> Moreover, one could play word games with the mathematical language, creating problematic statements like “this statement is false” (if it’s true, then it’s false; if it’s false, then it’s true) that indicated there were problems with the axiomatic system.
I hope this isn't too far off topic, but can someone clarify exactly how this problem indicts axioms? As an uninformed and naive musing, it occurs to me that an issue with the statement "this statement is false" is this. The whole of the statement, that is, the thing having truth or falsehood, cannot be addressed by one of its components.
He's alluding to the fact that unlike what one might naively intuit, it's impossible to formulate a set of axioms that can formally express all mathematics.
Because of Godel's incompleteness theorems, any formal system that's sufficiently powerful to formulate basic arithmetic (see for instance Robinson arithmetic) and consistent (that is, it cannot prove false statements) is both incomplete (meaning that it's possible to formulate a wff within that system that the system itself can neither prove nor disprove) and can't prove it's own consistency.
This fact is often portrayed in popular media as being a "bug" or a "missing foundation" of mathematics. That is inaccurate -- it's just a property of how logical systems work -- but it does prove that the search for the holy grail of a grand unified system of axioms for all of mathematics is destined to remain fruitless.
Modern mathematics is most often implicitly assumed to be formulated in a formal system called ZFC, but there are alternatives.
I think the second theorem is the "worse" result for mathematics. It shows that it's completely impossible to use a weaker system, which is hopefully self-evident, to prove the consistency of a stronger, but weirder system like ZFC.
Hilbert wanted to show that, even if you don't "believe" in ZFC, we could at least try to convince you that it doesn't lead to contradictions, but that fundamentally didn't work.
Oh this is interesting, I didn't know about this historical detail.
It makes sense, Godel's first could be mostly seen as a version of the halting problem, while the second is "game over" for a Hilbert-like foundation program.
p.s. Just wanted to point out that it's funny how we wrote two mostly identical sets of comments. Great minds think alike? :)
> Oh this is interesting, I didn't know about this historical detail.
That's the characterisation I got from Peter Smith's book about Gödel's theorems. I didn't verify original sources or anything, but it sounds very plausible to me.
And it also kind of answers the question about why we should care: if a system can't prove its own consistency, well that's not terribly interesting (even if it could, we would have to believe the system a priori to trust its own consistency proof). But if it also can't prove a stronger system consistent, then that's much more interesting.
> Great minds think alike?
That would be a bit too self-aggrandizing for me. :D I'm very curious about logic (and maths in general), that's why I know some stuff about it, but I'm no maths genius.
Is the fact that these systems cannot prove their own consistency actually a feature of the incompleteness theorem? I thought it effectively boiled down to "you can keep either consistency or completeness, not both". It's been a while since my metamathematics course ...
It depends on exactly what you mean with those terms.
When you say "you can keep consistency or completeness but not both" you are essentially stating Godel's first.
Morally speaking, Godel's first is "no system is complete", but there are two exceptions: if the system isn't powerful enough to formulate the Godel sentence then the proof doesn't work, and any inconsistent system is trivially complete because ex falso quodlibet sequitur, i.e. any wff is true.
The part about a system not being able to prove its own consistency is Godel's second theorem.
But the theorem only does what it says on the tin: the system not being able to prove its own consistency doesn't mean that it being inconsistent! [1]
This is the case for ZFC, for example. We can't prove ZFC's consistency within ZFC: that would violate Godel's second, and we know that either ZFC is inconsistent (that is, you can derive an absurd from the axioms) or ZFC is incomplete (that is, there exists a well-formed formula of ZFC that cannot be proved or disproved within ZFC).
We don't know which one it is.
[1] This implication and similar ones are often made in pop-culture presentations of the foundation problem. I generally vigorously object to them because they're not only imprecise -- that's understandable -- but they leave a layman with a completely wrong impression.
> the system not being able to prove its own consistency doesn't mean that it being inconsistent!
Another funny thing that can happen is that a system proves its own inconsistency, despite being consistent. The short summary is to never trust a system talking about its own consistency.
You're precisely right that it's the self-referentiality that gets us in trouble. Systems which don't have that don't suffer from Gödel-related problems (e.g. propositional logic, or arithmetic without multiplication).
Unfortunately, what Gödel showed is that any logical system that is strong enough to capture addition and multiplication over natural numbers - and nothing more - includes self-referentiality.
That includes the language of arithmetic - you can write down formulas that refer to themselves - and Turing machines (via the recursion theorem, any Turing Machine is equivalent to a machine that knows its own source code). The constructions are a bit technical, but it's possible.
Yeah I always thought that the construction of Gödel numbers was always the weakest part of the proof / the biggest leap of faith / the part your prof would just hand-wave as being a valid move.
Of course once you get into Turing machines it all flows more naturally, what with all of us being accustomed to "code is just data".
I agree that Turing Machines feel more natural to programmers than first-order logic (although "natural" doesn't necessarily mean "rigorously proven"), but there are no leaps of faith involved in Gödel's construction.
You can write down "P is provable" as a first-order sentence of arithmetic (which involves some number theoretic tricks), and you can also do the diagonalisation trick that gives you self-referentiality. That's really all you need.
The article is a bit oversimplifying in summarizing the axiomatic crisis as being problem with sentences like “this statement is false“.
This being said, your intuition is absolutely correct, the crux of the issue is with ‘this‘. What mathematicians realized is that if you are not careful with your choice of axioms, the resulting logical system becomes too “powerful” in the sense that it becomes self-referential: you can construct sentences that refer to themselves in a self-defeating manner.
As others have mentioned, this is the idea underlying Gödel's incompleteness theorem but also, to some extent, Russel's paradox that came before and is what the article is referring to. In Russel's paradox, the contradiction comes from constructing the set of all sets that contain themselves.
Maybe in analogy to Russell's paradox and how you can "fix" it by distinguishing sets and classes, you can "fix" a Gödel sentence by adding it as an axiom, but then you'll just get a new Gödel sentence... and so on.
We would hope that the axioms fully characterised the thing they're meant to describe.
Can we even talk about "the" natural numbers at all? If you have two copies of the natural numbers, one red and one blue, are they both the same? Well, trivially they're not: one's red and one's blue. But (we'd hope) all the statements in the language of the natural numbers (which doesn't talk about redness or blueness) that are true of the red copy will be true of the blue copy, so it doesn't matter which copy you use, they're both "the" natural numbers.
E.g. why did we care about the parallel postulate? Why not just have geometry without a fith axiom? Well, because without it the axioms are, well, incomplete: there are statements in the language of geometry that cannot be proven from the other four axioms. You can have spherical geometry, planar geometry, and projective geometry, and they all conform to the first four axioms, but sometimes one of these geometrical statements will be true in one and false in another. So neither is "the" geometry, it matters which specific version of geometry you work in, and we need that fifth axiom to have a complete theory of (planar) geometry.
We'd hope to avoid that kind of situation with the natural numbers - if we need any additional "parallel postulate", we'd like to know about it, and if we don't, we'd like to be able to prove that we don't need one rather than just assuming it because we haven't stumbled across it yet. But Goedel proved that we cannot have a provably complete theory of the natural numbers with addition and multiplication, because he found a way to encode a statement akin to "this statement is false" in the language of the natural numbers. Which indicts any axiomatization of arithmetic - either your axiomatization proves this statement is true (in which case it's inconsistent), it proves this statement is false (in which case it's also inconsistent), or it doesn't prove this statement one way or another (in which case it's incomplete).
> As an uninformed and naive musing, it occurs to me that an issue with the statement "this statement is false" is this. The whole of the statement, that is, the thing having truth or falsehood, cannot be addressed by one of its components.
Well, yes, getting around that is the clever part of Goedel's proof :).
IANA mathematician, but I read "axiomatic system" broadly as including not just the axioms but also the logic in which they are based. My understanding is that a common interpretation is that ZFC axioms are a list of 10 strings of symbols, which only have some sort of meaning when you pick a logic that gives meaning to these symbols. But I think also that this particular understanding of what axioms are ("formalism") is just one way of understanding truth in mathematics, and there are others. https://en.wikipedia.org/wiki/Philosophy_of_mathematics
This idea of giving "meaning" to a set of axioms is precisely captured by the notion of "interpretation" in logic [1]. The rough idea is to map the symbols of the formal language to some pre-existing objects. As you say, this gives one way of formalizing truth: a sentence (string of symbols that respect the syntax of your language) is true if it holds for the objects the sentence is referring to (via a chosen interpretation). This notion of truth is sometimes referred to as semantic truth.
An alternative approach is purely syntactic and sees a logical system as collection of valid transformation rules that can be applied to the axioms. In this view, a sentence is true if it can be obtained from the axioms by applying a sequence of valid transformation rules. This purely syntactic notion of truth is known as “provability”.
Then the key question is to ask whether the two notions coincide: one way to state Godel's first incompleteness theorem is that it shows the two notions do not coincide.
> Then the key question is to ask whether the two notions coincide: one way to state Godel's first incompleteness theorem is that it shows the two notions do not coincide
It's even more subtle than that. They do coincide in a sense, which is proven by Gödel's completeness theorem (well, at least in First-Order Logic). That one just says that a sentence is provable from some axioms exactly iff it's true in every interpretation that satisfy the axioms.
So one thing that Gödel's first incompleteness theorem shows it's that it's impossible to uniquely characterise even a simple structure such as the natural numbers by some "reasonable"[0] axioms - precisely because there will always be sentences that are correct in some interpretations but not in others.
Unless you use second-order logic - in which case the whole enterprise breaks down for different reasons (because completeness doesn't hold for second order logic).
[0] reasonable basically means that it must be possible to verify whether a sentence is an axiom or not, otherwise you could just say that "every true sentence is an axiom"
Agreed, thanks for the clarifications. Another result worth mentioning, which also shows that you cannot hope to uniquely characterize a structure by "reasonable" axioms is the Löwenheim–Skolem theorem which predates Godel's incompleteness (although the history of these results is somewhat convoluted).
There, the obstacle is in some sense of a simplest nature: if your set of axioms admits a countable model, then it admits models of all infinite cardinalities. In other words, it shows that there is something fundamentally impossible in trying to capture an infinite structure (like numbers) by finite means (e.g. recursively axiomatizable).
> In other words, it shows that there is something fundamentally impossible in trying to capture an infinite structure (like numbers) by finite means (e.g. recursively axiomatizable).
Let me just point out for other readers that Löwenheim-Skolem applies to ANY first order theory (in a countable language, or also in an uncountable language if stated in the form that a theory with an infinite model with cardinality at least that of the language has infinite models in all cardinalities at least as big as that of the language), it doesn't care about how complex the axioms are from a computability point of view
You're correct, but I think the spirit of what GP wrote is still true.
You can't capture an infinite structure fully by "finitary" methods, either you use FOL and then you run into L-S, or you use higher-order logic (for which L-S doesn't apply) but then you don't have a complete proof system anymore.
To tie it all together, L-S and incompleteness are about different flavours of "not being able to capture something". L-S is about models of different cardinalities. These models do still all satisfy exactly the same sentences. Incompleteness is about different models actually satisfying different sentences.
I hope this isn't too far off topic, but can someone clarify exactly how this problem indicts axioms? As an uninformed and naive musing, it occurs to me that an issue with the statement "this statement is false" is this. The whole of the statement, that is, the thing having truth or falsehood, cannot be addressed by one of its components.