) + x

It hardly seems enough. Surely the resulting theory is logically weaker than the theory of real closed fields? Well, it turns out, the answer is no. There is enough deductive power here to deduce Godel’s theorem.

This theory therefore can’t be proven to be consistent except in a “stronger or equally strong theory”.

Now, this doesn’t mean that it is inconsistent. Indeed, it doesn’t actually rule out consistency proof. Indeed Gentzen did a consistency proof for Peano arithemtic, using a slightly different and in some ways simpler axiomatization of arithmetic called “primitive recursive arithmetic” plus something called “transfinite ordinals”. All of this is very techy, but mathematicians find this reassuring because Peano arithmetic with its induction axiom seems a bit uncomfortably like the powerful theory of Frege’s axiomatization of set theory, but this other axiomatization is in a way more straightforward. So it’s good evidence, perhaps, that we won’t get into any trouble by treating Peano Arithmetic as a consistent theory, even though we can’t prove this. Gentzen's consistency proof - Wikipedia

Rather, I think a better way of looking at it, by Godel's incompleteness theorem, any powerful enough system of axioms can never capture all the maths that is implied by those axioms.

If you've described it clearly enough so that it's totally clear how theorems are proved - then by Godel’s ingenious methods, that leaves it open to processes of adding new axioms to the system, which a mathematician can see must be true, but which are not included in your original list of axioms. It also leaves you open to the possibility of adding in the negations of those new axioms as well of course, if you want to explore systems that are omega inconsistent -if you are interested in these strange theories, where you can say “There is a number with property P” and yet can say “1 doesn’t have property P, nor does 2, nor 3, nor 4, nor 5, …” and every single statement of that sort is false. It seems inconsistent, but actually mathematically it isn’t possible to make a finite proof of an inconsistency. So it’s a bit weaker than normal inconsistency, and the techy word for this is “ω inconsistent” (there ω is a symbol used for the sequence of all the numbers 1 2 3 …. - it’s consistent unless you string together infinitely many statements which we can’t do in practice - so you can never prove it is inconsistent)

So anyway - you are not prevented from exploring those strange theories either in full knowledge of what you are doing. Indeed some mathematicians are also interested in “paraconsistent” theories - theories that have proofs of inconsistencies that are finite, but rather large, so you can work with them for a long time without hitting an inconsistency. These are not unlike ordinary logic that we use in everyday life. We are actually able to work perfectly well with inconsistent theories. E.g. - if you want to take something out of your house, you might think it is small enough to fit through the door upright, and actually try to take it out that way, only to find out it won’t fit. This means you had an inconsistent theory about that object. No problem. Modify your theory to say “oh I get it now, it has to be turned on its side” and now you can take it out of your door. In more complex situations you may work with beliefs or ideas you know to be inconsistent, but just avoid the situations where you have to face the inconsistencies. It might work just fine. In some areas, e.g. law, you may have to work with case decisions that are inconsistent with each other, and try to find a way through the situation. Basically we’d be hugely handicapped in our everyday life if we had to work only with consistent sets of ideas. So - sometimes it’s interesting to work with inconsistent sets of axioms in maths too.

So you can do all that. And you can also work with consistent theories of course. Or at least, theories that you have every reason to believe are consistent even though you can’t prove this.

Just use the axioms you can see to be true of numbers, based on Peano’s axioms, and then Godel’s sentence - see that it’s true - and so unprovable. Add it as an axiom. Just keep going. And you can then expand your axiom systems as needed in creative ways if you need to go beyond their limitations. Just on and on, as much as you like.

So - it means maths has to be creative, and never ending. And we can never know for sure that it is consistent once it reaches a certain level of complexity and power. But it doesn't have to be inconsistent.

Yes, there is a case of “once bitten twice shy” that when Frege published his life’s work, a foundation for all of set theory, then Russell found a mistake in it, he proved that it was inconsistent through Russell’s paradox. It could be fixed, but only in clumsy ways. This suggests that it’s not as easy to come up with a theory that you know for sure is consistent as one might think. Indeed Godel’s theorem shows we can’t prove that even the Peano axioms are consistent.

But I think most mathematicians would say that there is no real risk that the axioms are inconsistent in the sense of Frege’s set theory. We don’t need to worry that some ingenious fellow will pop up, like Russell did for Frege, and say “look, here is a proof of an inconsistency from the axioms of Peano arithmetic”. Most mathematicians would say we don’t need to worry about that possibility.

They are just impossible to encapsulate completely in an axiom system that captures all their properties. That’s why they aren’t decidable, are incomplete, and can’t be proven to be consistent.

Now, I should mention Gödel's completeness theorem because this confuses many people. It’s a different notion of completeness from the one used in Godel’s “incompleteness theorem”. It’s about the formal consequences of the axioms, not the informal consequences that you can see by reasoning about the theory in a meta mathematical way.

It says that if you enumerate all the proofs that you can make using the axioms of the theory - those are all the results you can deduce from the axioms and the only results you can deduce. There are no hidden extra axioms you need to add to “complete” the theory - it’s all there.

Put another way, then it says that if you interpret truth as “true in every possible model” then all such truths can be deduced from the axioms of your theory.

I should say - this is stuff I researched into back in the 1980s. I haven’t done any work on it at all since then, and if you asked me to go into details about some of these things, I’d have to “look it up and get back to you”. But hopefully it gives a reasonable idea of how it works.

It’s rather baffling, for sure. But I think the best way to look at it myself is that Godel proved that mathematics can’t stagnate and that there is need for endless creativity as it can’t be “cut and dried and diced” into a single overall theory that encompasses everything.

But some theories, notably geometry, can be made completely decidable, consistent and complete.

Also all the problems here are to do with infinity. If you are studying something finite, for instance a finite group, with a finite number of elements, so all questions can be answered by doing finite, if very complicated calculations - then that’s going to be a complete, consistent and decidable theory too. It’s just when you start having infinity come in in some way, that you may (but not always) hit this.

You might also be interested in my Russell's paradox