What Are the Deeper Implications of Gödel's Theorems?

[k3N70n]

On Gödel (revised)

Hi. I'm writing a paper on Gödel for a math class (called Foundations of Math). I've been working on it for quite sometime but I've found it to be quite a hard topic. I'd appreciate any advice on further research to do, any errors I've made in logic or any other advice you could give me.

One thing I should note is that I go to a liberal Christian Arts school so I've written my paper a bit differently than I guess would be expected at other places.

Lastly the exact topic that I was given for this paper is "The Deeper Implications of Gödel's theorems"

Anyway, thank you kindly for taking the time to read and criticizes.

-kentt

PS. I'm not entirely sure if this is the right place, whether it should have been in 'Homework' or 'Philosophy' so please feel free to move it to the correct place.

EDIT: Revised paper attached. Thanks again for any insight.

 

[Hurkyl]

The first time you use the word "arithmetic", and possibly everywhere else, you should qualify that you mean "integer arithmetic" (or "natural number arithmetic" or "peano arithmetic", or something else equivalent). As Tarski showed, the (first-order) arithmetic of real numbers (called the theory of real closed fields), for example, is consistent and complete. This isn't a contradiction, because real number arithmetic doesn't allow you to formalize the notion of "integer", and thus it cannot fully express integer arithmetic. Tarski also gives an axiomization of Euclidean geometry which is equivalent to the theory of real closed fields, and so is also consistent and complete.


Remember that there are two Gödel incompleteness theorems: the second says that if a consistent formal system can express the notion of its "own" consistency, it cannot prove itself to be consistent. Some of your later comments seem to be referring to this second incompleteness theorem, but you never mention it in your paper.


Gödel's incompleteness theorems do not prove the independence of the Continuum Hypothesis.

Many look at Gödel’s Incompleteness theorem as quite remarkable and surprising. Not to trivialize what Gödel proved but in some respects Gödel’s results could have been expected. ...

This is a erroneous philosophical perspective -- look at chess as an analogy. The rules of chess are completely known and the game is finite. In principle, chess is known to be completely "computable". But, can you say that there is no creativity nor intelligence in the playing of chess?

 

[JasonRox]

I really don't see how Godel's results are anywhere near trivial!

 

[Chris Hillman]

k3N70n, I hope you realize that stuff like this has nothing whatever to do with logic, mathematics, or Goedel:

Gödel’s has also been extrapolated to invalidate Christianity. Some would say [find actual quotation] that while Christians believe in God with absolute certainty, Gödel has proved that we cannot know anything for certain.

These statements are in fact terribly misleading, in fact I am fairly aghast that anyone would write such things even as a "straw man" argument! I'd advise you to leave religion out of the paper entirely.

I didn't read your paper in any detail and I am not sure it is appropriate for you to seek explicit editing advice here, although at and earlier stage it would have been appropriate (IMO) to ask for references to good books. I have to say I find it troubling that PF seems to be helping you write your paper, which presumably is intended to be original work

I can suggest a nice article I think should prove intriguing: Martin Davis, "What is a Computation?" in Mathematics Today, ed. by Lynn Arthur Steen, Vintage, 1980. In this article, Davis briefly but very clearly explains Chaitin's form of Goedel's incompleteness theorem.

When you cited "Hawkins" (Hawking?) I really hope that was not Stephen Hawking!

 

[k3N70n]

I'm extremely grateful for your help. It's not easy to find people who can understand this paper and critic it.

The first time you use the word "arithmetic", and possibly everywhere else, you should qualify that you mean "integer arithmetic" (or "natural number arithmetic" or "peano arithmetic", or something else equivalent). As Tarski showed, the (first-order) arithmetic of real numbers (called the theory of real closed fields), for example, is consistent and complete. This isn't a contradiction, because real number arithmetic doesn't allow you to formalize the notion of "integer", and thus it cannot fully express integer arithmetic. Tarski also gives an axiomization of Euclidean geometry which is equivalent to the theory of real closed fields, and so is also consistent and complete.

Thanks. I guess I have been a bit sloppy. Now that I have been reminded of Tarski I might go back and add him in.

Remember that there are two Gödel incompleteness theorems: the second says that if a consistent formal system can express the notion of its "own" consistency, it cannot prove itself to be consistent. Some of your later comments seem to be referring to this second incompleteness theorem, but you never mention it in your paper.

I realized I missed that while I was waiting someones response. I've got it in there now

Gödel's incompleteness theorems do not prove the independence of the Continuum Hypothesis.

ummm...crap I suck.

This is a erroneous philosophical perspective -- look at chess as an analogy. The rules of chess are completely known and the game is finite. In principle, chess is known to be completely "computable". But, can you say that there is no creativity nor intelligence in the playing of chess?

No of course not. If I remember correctly this was something that I got from Boyer. I must have misinterpreted what he said/implied.

I'm kind of having a hard time objectively finding applications of Gödel's theorems and drawing any sort of conclusions (philosophical or not). Could you perhaps suggest some place where I could look?

Thanks again to Hurkyl or any others who can give me some direction or insight.

-kentt

 

[robert Ihnot]

Many look at Gödel’s Incompleteness theorem as quite remarkable and surprising. Not to trivialize what Gödel proved but in some respects Gödel’s results could have been expected. Let’s look at the possibility of Gödel’s theorem being false. If Gödel was wrong then all of mathematics could be reduced to axioms. Thus by using these axioms we could derive all truths about mathematics from these. In many ways this takes all creativity and intelligence out of mathematics. Mathematicians would then simply be theorem proving machines giving unremarkable results because all of math anyone else could have come across the same result. There would be no creativity involved, minimal intelligence.

This is overdone, there were outstanding mathmaticans who attempted to formalize all math, reducing it to axioms. They saw this, clearly, as the next step and wanted to take it.

Hilbert is a chief example, who said, "I believe that everything which may be the object of scientific thinking in general, as soon as it is ripe for the formation of a theory, falls into the axiomatic method, and thereby indirectly into mathematics."

Bertrand Russell, who called math, "Marks on paper,"and was like many, a Formalist, also made the interesting statement, "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

 

[DeadWolfe]

"Thus by using these axioms we could derive all truths about mathematics from these. In many ways this takes all creativity and intelligence out of mathematics. Mathematicians would then simply be theorem proving machines giving unremarkable results because all of math anyone else could have come across the same result"

But mathematics still IS done using axioms, it's just that there a few problems which can be shown to be undecidable, and mathematicians then systematically avoid discussion of them. The incompleteness limits the scope of math, not the nature.

 

[robert Ihnot]

DeadWolfe: "Thus by using these axioms we could derive all truths about mathematics from these. In many ways this takes all creativity and intelligence out of mathematics.

I can't agree with that because I see it as a personal observation, a philosophical viewpoint. And, even if I thought as much, there have been many who see it otherwise. As my logic professor put it, "If we are not going to use axioms, how are we going to prove anything?"

There has been a division of Formalists and Intuitionists. Obviously the Formalists see Math as "marks on paper," which are moved around according to a set of rules. The Intuitionists are more into your ideas. In fact, Godel was an Intuitionist who believed that mathematical objects were as real as physical ones. He thought like Plato. I quote:

Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things 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. http://www.friesian.com/goedel/chap-2.htm

A key remark, I think, is "Exist independently of our definitions and constructions." But the whole thing is just a philosophical argument without a real answer.

 

[Cincinnatus]

Bertrand Russell wasn't really a formalist. There is definitely a tension in his writings between his British Empiricism and some form of quasi-platonism about logic. He believed mathematics could be derived from logic and demonstrated this in the Principia Mathematica.

 

[robert Ihnot]

Cincinnatus: Bertrand Russell wasn't really a formalist.

Russell did have along career and said quite a few things, so that could be true. However, the "Marks on paper" remark sounds like Formalism.

 

[matt grime]

I think you're misreading Deadwolfe, Robert. It was the OP's opinion that you disagree with on the removal of creativity from mathematics.

 

[DeadWolfe]

matt is correct. I was quoting the OP and making much the same point as you are.

 

[robert Ihnot]

I see. I am sorry. I incorrectly quoted the source.