Godel's Incompletelness Theorem

[detrop]

A typical Saturday night at Ed’s house:

[Ed3]

Hi debaser;

In the book “Godel’s Proof on page 110 the author, I believe it is Hofstadter, writes “… as Godel himself appears to believe, only a thoroughgoing philosophical “realism” of the ancient Platonic type can supply an adequate definition, are problems still under debate and too difficult for further consideration here.”

In addition there is an extensive note (I love reading the notes, sometimes I think it is the best part of a book) that reads:

“Platonic realism takes the view that mathematics does not create or invent its “objects” but discovers them as Columbus discovered America. Now if this is true, the objects must in some sense “exist” prior to their discovery. According to Platonic doctrine, the objects of mathematical study are not found in the spatial-temporal order. They are disembodied eternal Forms or Archetypes, which dwell in a distinctive realm accessible only to the intellect. On this view, the triangular or circular shapes of physical bodies that can be perceived by the senses are not the proper objects of mathematics. These shapes are merely imperfect embodiment of an indivisible “perfect” Triangle or “perfect” Circle, which is uncreated, is never fully manifested by material things, and can be grasped solely by the exploring mind of the mathematician. Godel appears to hold a similar view when he says “Classes and concepts may…be conceived as real objects…existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence” (Kurt Godel “Russell’s Mathematical Logic,” in the Philosophy of Bertrand Russell (ed. Paul A. Schilpp, Evanston and Chicago, 1944), p 137)."

I think that all of this is very straightforward and gives credence to the view that Godel was a Platonist.

I do not have any additional information. Just speculation.

I am curious about your comment that this looks like set theory. Do you think that there is some isomorphism between Platonic numbers 0, 1, 2, 3,… and the sequence Φ, {Φ},{Φ,{Φ}}, {Φ,{Φ}, {Φ,{Φ}}},… which would allow us to identify these concepts as equivalent? I personally think there is a strong likelihood that this is true and the isomorphism is exactly as I have shown it. If this is so, I think that any effort by the Platonists to avoid incompleteness is completely doomed.

On the other hand, I am not sure that the incompleteness is really all that crippling. As a practical matter it seems like everything in life has flaws; and it allows us the opportunity to explore alternative, and possibly enriching, possibilities.

As a final thought, Russell’s paradox, famous though it is, seems to me like just another self referential statement. Can not set theory be constructed without these types of statements analogously to PM? The answer may well be no because Godel basically constructed his unproveable statement as a variation to the Richard paradox. Though, as I understand it, he went out of his way to insure that it was in fact not self referential. Additionally my initial reading of Set theory indicates that there are restrictions on the variables to avoid this type of thing.

Hi detrope,

I was serious about what I consider a merited compliment.

If you are curious about my personal opinions, and personality I would be happy to discuss them with you in a PM. I would draw the line on my thoughts about the Karma Sutra however.

[debaser]

Hi Ed,

Thanks for the info.

I was thinking about Godel and the platonic realm. Rather than worry about any isomorphic relationships we can come to two conclusions about Godels belief about it. Assuming the platonic realm is consistent:

1. The platonic realm is complete but any attempt to represent it as such is doomed to fail.

2. The platonic realm is incomplete and this incompleteness is reflected in our attempts to represent it.

3. is quite frankly weird and probably unjustifiable in the scientific sense but not in a faith sense. He could have believed it I suppose… but why?! (If you could hear the ‘but why?’ bit it would have sounded like a pondered whisper, not unlike James T Kirk [i’m not a trekkie])

I don’t think Plato would appreciate number 2. It’s like using his name in vain… I just find the platonist thing strange: it reminds me of the parllel clock thing (Leibniz??) with God and the world… what’s the point in even considering it? I suppose we would have to define ‘exist’ before we got onto that.

I don’t think incompleteness is crippling at all… it’s just weird. By weird i suppose i mean not intuitive. I thought about comparing it to the dilemna discussed in Wittgenstein’s skeptical Paradox - until my supervisor shot me down in flames and i dropped it. But if you were to think about it we can question our knowledge of our own consistency and we end up in the same boat - we could assert our own consistency but based on what fact?

Russell’s paradox is another statement about self reference but the point about Godel’s theorem is that this kind of sentence is inevitable. The paradox is like a sympton of a disease, whereas GIT tells us what the disease is.

[Twiffy]

Debaser,

I’d like to correct a few things you’ve said / misunderstandings that may have resulted from things you’ve said.

The method involved with cantors diagonal argument is extremely important to the Godel result:

Everything you say is true, but Godel never used anything attributable to Cantor’s diagonal argument. Godel created Godel numbering to achieve the same result. It’s true that the equivalent proof in algorithmic logic does use something that can plausibly be said to be the same style argument, but 1) that’s not what Godel himself originally did, and 2) the diagonal argument really isn’t all that ground-breaking. It’s a very general style of argument, and the mere fact that it’s sort of used in a Godel-equivalent proof isn’t all that important conceptually.

More importantly:

It is my assumption that Godel and the other Platonists regard numbers as platonic ideals. The number 1 is the abstraction of the concept of one of something, and 2 is the abstraction of 2 of something and so on. From a purely Platonic point of view, numbers are a pre existing ideal that we may discover.

Thhis sounds dangerously close to Set theory which Russell turned upside down with his paradox, and Godel subsequently proved why Russell was able to do that and why any similar system would be doomed to the same fate.

There are a few things wrong with your response here. First, what Russell turned upside-down with his paradox was Naive Set Theory, a poorly-formulated theory in which it was naively thought that a set could contain anything (and any number of things) as a member. Russell proved that naive view wrong; we now use Zermelo-Fraenkel Set Theory, usually abbreviated ZFC. It is a very powerful and very precisely axiomatized system which is, to the best of our ability to tell, consistent. It hasn’t yet been “turned upside down” in any way whatsoever (including by Godel’s Theory.)

Second, Godel’s proof (in fact, his proofs) had nothing to do with how Russell’s paradox was able to work. Thirdly, it is not true that systems “similar” to Naive Set Theory would be provably inconsistent at all, let alone in the same way. You’re thinking of the fact that systems “similar” to Set Theory (by which we mean, precisely, able to formulate the existence and properties of the integers, such as Peano Arithmetic) are also necessarily incomplete.

[debaser]

Thanks Twiffy,

that’s really useful. Will post later

[debaser]

Twiffy,

Thanks for setting me straight I appreciate it. Actually I should have known at least some of what you said already but nevermind.

Can I ask you a question? Penrose makes a lot about the fact that we can understand GIT, which he claims would not be possible if we operated formally. He uses other arguments aswell but to be honest the GIT one worries me.

Actually I’m not that worried as his argument is not strictly applicable: he confuses ‘provable’ in the computational sense with ‘arguable’ in the human sense and so on. Therefore his argument is strictly speaking a non sequitor. , at least i think so. However what do you think about Penrose’s argument?