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Gödel, Escher, Bach: an in-depth explainer

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Re: Gödel, Escher, Bach: an in-depth explainer

#211

Post by allturtles »

davesque wrote: post #10

I've read GEB over many years rather in the way someone would read the Bible. I pick it up from time to time and enjoy chewing on one or two chapters of material. But I've yet to figure out if the book actually has a specific thesis. I know it's all about the power of interpretation and the way in which interpreting a formal system as self-referencing has the effect of completely blowing up the intended design of tha…

Others have pointed out what the core argument of the book is, so I'll point to the spot where I believe the thesis is stated most clearly (p. 709 in my copy, the 1980 Vintage Books paperback edition):

"My belief is that the explanations of 'emergent' phenomena in our brains--for instance, ideas, hopes, images, analogies, and finally consciousness and free will--are based on a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level. ...The self comes into being at the moment is has the power to reflect itself."

Re: Gödel, Escher, Bach: an in-depth explainer

#212

Post by SkyBelow »

layer8 wrote: post #73
SkyBelow wrote: post #48

Earlier quoted context omitted.

Is there a distinction between saying a system has something that is provable and untrue and saying the system is self contradictory (and the more generalized layman interpretation that the system is wrong).

See https://en.wikipedia.org/wiki/Consistency . Under the syntactic definition of consistency, a self-contradiction simply means that a particular statement and its logical negation can both be proved in the system. That doesn’t say anything about the truth of the statement.

Wouldn't the 'truth' of a statement be depending upon the axioms of where you are making the statement. Thus something proven is true and if the negation is proven it is false, and thus a system able to prove both is self contradictory and even basic logic no longer applies, thus we lose any real world application.

The opposite a system which has true statements we can't prove is still useful, even if there might be some problems we won't ever be able to solve. But a system which can prove both a statement and its negation loses meaning.

Or does it? Naïve set theory, despite Russel's paradox, and language in general, despite the local equivalent of "This statement is a lie." is still a useful tool, so maybe the same applies even to more formal systems?

Re: Gödel, Escher, Bach: an in-depth explainer

#213

Post by black_knight »

trashtester wrote: post #205
black_knight wrote: post #63

Earlier quoted context omitted.

> How can a statement that is unprovable be true? If you are a platonist and believe in some preferred model where every statement is decided this makes perfect sense. For the rest of us, this just means that in a sufficiently complex system there will be undecided statements. Which is not such a big surprise – but a rather awesome technical exercise!

Undecided is not quite the same as true. My understanding is that there can be statements that are necessarily true within a given set of axioms, but still unprovable using a proof of limited length.

Not sure what you are thinking of here. In first order logic if something is true in all models it is also provable. This is called completeness. And is one of the sanity requirements of a semantics.

Re: Gödel, Escher, Bach: an in-depth explainer

#214

Post by wikfwikf »

adastra22 wrote: post #203
wikfwikf wrote: post #188

Earlier quoted context omitted.

This is the best example - before it was shown that the Continuum Hypothesis was unprovable, many people believed, like the GP, that only uninteresting and artificial statements could be shown to be unprovable. The CH is undoubtedly meaningful, natural, and of huge interest to (a subset of) mathematicians.

What are the practical applications of CH?

lol

Re: Gödel, Escher, Bach: an in-depth explainer

#215

Post by cruegge »

bedman12345 wrote: post #208
cruegge wrote: post #196

Earlier quoted context omitted.

What I meant to say is that multiple models are not the only reason for something to be true but unprovable, the incompleteness theorem also holds in more general conditions. Concerning multiple models of ZFC: I'm always confused by such statements about the foundations of set theory itself, they seem weirdly self-referential. ZFC certainly can't prove that it has multiple (or even any) models. Does such a statement…

You simply use "intuitive mathematics", in other words: no formalization. That's at least what I got when I read books on set theory. Löwenheim-Skolem implies the existence of a countable model of ZFC. https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skole... "What I meant to say is that multiple models are not the only reason for something to be true but unprovable, the incompleteness theorem also holds in mor…

I see. Checking the formulation of the incompleteness theorem again, I noticed that I probably misunderstood something here: it indeed essentially requires proofs to be verifyable, which second order theories do not provide. So second order PA can (and in fact does) have a proof of every statement or its negation, without contradicting incompleteness, but provability is somewhat useless in this case. For theories that fit the conditions you listed, unprovability is indeed due to multiple models. Does that make sense?

Edit: however, consistent second order theories don't always have a model. In first order, if S is an undecideable statement in theory T, then both T+S and T+!S have models, both of which are models for T, so undecideable statements always come from multiple models. But that does not need to be the case in second order, so your claim "it is independent from the axioms, i.e. there exist multiple models" is not neccessarily true, i.e. there may be cases when decideability of some statement fails in a theory with a unique model. Or is there another argument for your claim?

Forgive me for spamming questions, I just try to understand how these things fit together. But maybe we should just stick to first order, since everything else is too weird anyway.

Re: Gödel, Escher, Bach: an in-depth explainer

#216

Post by erodommoc »

jcranmer wrote: post #101
red_trumpet wrote: post #51

Earlier quoted context omitted.

How can a statement that is unprovable be true? I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with

I'll explain it in a different way than normal. We can define a notion of complexity for any given integer as being the size of the smallest program that returns that integer. Obviously, I'm being imprecise here, but it should hopefully be clear that it is possible to get the details right and the precise nature of those details aren't going to be relevant for what follows. Now suppose we have a program that can find…

I feel like I'm missing something here - wouldn't the shortest program that returns the integer i just be "return i"? The length of that seems pretty easy to compute.

Re: Gödel, Escher, Bach: an in-depth explainer

#217

Post by widowlark »

krsw wrote: post #139
widowlark wrote: post #65

Earlier quoted context omitted.

The insult is the implication that thinking like this is somehow due to use of and exposure to psychedelics. Maybe some people need assistance in thinking deeply, Hofstadter certainly is not one of them.

Wouldn't it be insulting that you would somehow proclaim to know what did or did not lead to Hofstader's thinking? Cut the puritanical bullshit.

You are free to look up his views on the use of drugs

Re: Gödel, Escher, Bach: an in-depth explainer

#218

Post by denton-scratch »

gerdesj wrote: post #69
red_trumpet wrote: post #51

Earlier quoted context omitted.

How can a statement that is unprovable be true? I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with

"This statement is false". GEB is a marvellous work that is accessible to anyone with reasonably good school grade maths. I chanced upon it by accident in the school library one day and was hooked after a few pages. Anyway the crux of the matter is that you can very carefully construct a statement about a system that can't be either proven or disproven by that system! I don't have anything like the formal knowledge t…

I had to study WVO Quine at university. I found his fussy, fancy prose an obstacle to getting at his meaning.

> the links between maths, music and art.

Mr. H. is by reputation a very competent violinist; even though he's a mathematician, he can pronounce with some authority on subjects like Bach.

Re: Gödel, Escher, Bach: an in-depth explainer

#219

Post by denton-scratch »

davesque wrote: post #10

I've read GEB over many years rather in the way someone would read the Bible. I pick it up from time to time and enjoy chewing on one or two chapters of material. But I've yet to figure out if the book actually has a specific thesis. I know it's all about the power of interpretation and the way in which interpreting a formal system as self-referencing has the effect of completely blowing up the intended design of tha…

I think his "specific thesis" is an argument that intelligence (and he includes consciousness) are algorithmic. I think he fails to make that case; but the journey he leads us on is extremely entertaining. The book's a masterpiece.

Re: Gödel, Escher, Bach: an in-depth explainer

#220

Post by pdonis »

jstanley wrote: post #39

> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…

> or has strings which are provable but untrue.

Where are you getting this from?

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