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I thought that Godel demonstrated that there are theorems in a formal system that cannot be proved in that system. Physics is not expressed with the formalism t

by sgt101 6y ago

I thought that Godel demonstrated that there are theorems in a formal system that cannot be proved in that system. Physics is not expressed with the formalism that is reality, and there are different choices of formalism that can be used to express the concepts of physics. I don't think that Godel showed that there were statements that can't be proven in any system.

syops 6y ago
Assume consistency. Take the collections of all true statements about the standard model of the natural numbers and make these statements your axioms. Now every true statement about the standard model of the natural numbers is provable. It’s a useless axiomatic system since there is no procedure for knowing if a statement is an axiom of not (that works for every statement). The second order Peano axioms are categorical and assuming consistency every true statement is ‘provable’.
- ProfHewitt 6y ago
  Higher-order Dedekind/Peano axioms are categorical in the sense that they characterize Natural Numbers up to a unique isomorphism. However, there is still a proposition about Natural Numbers that true but unprovable using the Dedekind/Peano axioms. Further information can be found here: "Physical Indeterminacy in Digital Computation" https://papers.ssrn.com/abstract=3459566 https://papers.ssrn.com/abstract=3459566
  - syops 6y ago
    So the last sentence I made is false. The rest is correct, right?
    - ProfHewitt 6y ago
      Yes, the last statement that you made is indeed false, that is, it is not the case that every true proposition is provable in Dededkind's theory.
thaumasiotes 6y ago
> I don't think that Godel showed that there were statements that can't be proven in any system. He definitely didn't, since we know of systems that are capable of proving any statement. It's sufficient for the system to be internally inconsistent. Outside of that, we'd need to be a little more clear about what we mean by "there are statements that can't be proven in any system". I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving. In the general case, I would expect that a statement, taken by itself, would constitute a consistent (if uninteresting) system. That system would be easily able to prove its only axiom. In my mind, the incompleteness theorem says that a consistent system of "sufficient power" cannot prove its own consistency. We could reify that into a statement that looks like "This system is consistent"; that's what I'm trying to rule out above by referring to "a statement with a fixed meaning". "This system" would refer to different things within different systems. Wikipedia says this: > The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. > For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. > The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. This allows for a single statement (say, a result about the natural numbers) that no sufficiently powerful consistent system can prove, but it doesn't require it. The sets of true-but-unprovable statements might be disjoint between two particular systems.
- sgt101 6y ago
  That's a really good framing. I like it.
- Viliam1234 6y ago
  > I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving. A typical example -- now that I think about it, the only example I have seen -- is the statement "this statement cannot be proved". Technically, it is not a single statement, because for different axiomatic systems A1, A2... you get different statements "this statement cannot be proved in A1", "this statement cannot be proved in A2" etc. For any given axiomatic system A, if the system can prove "this statement cannot be proved in A", it contains a contradiction: it just proved something that is not true. But if the system cannot prove "this statement cannot be proved in A", we have an example of a statement that is true but unprovable (in A). Gödel's trick was inventing a way how a statement can talk about itself. Like, if you say "it is forbidden for a statement to talk about itself", Gödel asks "but surely it is okay for a statement to talk about another statement that it can define unambiguously?" and then creates a statement that sounds like "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A", but when you actually unpack "7461287618276312876873", you get "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A" again. So it talks about itself without actually using the words "this statement". What exactly this all means... I wish I would grok, but I don't. But I am pretty sure it means none of the usual "human brain has mystical magical quantum powers beyond mere computation".
wedesoft 6y ago
In math you can prove statements given a set of axioms. But in physics you can just validate things with experiments.
- sgt101 6y ago
  Odd you were downvoted ! I think that this is a good insight because it underlines physics!=math!=physics.

by sgt101 6y ago

syops 6y ago

Assume consistency. Take the collections of all true statements about the standard model of the natural numbers and make these statements your axioms. Now every true statement about the standard model of the natural numbers is provable. It’s a useless axiomatic system since there is no procedure for knowing if a statement is an axiom of not (that works for every statement). The second order Peano axioms are categorical and assuming consistency every true statement is ‘provable’.

ProfHewitt 6y ago
Higher-order Dedekind/Peano axioms are categorical in the sense that they characterize Natural Numbers up to a unique isomorphism. However, there is still a proposition about Natural Numbers that true but unprovable using the Dedekind/Peano axioms. Further information can be found here: "Physical Indeterminacy in Digital Computation" https://papers.ssrn.com/abstract=3459566 https://papers.ssrn.com/abstract=3459566
- syops 6y ago
  So the last sentence I made is false. The rest is correct, right?
  - ProfHewitt 6y ago
    Yes, the last statement that you made is indeed false, that is, it is not the case that every true proposition is provable in Dededkind's theory.

thaumasiotes 6y ago

> I don't think that Godel showed that there were statements that can't be proven in any system. He definitely didn't, since we know of systems that are capable of proving any statement. It's sufficient for the system to be internally inconsistent. Outside of that, we'd need to be a little more clear about what we mean by "there are statements that can't be proven in any system". I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving. In the general case, I would expect that a statement, taken by itself, would constitute a consistent (if uninteresting) system. That system would be easily able to prove its only axiom. In my mind, the incompleteness theorem says that a consistent system of "sufficient power" cannot prove its own consistency. We could reify that into a statement that looks like "This system is consistent"; that's what I'm trying to rule out above by referring to "a statement with a fixed meaning". "This system" would refer to different things within different systems. Wikipedia says this: > The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. > For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. > The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. This allows for a single statement (say, a result about the natural numbers) that no sufficiently powerful consistent system can prove, but it doesn't require it. The sets of true-but-unprovable statements might be disjoint between two particular systems.

sgt101 6y ago
That's a really good framing. I like it.
Viliam1234 6y ago
> I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving. A typical example -- now that I think about it, the only example I have seen -- is the statement "this statement cannot be proved". Technically, it is not a single statement, because for different axiomatic systems A1, A2... you get different statements "this statement cannot be proved in A1", "this statement cannot be proved in A2" etc. For any given axiomatic system A, if the system can prove "this statement cannot be proved in A", it contains a contradiction: it just proved something that is not true. But if the system cannot prove "this statement cannot be proved in A", we have an example of a statement that is true but unprovable (in A). Gödel's trick was inventing a way how a statement can talk about itself. Like, if you say "it is forbidden for a statement to talk about itself", Gödel asks "but surely it is okay for a statement to talk about another statement that it can define unambiguously?" and then creates a statement that sounds like "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A", but when you actually unpack "7461287618276312876873", you get "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A" again. So it talks about itself without actually using the words "this statement". What exactly this all means... I wish I would grok, but I don't. But I am pretty sure it means none of the usual "human brain has mystical magical quantum powers beyond mere computation".

wedesoft 6y ago

In math you can prove statements given a set of axioms. But in physics you can just validate things with experiments.

sgt101 6y ago
Odd you were downvoted ! I think that this is a good insight because it underlines physics!=math!=physics.