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Mathematics self-proves its own Consistency (contra Gödel et. al.)

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Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#21

Post by jfarmer »

EGreg wrote: post #12

"(There is a very weak theory called Provability Logic that has been used for self-referential propositions coded as integers, but it is not strong enough for the purposes of computer science.) " weak theory? not strong enough for computer science, eh? strange ways to talk about mathematical theorems that are either true or not. let's see him define what he means by any of these terms lol

"Strong" and "weak" have well-understood meanings in mathematical logic and model theory.

For example, imagine there's a true and "important" statement S about Turing machines for which we could prove "it is impossible to prove S using provability logic." In that sense PL would be "too weak" to do computer science.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#22

Post by ghswa »

acchow wrote: post #13
GuiA wrote: post #4

Carl Hewitt is an interesting character [1] and I'm not really sure what to make of it. There are signs of a bright mind gone completely bonkers, and there are signs of someone being rejected for purely political (and not academic) reasons. Both have happened numerous times in the past which is why it's hard to tell. If someone more enlightened on the topic cared to comment about the nature of things, that'd be great…

Why is LtU even bringing up this abstract? Are they just trying to further embarrass him?

Hewitt posted it himself.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#23

Post by gphilip »

stiff wrote: post #15

Carl Hewitt isn't just some random crank, he is the author of PLANNER, a breakthrough programming language in the early AI days, that gave rise to Prolog later on. This is a pretty interesting paper and it doesn't claim any big breakthrough in the sense of invalidating any great mathematical result. I can try to explain what it is about, at least the beginning of the quoted fragment: In the first paragraph, he shows…

> In the first paragraph, he shows that if you allow proofs by contradiction, you can use them for every theory claimed to be inconsistent to show it indeed is consistent. This is so, because to prove by contradiction means to infer "not A" if from "A" follows "B" and "not B". So, if from "mathematics is inconsistent" follows "theorem A is true" and "theorem A is false", it follows that "mathematics is consistent". He claims that this shows consistency isn't rigorously proved in classical mathematics but is just assumed, since classical mathematics allows proofs by contradiction.

So the first paragraph is about showing that if one allows statements about mathematics (e.g: "Mathematics is consistent") and statements in mathematics (e.g: "proposition A is true", for some statement A which can be expressed entirely "within" mathematics) to freely mix together on the same footing, then things become inconsistent? But this is well-known for over a century; see, e.g: Russell's Paradox [1]. And (if I understand it correctly), one doesn't need Proof by Contradiction for this inconsistency to occur---it suffices to allow meta-statements and "plain" statements to mix together; see Curry's Paradox [2]. And ways to work with metamathematical statements while avoiding this paradox where also worked out a long time back [3]. That the first paragraph of the OP mentions none of this seems ... surprising, to say the least.

[1] https://en.wikipedia.org/wiki/Russell's_paradox [2] https://en.wikipedia.org/wiki/Curry's_paradox [3] https://en.wikipedia.org/wiki/History_of_type_theory

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#24

Post by stiff »

gphilip wrote: post #23
stiff wrote: post #15

Carl Hewitt isn't just some random crank, he is the author of PLANNER, a breakthrough programming language in the early AI days, that gave rise to Prolog later on. This is a pretty interesting paper and it doesn't claim any big breakthrough in the sense of invalidating any great mathematical result. I can try to explain what it is about, at least the beginning of the quoted fragment: In the first paragraph, he shows…

> In the first paragraph, he shows that if you allow proofs by contradiction, you can use them for every theory claimed to be inconsistent to show it indeed is consistent. This is so, because to prove by contradiction means to infer "not A" if from "A" follows "B" and "not B". So, if from "mathematics is inconsistent" follows "theorem A is true" and "theorem A is false", it follows that "mathematics is consistent". H…

This is just an abstract. I think the novelty he perceives is combining all this into a proof of consistency of mathematics. Page 6 of his paper mentions everything you have just written about:

https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/...

This guy spent a few decades working in mathematical logic. He might overestimate the importance of some things, but I would be careful assuming he suddenly stopped understanding simple technical arguments.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#25

Post by losethos »

To the honest, You are honest; to the perverse, You are devious.

http://www.biblegateway.com/passage/?search=1 Kings+22&version=NIV

http://www.biblegateway.com/passage/?search=numbers%2011&ver...

http://www.biblestudytools.com/2-samuel/22-27-compare.html

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#26

Post by runarberg »

I've always been fascinated with naïve set-theory, and how it was refuted after it's apparent inconsistency, the famous Russell's paradox of the set of all sets that don't contain it self, and only contains it self if it doesn't etc. So supposing it's False, yields True, and supposing it's True yields False.

But what fascinated me was not the Russell's set, but it's inverse, the set of all sets that contain it self, lets call it S.

For the statement S is a member of S —the set of all sets that contain it self, really contains it self— yields True if and only if you suppose it's True, and it yields False if and only if you suppose it's False.

S is therefore always consistent no matter the value. Both values are correct, but never at the same time.

I don't know particularly what it means, but the fact that you can do this, makes me wonder if this super-consistency is applied anywhere else in the philosophy of science, but nobody has ever realized the equivalent inverse, and therefor never demonstrated the inconsistency.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#27

Post by zepolud »

stiff wrote: post #24
gphilip wrote: post #23

Earlier quoted context omitted.

> In the first paragraph, he shows that if you allow proofs by contradiction, you can use them for every theory claimed to be inconsistent to show it indeed is consistent. This is so, because to prove by contradiction means to infer "not A" if from "A" follows "B" and "not B". So, if from "mathematics is inconsistent" follows "theorem A is true" and "theorem A is false", it follows that "mathematics is consistent". H…

This is just an abstract. I think the novelty he perceives is combining all this into a proof of consistency of mathematics. Page 6 of his paper mentions everything you have just written about: https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/... This guy spent a few decades working in mathematical logic. He might overestimate the importance of some things, but I would be careful assuming he suddenly stopp…

He mentions those in relation to his own pet theories but in no way does he address GP's argument.

What his theorem boils down to is:

1. Let A be a formal system capable of describing the natural numbers. 2. Assume A is inconsistent. It follows that there exists such Φ that A ⊢ Φ and A ⊢ ¬Φ 3. ????? 4. A is consistent!

At step 3. he appears to conflate A⊢Φ ∧ A⊢¬Φ with Φ∧¬Φ and use it to refute the assumption in 2. But Φ∧¬Φ is not generally true outside the context of A and therefore the refutation does not follow.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#28

Post by asayers »

The proof offered in the abstract demonstrates a simple link between consistency and the validity of proof by contradiction. It shows that if mathematics is consistent (ie. ⊢Φ and ⊢¬Φ is impossible) then mathematics is consistent.

This is NOT a self-proof - it is a meta-proof. Taking arithmetic as an example, a self-proof of consistency would be a derivation of the consistency sentence (ie. "I am consistent", or "¬◻(0=1)") from the arithmetic axioms. That is not what we have here.

The existence of a proof of the consistency of a theory does not put it at risk from Godel's 2nd - for that we require that a theory prove its own consistency.

As far as I can tell, Hewitt begins with a proof that consistency and the validity of proof by contradiction are equivalent, and then proceeds on the grounds that consistency is proven internally to the theory - which it is not.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#29

Post by stiff »

zepolud wrote: post #27
stiff wrote: post #24

Earlier quoted context omitted.

This is just an abstract. I think the novelty he perceives is combining all this into a proof of consistency of mathematics. Page 6 of his paper mentions everything you have just written about: https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/... This guy spent a few decades working in mathematical logic. He might overestimate the importance of some things, but I would be careful assuming he suddenly stopp…

He mentions those in relation to his own pet theories but in no way does he address GP's argument. What his theorem boils down to is: 1. Let A be a formal system capable of describing the natural numbers. 2. Assume A is inconsistent. It follows that there exists such Φ that A ⊢ Φ and A ⊢ ¬Φ 3. ????? 4. A is consistent! At step 3. he appears to conflate A⊢Φ ∧ A⊢¬Φ with Φ∧¬Φ and use it to refute the assumption in 2. Bu…

My understanding is that if proof by contradiction is allowed in A, Φ ∨ ¬Φ is a true sentence in A, so if you try to include the sentence "A is inconsistent" in A, it follows that A ⊢ Φ and A ⊢ ¬Φ, therefore in A Φ ∧ ¬Φ is true, but this contradicts Φ ∨ ¬Φ, therefore A is consistent. In other words, it is impossible to show the inconsistency of any system that includes the law of excluded middle.

This is really trivial in the end. Inconsistency is "Φ ∧ ¬Φ", law of excluded middle is "Φ ∨ ¬Φ", so in the assumption of law of excluded middle is hidden the assumption of consistency. That's what he means by

The above theorem means that the assumption of consistency is deeply embedded in the structure of classical mathematics

Edit: I agree with you in the end, the proof is valid, but the conclusion does not hold in A, but one "level" above, so it does not contradict Goedels theorem. Am I reading this right?

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#30

Post by voyou »

tlb wrote: post #19

I know Carl a little from the late 90s. I figured there was a 50% chance he would come up with something historically important, and a 60% chance he was crazy. (The two options are not quite mutually exclusive.) Indeed, most examples illustrating the Incompleteness Theorem involve self-reference. But, it seems hard to prove anything interesting in a system that precludes self-reference. BTW, the Y Combinator is a way…

"It seems hard to prove anything interesting in a system that precludes self-reference."

Indeed, isn't that one of the important features of Goedel's proof: that any system that can express basic arithmetic is capable of self-reference.

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