If I wanted to state Godel's Theorum using ONLY math symbols and (widely accepted) math abbriviations, what would it look like?

It would also be helpful to know what the symbols are and what they mean.

(Godel's Theorum: "To every [omega]-consistent recursive class [kappa] of formulae there corespond recursive class-signs r, such that neither [ypsilon] Gen r nor Neg([ypsilon] Gen r) belongs to Flg([kappa]) (where [ypsilon] is the free variable of r)")

(Note: I don't know how to make the Greek symbols "omega" or "kappa" or "ypsilon", thus the spelling out of those three above in brackets)

Your question is a little confusing, since it seems like you've already answered the first part. Except for coding the Greek letters in HTML, you've already stated Gödel's theorem using mathematical notation. Here it is with the Greek letters:

To every ω-consistent recursive class Κ of formulas, there correspond recursive class-signs r such that neither (ν Gen r) nor Neg(ν Gen r) belongs to Flg(Κ), where ν is the free variable of r.

The version I'm familiar with has a nu (ν) rather than an upsilon (υ), but these may look the same, depending on your browser. To make Greek letters in HTML, use the following format:

<font face="symbol">K</font>
where the corresponding English letter (in this case K for kappa) goes between the bracketed codes. I used w for omega, K for kappa, n for nu, and u for upsilon.

As for what the symbols mean, that's really beyond the scope of this forum. Unless you have a strong background in logic and set theory, the explanation would be meaningless to you, even if I could explain it (which I can't). I have a bachelor's degree in math, and I still don't carry that kind of information around in my head.

It's not something that's used very often in applied mathematics. This falls more under the scope of metamathematics, which is using logic to examine mathematical systems themselves. This is generally the field of logicians (mathematicians who specialize in logic), and not something that most mathematicians study.

If you're content with a less formal translation of Gödel's theorem (without getting into the specific meaning of each symbol), then there are plenty of sources available, in print and on the Internet. Computer programming is a great paradigm for explaining some metamathematical concepts; here's a site that uses this technique, and includes a discussion of Gödel's Incompleteness Theorem: Mathematical Structure. I would recommend reading the whole section, rather than jumping right into Gödel's Theorem.

Another great reference is the book Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter, but it's definitely not for the intellectually timid. It has a great discussion of logic, metamathematics, and artificial intelligence, but it can be tough to read, even at the college level.

This message has been edited. Last edited by: silenteuphony,

I am quite familiar with the book you mentioned (I have 2 copies of it). What I was asking was about the symbols that could be used for the other parts (the words). For instance, "To every" might be shown as an upside down "A" (which means "for all"). I was wondering if there are any such symbols that might be used for the other words.

Thanks for the help,

Shaggy

Sorry for the confusion. We get questions from so many different people in this forum, sometimes it's hard to know what people are really asking for, and what they already know. Now that I understand your question, I do know some logical symbols you could use to abbreviate this theorem.

As you already mentioned, we can use the "for all" symbol . We can also use the "there exists" symbol . These are known as the logical quantifiers, respectively the universal and existential quantifiers.

There's also a logical symbol for "such that", but whenever a logical statement follows the existential quantifier, the "such that" is assumed, so we can drop that phrase. Finally, we can use the standard logical symbols for "not" (¬) and "and" ( ), and the symbols for set membership ( ) to abbreviate the last part of the theorem.

[Author's note: Originally, this post referred only to HTML; given the current use of UBB in many bulletin boards including this one, I revised the post to include more UBB-compatible information.]

These characters are all in the Symbol font in HTML. Depending on the forum, you may be able to use HTML codes or Unicode references to generate these characters. Here's a link that provides all of these character codes:
Using Greek and special characters from Symbol font in HTML
In this post, I used the HTML codes.

Putting it all together, we get:

ω-consistent recursive class Κ of formulas recursive class-signs r
[ (ν Gen r) Flg(Κ)     ¬(ν Gen r) Flg(Κ) ], where ν is the free variable of r.

This message has been edited. Last edited by: silenteuphony,

silenteuphony,

That was EXACTLY what I was looking for!

Thanks,

Shaggy

What do "Gen" and "Flg" mean?

Shaggy

Ah, now we're back to my original reply. That question gets into an area that's out of my expertise, but if you can find a friendly logician, maybe they'll explain it to you. As I said before, there are plenty of informal explanations of the theorem in print and on the Internet, but not many sources that explain the actual wording of the theorem. Sorry I couldn't help, and good luck in your quest for knowledge.

silenteuphony,

Although you were unable to answer my last question, you have given me more info than I have been able to find myself (through books, the internet, and friends combined).

Thanks,

Shaggy