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Comment by infinity0 | Hacker News Reader

> 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".

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> 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.

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.

That's the joke.

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.

[1] https://en.wikipedia.org/wiki/True_arithmetic

You are invoking a classical notion of existence.

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.

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

You seem to be forgetting that constructive truth means provability.

Also, it doesn’t seem right to use the informal “take” when the object in question is not computable.

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:

https://news.ycombinator.com/item?id=18756974

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.

"Gödel's proof is lovely and elegant, and can be understood with minimal knowledge of logic"

I suspect you haven't gone through the actual rigorous proof which is highly technical and requires more than a "minimal knowledge of logic"

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.

I’m not used to seeing “axiomatic model”. When I read that I thought you meant axiomatic system and not model. Sorry.

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).

What’s the problem with the disjunction?

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.

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.

- 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.

- 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'.

> 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.

> That doesn't mean "there are truths that humans can never prove"

Well, if there’s an analogue of a Gödel sentence for humans...

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