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> 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 irration

> 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".

lukaa 8y ago
"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.
joe_the_user 8y ago
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).
- madmax96 8y ago
  What’s the problem with the disjunction?
  - joe_the_user 8y ago
    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.
- james_s_tayler 8y ago
  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?
  - joe_the_user 8y ago
    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.
truantbuick 8y ago
> 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.
- skh 8y ago
  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.
  - whatshisface 8y ago
    How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.
    - AlexCoventry 8y ago
      That's the joke.
skh 8y ago
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.
- infinity0 8y ago
  > 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.
  - skh 8y ago
    I’m not used to seeing “axiomatic model”. When I read that I thought you meant axiomatic system and not model. Sorry.
KMag 8y ago
> 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.
- paganel 8y ago
  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”.
antidesitter 8y ago
> That doesn't mean "there are truths that humans can never prove" Well, if there’s an analogue of a Gödel sentence for humans...

lukaa 8y ago

"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.

joe_the_user 8y ago

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).

madmax96 8y ago
What’s the problem with the disjunction?
- joe_the_user 8y ago
  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.
james_s_tayler 8y ago
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?
- joe_the_user 8y ago
  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.

truantbuick 8y ago

> 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.

skh 8y ago
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.
- whatshisface 8y ago
  How would you be able to take every true statement as an axiom? Without proving anything I don't see how you could identify any statements as true.
  - AlexCoventry 8y ago
    That's the joke.

skh 8y ago

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.

infinity0 8y ago
> 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.
- skh 8y ago
  I’m not used to seeing “axiomatic model”. When I read that I thought you meant axiomatic system and not model. Sorry.

KMag 8y ago

> 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.

paganel 8y ago
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”.

antidesitter 8y ago

> That doesn't mean "there are truths that humans can never prove" Well, if there’s an analogue of a Gödel sentence for humans...