Opinions on Gödel's Theorems of Incompleteness - Logics/PoMa

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By me, from http://www.t-lea.net/philosophical_notes.html#OGTI. Over Logics an Philosophy of Mathematics.

It's my opinion that Gödel's Theorems on this matter make either unreasonable assumptions on axioms or try to say too much, particularly on a system's axioms which may themselves, ultimately be hinged on a nature of infinity. It seems strange to me that Gödel's "Incompleteness" is about not being able to prove the axioms from within the given system. In my opinion, every "idiot" goes only for consistent and Gödel "incomplete" systems. This should be clear! I see no problem with the descriptive power of this system as a consequence of Gödel "incompleteness".
If Gödel's two theorems are to kick in, the human viewpoint would have to be completely different and the ontological status of infinity soundly removed, but this is clearly not the case today.
The theorems of Incompleteness should thus be renamed Theorems of Non-Self-Reference or Theorems of Non-Tautology.

Drawing from the Philosophy of Science, I see the creation of systems like non-Euclidean geometry and Fuzzy Logic, being only two examples, as sliding in nicely with existing systems and this should also be kept in mind when you regard the whole story of various systems through the course of human evolution.

(Wild) questions:
Are the Gödel theorems of incompleteness contradictions? Are they begging for the impossible, implicitly?
Are the theorems controversial?
What kind of system is it the theorems ask for?
I've been thinking that you can add as many axioms to a system you'd like in order to have the useful scope of descriptive tools you'd like. That these axioms can't be proved by the very same system, can't hardly be a problem, no?

For educational purposes: How do you build something without having a world to build something in first? Why question the building materials you've selected when you're making a building? How do you prove your "Universe of Discourse"? I sense there's something "sick" about imposing a requirement of being able to prove the establishment of the world that's going to support your descriptions. What I'm saying is that Gödel's "incompleteness" is negligible as opposed to other possible meanings of "incompleteness".

Originally written 07.02.2010 and 11.02.2010.

What do you think? I'm thinking it's a small victory to identify the Gödel's "incompleteness" as just that kind of incompleteness and not the "ordinary" incompleteness that "normal" people think of.

Opinions of yours?

[Richard Baron]

The theorems are not controversial, but their interpretation (going beyond the mathematics) is controversial. And yes, the sense of "incompleteness" involved is not the everyday sense.

Adding axioms is fine, and plugs one negation-incompleteness, but gives rise to another.

I recommend Francesco Berto's book There's Something About Gödel (Wiley-Blackwell, 2009), a very good semi-technical introduction to the theorems and the controversies.

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Richard Baron
Thanks a lot! Class! I'll surely buy the book! Pleasure to have your reply.

[Edit:] Isn't there a possibility that Hilbert's 2nd problem is resurrected by naming the kind of incompleteness Gödel presents as Gödel "incompleteness"?

Hilbert's 2nd problem is to "Prove that the axioms of arithmetic are consistent."

[Edit2:] I'd like to add that Gödel's "incompleteness" can be combined with Tarski to be Tarski-Gödel "incompleteness"-"undefinability"! This point is modified thanks to the reading of Raymond Smullyan on Wikip. where Smullyan refers readers on to Tarski in being fascinated by Gödel! Although being uncertain about both, I'm now on some material of Gödel, at least, and work will continue.

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Added 17.03.2010, Hilbert's 2nd problem:
I also find these 2 possible interpretations of Hilbert's 2nd Problem. The first is the one that is answered by Gödel that axioms can't be proved by the system they establish, but the 2nd one is that the scope in some future may be prove to be inconsistent by the very application of these axioms. I've gotten the word that one of Euclid's axioms has been either proven to be false or to be excessive and it's in this line of thought I'm thinking of the 2nd interpretation of Hilbert's Problem and it's possible resurrection.

Cheers!

[Richard Baron]

I've gotten the word that one of Euclid's axioms has been either proven to be false or to be excessive

Are you thinking of the parallel postulate? It is independent of the other axioms. If it holds, you have Euclidean geometry. If its negation holds, you have a non-Euclidean geometry. Perhaps these facts have led to the rumour that it is false, or to the rumour that it is excessive.

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Richard Baron
Thanks a lot for your input! Always cool!

I'll look deeper into it. Hopefully, this may be a good example. Either way, I think the Riemann alternation from traditional Euclidean geometry provides an angle to this Tarski-Gödel issue in light of Hilbert's 2nd Problem.

Is there any possibility you can get something definite from a mathematician friend? At least, I'll investigate more thoroughly.

Cheers!

[Richard Baron]

I don't think I can help more without a better understanding of how you see Tarski as coming into this. Are you thinking of his work on the undefinability of arithmetical truth within arithmetic? Or about his work on the definition of truth in general? Could you expand on your thoughts so far?

[Aetixintro]

I can at least give you these links from where I've gotten some information:

Raymond Smullyan: http://en.wikipedia.org/wiki/Raymond_smullyan Note this: While a Ph.D. student, Smullyan published a paper in the 1957 Journal of Symbolic Logic showing that Gödelian incompleteness held for formal systems considerably more elementary than that of Gödel's 1931 landmark paper. The contemporary understanding of Gödel's theorem dates from this paper. Smullyan later made a compelling case that much of the fascination with Gödel's theorem should be directed at Tarski's theorem, which is much easier to prove and equally disturbing philosophically.

Then:

The specific Tarski "undefinability": http://en.wikipedia.org/wiki/Tarski%27s ... ty_theorem All of the text on this page should be relevant.

And the doors open...

PS: Francesco Berto's book There's Something About Gödel (Wiley-Blackwell, 2009) is in the house, sold and bought!

[Richard Baron]

OK, so it's the indefinability part of Tarski's work that you are bringing in. That's helpful. But I still won't have anything useful to add without more of your thought process. Sorry about that.

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Richard Baron
It's alright! Relax! I'll just let time run its course and add more when there's more to add. Thank you so far!

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I'm thinking about adding this to my website that takes my thoughts further on the topic and my angle to it:

(I've visited Wikip. on Tarski some time ago and then it says that Tarski has proved both geometry and algebra to be both complete and consistent. The page seems heavily edited today, 02.07.2010 and it doesn't say so anymore. So, this is a pause for me and an angle I need to investigate if I'm to get somewhere with my writing on this. This is just a notice for what follows.)

* Geometry - Complete and fulfills all descriptive tasks.
* Algebra - Complete and fulfills all descriptive tasks.
* Arithmetics - Incomplete and fulfills all descriptive tasks (yet missing its Gödel "completeness" by Tarski).
Further:
* Language - Incomplete, I think, and fulfills all descriptive tasks (yet missing its Gödel "completeness" by Tarski).
* HDM and science - Incomplete, I think, and fulfills all descriptive tasks (yet missing its Gödel "completeness" by Tarski).

Also, in the light of all this, we have the issue of computability. Can everything be computed? Can one set up a complete computer system that replaces the human being as the scientist and what have you, except its functioning of consciousness? What is "completeness" or "incompleteness" in this regard?

This also goes for the semantic/syntactic divide. If we can replicate the human reasoning by computing, then what is left of this "semantic" ability?

It should be exciting to see how this develops, but it will demand quite a lot of effort and time.

Well, project is ongoing. Please, be patient. This is one of my weakest writings and I lack (a lot of) knowledge on the issue.

This is the update for now! Cheers!

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Richard Baron, especially, and all others for that matter

I've now supplied a review of the book you've recommended. It's here: viewtopic.php?f=15&t=4592.

Just make your input accordingly to that thread, please, if you like. The project is ongoing and I'll be looking for more literature! Cheers!

[Edit, 21.07.2010:] I've forgotten to mention that the input would be all voluntarily! I see that you've replied and I thank you for that. Excuse me if I have appeared rude! (It was certainly not the intention! )

[Edit, 22.08.2010:] please, if you... [End of edit.]