Actually, Gödel's theory is fairly accessible compared to, say, the General Theory of Relativity.
All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.
Then you introduce the idea of a model and make the concept fully exact.
The thing to remember is Gödel himself never really liked this simple view and wanted truth to be more transcendent. He shared with Einstein the quality of not liking the results of his discoveries, perhaps a source of their friendship.
It's true, a glimmer of the proof can be easily grasped. The analogy between the Godel sentence and the Liar's Paradox, "this sentence is false", is not perfect but almost isomorphic. I think this is rather the most curious feature of the theorem, that a 5-year old could understand it, at least a glimmer of it. Now there is a big gap between grasping it somewhat and being able to consider its full ramifications.
It's definitely easier than General Relativity. I read Goedel's original paper when I was 19, whereas there are aspects of GR I'm still getting my head around.
The quality of being “true” is dependent on the model one is using. One can not talk about “truth” without being in a model. (Assuming we are talking about standard mathematical logic.). A statement in a first order system is provable if and only if it is true in all models for that system.
You either don't understand the point of the theorems or are being contrarian.
In the context for the Incompleteness theorems, there are two 'kinds' of truths: a more informal kind used by all of us everyday and the mathematical kind as in, proven true under a given system.
The entire purpose of the theorems is to establish that there exists theorems in the first set that are not in the second set, while being expressable in the system. To deny the existence of the first kind of true statements ignores the entire purpose of it all.
I used the term ‘truth’ as used in mathematical logic. In a given model a statement can be true or false. Under a given axiomatic system a statement is either provable or not. We don’t use the word “true” when dealing with statement under an axiomatic system. We do use the word when dealing with a statement in a given model.
Your third paragraph doens’t make sense. In first order logic a theorem is a statement that is true in all models and is one that is provable. This is a result of the Completeness Theorem.
i don’t know if it’s accurate, but i get the sense godel liked finding pathologies. for example, his “godel metric” as a solution to einstein’s general relativity showcased “unphysical” possibilities but was still a legitimate solution.
" He announced that he had studied the US constitution in detail, and—no doubt, forensically examining its propositions one at a time and perhaps testing it against thought experiments against wild possible futures in which the president was allowed to get out of control—he had discovered how the US could legally be turned into a dictatorship."
sadly this article skips a lot of detail on what happened during and before the hearing and how Einstein tried to coach him. The New Yorker had a much better summary on this. Here in all its hilarity:
So naïve and otherworldly was the great logician that Einstein felt obliged to help look after the practical aspects of his life. One much retailed story concerns Gödel’s decision after the war to become an American citizen. The character witnesses at his hearing were to be Einstein and Oskar Morgenstern, one of the founders of game theory. Gödel took the matter of citizenship with great solemnity, preparing for the exam by making a close study of the United States Constitution. On the eve of the hearing, he called Morgenstern in an agitated state, saying he had found an “inconsistency” in the Constitution, one that could allow a dictatorship to arise. Morgenstern was amused, but he realized that Gödel was serious and urged him not to mention it to the judge, fearing that it would jeopardize Gödel’s citizenship bid. On the short drive to Trenton the next day, with Morgenstern serving as chauffeur, Einstein tried to distract Gödel with jokes. When they arrived at the courthouse, the judge was impressed by Gödel’s eminent witnesses, and he invited the trio into his chambers. After some small talk, he said to Gödel, “Up to now you have held German citizenship.”
No, Gödel corrected, Austrian.
“In any case, it was under an evil dictatorship,” the judge continued. “Fortunately that’s not possible in America.”
“On the contrary, I can prove it is possible!” Gödel exclaimed, and he began describing the constitutional loophole he had descried. But the judge told the examinee that “he needn’t go into that,” and Einstein and Morgenstern succeeded in quieting him down. A few months later, Gödel took his oath of citizenship.
EDIT: something I also missed in this piece was that Gödel developed rheumatic fevers as a child and started reading medical books with the age of 8 to learn more about the condition. He concluded that he had a weak heart :D
Gödel is really worth studying closer and this article just leaves out a lot.
Coincidentally, I'm reading When Einstein Walked with Godel, and this anecdote is in there. Such an interesting man -- he's up there with Von Neumann for historical figures I'd like to learn more about, as the work they develop is a form of first principles that a lot of modern thought is derived from.
Any reccs on books to read, more focused on their theories than their lives?
> rescued the idea that there are truths that humans can never prove
This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic.
What Gödel showed was that proofs are relative to some underlying axiomatic model and that for any particular axiomatic model there are always truths unprovable by it.
That doesn't mean "there are truths that humans can never prove", all it means is that we have to extend our axiomatic systems in order to prove some truths. (And whether they are true or false "in reality" is another question, to be determined empirically.)
A more accurate (and just-as-click-baity) way of interpreting Gödel's theorem is that "mathematicians and logicians will always have a job".
> That doesn't mean "there are truths that humans can never prove", all it means is that we have to extend our axiomatic systems in order to prove some truths.
If you believe that the only consequence to Gödel's theorem is we need to "extend our axiomatic system", I do think you've missed the point.
For one thing, I think Gödel's theorem and Gödel's proof are unfortunately conflated.
Gödel's proof is lovely and elegant, and can be understood with minimal knowledge of logic, but, ultimately, all it does is provide a counter example. So when people read his proof and understand it, they tend to be unimpressed with its power because the counter example is very generic and seems an uninteresting barrier to our ability to discern "truth". But that says nothing about there may be lots of other kinds of unprovable statements.
Ultimately, Gödel's theorem tells us one absolute kernel of truth by saying all but very basic axiomatic models are necessarily incomplete. However which way you want to make the philosophical leap to connect that to our notion of "truth" seems far more up to interpretation, but it annoys me when people disrespect the theorem because they're unimpressed with the counter examples the proof constructs.
An important nitpick. His Incompleteness Theorem deals with recursively enumerable axiomatic systems. The second order Peano Axioms are categorical. That is, they have only one model up to isomorphism. It’s easy to come up with a complete axiomatic system for the standard model of the natural numbers. Just take as your axiomatic system the collection of all true statements. This ins’t a useful system since there is no procedure for determining if a statement is an axiom or not.
Whenever you have a structure M of some language L you can take the so called complete theory of the structure, denoted Th(M), which is just the set of all L-sentences true in M. In particular if L is the language of PA and M are the standard natural number with the standard operations Th(M) is a theory called true arithmetic. This theory is complete (that's because Th(M) is always complete) and clearly enough to talk about the integers, but it escapes Gödel's theorem since it's axioms are not recursively axiomatizable.
You seem to be confusing "true" and "provable", as far as first order logic is concerned those are equivalent (by another theorem of Gödel, the completeness theorem), but the first in defined in terms of models and the second is purely syntactic
Informally speaking we can take the set of all real numbers. Assuming we are using standard mathematics that most working mathematicians use then this includes non-computable numbers. Sets don’t have to be computable to be used unless you are someone who works with non-standard models of set theory.
I specified I'm working in standard first order logic. And I'm not sure what do you mean with the informal take.
Also intuitionist logic is not my area, but isn't it divided in inference rules for provability and (Heyting or Kripke) semantic for model theory and truth just like FOL?
You may be working in classical logic, but that doesn’t mean the parent poster is. You asserted the parent poster seems to be confused, but what the parent poster said makes sense in a constructive setting. Please see my other comment:
The point is that the set of all true statements about arithmetic [1] exists “out there” (and thus as a “theory” in a very general sense) even if we can’t identify it.
This is not acceptable to a constructivist, for whom “a statement is true if we have a proof of it, and false if we can show that the assumption that there is a proof for the statement leads to a contradiction”[1]. The parent poster, whatshisface, may be a constructivist.
[1] Troelstra A., D. van Dalen (1988) “Constructivism in mathematics: an introduction”
In the case of True Arithmetic, the true statements are the axioms. Thus a proof of any true statement consists of just invoking the corresponding axiom, which is a valid proof in any formal system (constructive or otherwise).
The weirdness of True Arithmetic comes not from proofs (which are trivial) but from the axioms themselves. You can object that it’s not a “reasonable” or “proper” theory because it’s not recusively axiomatizable (i.e. there’s no effective procedure for even deciding whether a statement is an axiom), but I don’t think that objection is specifically constructivist.
Indeed, the objection is against the method by which this system is supposed to be defined. As you say, we start with an implicit assumption that we don’t have an effective procedure to construct the set of statements that are true in the ambient metatheory. And so, this set exists in a classical sense, or “weakly exists”, and not in a constructive sense.
You are using the word “truth” but it is more correct to use provable/non-provable. What Godel showed is that -limiting the discussion to the natural numbers for simplicity - there are statements that are true in the standard model of the natural numbers that are not provable in the first order Peano Axiomatic system for the natural numbers. What this means is that such a statement will be false in some non-standard model of the natural numbers.
It’s important that we are talking about models of the first order Peano Axioms. One can always find a system of axioms in which all true statements are provable. To do this just take the collection of all true statements in the standard model. Now every true statement is a theorem. It’s easy to have a complete set of axioms. What can’t happen is a recursively enumerable set of axioms that is complete and consistent.
Recursively enumerability is needed so that one can have an effective means of determining if a statement is an axiom. Think computable when I say effective.
When talking about truth we need to be careful because this is tied to a model of an axiomatic system. By the Completeness Theorem a statement that is true in all models of the system is provable.
> You are using the word “truth” but it is more correct to use provable/non-provable.
I used both the words "truth" and "provable" in the correct and appropriate ways. Both are distinct concepts that form an important part of the theory.
This is a gross misinterpretation of Gödel's actual theorem that helps perpetuate irrational superstitious attitudes against science, mathematics, and logic.
It is, it really is. Yet is also an interpretation that Gödel himself would indulge in.
Consider his most famous quote: "Either mathematics is too big for the human mind, or the human mind is more than a machine." (I remember reading this statement in the Time-Life book on mathematics when I was a kid).
But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but that conjecture couldn't be proven under the given set of axioms... then in that case wouldn't it be that there were some truths about the universe that aren't possible to prove?
I don't think it's irrational or anti-science to say or think that. It just seems like it might be a possibility, which sure, why the hell not?
"There's this thing we can't prove under the given axioms"..."Ok, well just extend the axioms"..."Right but there is still this other thing we can't prove under that set of axioms."..."Ok, so rinse and repeat?"
How, if you can't prove it, do you figure out if it is an axiom in the first place?
Questions, question... I invite to learn some mathematical logic but I can give points.
But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but that conjecture couldn't be proven under the given set of axioms... then in that case wouldn't it be that there were some truths about the universe that aren't possible to prove?
- Well, the axioms of a given model generally can't be proven. Every model one builds begins with "unprovable things". However, the "truth about the universe" would have to be demonstrated by experimentation and mathematics would only provide the model. So every theory about the universe is using something unprovable (the axioms of its model). But Godel's theorem in particular (technically Godel's 1st incompleteness theorem) is still just a statement about proof processes work and how truth-assignments work. Whether are ineffable truths or not is a different question.
"There's this thing we can't prove under the given axioms"..."Ok, well just extend the axioms"..."Right but there is still this other thing we can't prove under that set of axioms."..."Ok, so rinse and repeat?"
- Yes, you can do that, well you can talk about doing it. In fact, Godel's "completeness theorem" is based on this procedure, more or less. The thing is you start with a set of axioms, add a proposition that can't be proven either true or false under the axioms, and decide arbitrarily whether to make it true or false. Continue forever (or transfinitely, depending how huge your system is) and you get a complete truth-assignment. Of course, there's no algorithm for finding all these undecidable propositions so this procedure is very theoretical. You or I or an AI couldn't do this to arrive pocessing this complete set but mathematically you can say "I hereby choose all the unprovable axioms and string them together thus (one of those weird "axiom of choice" things). So this approach is used, it's just it doesn't us to escape unprovable things.
Thanks for this answer. You helped me think about things a little deeper. I also read through the recent guide on how to self-study logic and learned a couple more things on the subject.
If I'm understanding things right it's that proving things with logic means you are showing that something is consistent within a system and that is all you are definitely able to say from that. Not whether something is 'true' in a wider sense outside of that system. And all axioms are assumptions, so all our proofs are saying is that 'assuming that these things are the case then we can show the following result is commensurate with those assumptions and completely internally consistent'.
Well, the only thing that Godel showed was that if you have consistent assignment of a truth value to every well-formed statement in a given logic language (at least as complex the Peano postulates), you will get some statements which are assigned "true" or "false" but whose truth cannot be deduced in a given axiom system for the language. However, the above statement very much involves a statement about "truth" in much transcendent sense - see the discussion of Godel's Platonism and religious beliefs in the article.
> That doesn't mean "there are truths that humans can never prove", all it means is that we have to extend our axiomatic systems in order to prove some truths.
But the very interesting part is that you might keep extending and extending two axiomatic systems until every true statement is provable in one or the other system. However, you're guaranteed that in that case, the two systems will be inconsistent with each other such that they cannot be combined into a consistent whole.
Genuine question: how will we know that we have reached the point where “every true statement” is provable? What about unknown unknowns? Is there a “concept” (for a lack of a better word) for these “unknown unknowns” in modern logic? I know neo-Platonism (Plotinus and his friends) was circling around this idea/concept, but truth be told I haven’t read all that much coming from them, plus they may be reguarded by modern logicians/mathematicians as maybe too “mystical”.
"And whether they are true or false "in reality" is another question, to be determined empirically."
Mathematical and logic truths have highest apstraction level.By definition they can
't be proved or disapproved in reality they exist only in apstraction.
Also of interest is Godel's AMS Gibbs Lecture, which unfortunately I have not been able to locate on-line. It can be found in Volume 3 of Godel's Collected Works, alongside his paper / lecture notes on closed time loops in general relativity.
The biggest qualm that I have with the example for the incompleteness theorem, is the use of two-valued, Boolean/Aristotelian logic, in which "not true" automatically becomes "false".
If you allow the use of a three-valued logic, for example, with (true, undetermined, false), the Gödel statement, “I am not provable" amounts to saying "My provability is false or undetermined".
The same problem occurs in Russell's paradox, "Does the set of all sets that do NOT contain themselves, contain itself?"
The use of the NOT-operator is degenerated in a cyclic group Z2. It is the only situation in which the NOT-operator is not set-valued.
In my opinion, if the "isProvable()" predicate allowed for multi-valued logic, Gödel's incompleteness theorem would look much less paradoxical. In other words, the paradoxical outcome could simply be the result of Boolean shoehorning.
The incompleteness result does not even mention "true" and "false". It says that "there are statements of the language of F which can neither be proved nor disproved in F." Whether those statements are true/false is left unsaid.
Gödel's result, being constructive, is buildable without the Law of Excluded Middle, and it should be buildable in any topos that has Boolean logic and LEM. (And probably some other features, like infinities.) Recall that provability from a set of axioms is a path or a chaining-together of many small deductions, each one machine-checkable, and thus we can construct a proof mechanically. Therefore we are in the realm of the Booleans since that is the realm in which the proof checker is operating. Crucially, for a given set of axioms and a given arbitrary statement, either the proof does exist, in which case we may machine-check it and use the proof statement as a witness to the proof's existence, or it does not exist, and it is equivalent to the Halting Problem to show so.
To be pithy, if we choose a (true, undetermined, false) compatible with topos logic, we should be able to encode Gödel's work using only the "true" and "false" values.
You have hit upon something important, though. Recall that, in constructive logic, (WFF) statements are either true, false, or not false, where "not false" is not an actual intermediate truth value, merely a potential truth value. If we take as an axiom that statements that are not false are true (double-negation elimination) then we can derive LEM. Nonetheless, "not false" is a wonderful truth value to assign to the classic paradoxes.
To continue to be pithy, and to paraphrase a category theorist, "not false" is something that we see from the outside, but it isn't visible from the inside, and Gödel's theorems are all about encoding things into the inside.
Suppose, indeed, that we write out the Gödel statement G in some formalized English, as "This statement is true and unprovable within Formal English." Can G be true? Yeah, sure, it's true in the integers, but not in a way that Formal English can show. (That's the incompleteness!) Can G be false? Meh, yes, but it gets nasty, because G is true in the integers, so ~G leads to a non-standard model. Could G be not false? Surprisingly, yes!
I wonder whether this is the line of thought that led Bishop to his terminology.
> The theoretical physicist and mathematician Roger Penrose, for example, has argued that Gödel’s theorem shows that “Strong AI” is false: our minds cannot be computers, and that by extension the intelligence of computers will never fully replicate them.
Note that there are camps that have been disputing this (e.g. McCullough’s Objection):
All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.
Then you introduce the idea of a model and make the concept fully exact.
The thing to remember is Gödel himself never really liked this simple view and wanted truth to be more transcendent. He shared with Einstein the quality of not liking the results of his discoveries, perhaps a source of their friendship.