A Mathematical System is based on a limited set of axioms, i.e. assertions or statements that are assumed to be true without the need for any proof."...given any consistent set of arithmetical axioms, there are true arithmetical statements that cannot be derived from the set."

Using the internal rules of the Mathematical System, these axioms can then be used as basis to determine whether other assertions within such a system are true or false. Assertions that are proven to be true are accorded the status of Theorems.

These theorems can in turn be used (together with axioms or other theorems) to prove or disprove other assertions via the same method of Deductive Reasoning. One would therefore expect that such an ongoing process of generating assertions and proving them would result in a mathematical system that is able to extend itself by uncovering or constructing more mathematical truths while, at the same time, avoiding false assertions that would allow inconsistency(-ies) to creep into the System. Gödel found out that this is not the case.

(Click on image to enlarge)

What Gödel proved in 1931 was that avoiding inconsistency within the system comes at the expense of

*completeness*, i.e. the system's characteristic of being

*self-contained*(in other words, '

*complete in itself*'). In a mathematical system based on axioms, there will always be assertions that are

*true*, but whose truth cannot be proven from within the system on the basis of its axioms (or by way of the system's theorems). The truth of these assertions can be determined from outside the system [aka the 'Environment'], but not from within.

(Click on image to enlarge)

Gödel's Incompleteness Theorem has a number of implications that have been explored at length, but i believe that one of the most significant, as explained by Douglas Hofstadter* is that:

A timely reminder to those who count on a given system to always come up with"...provability is a weaker notion than truth** no matter what axiomatic system is involved."

*evidence*to establish

*truth*. There are situations when lack of evidence does not negate the truth of an assertion. Sometimes, you just have to step outside the system to see that this is indeed the case.

**Update April-16-2008:**Here's an excellent discussion on Douglas Hofstadter's book Gödel, Escher, Bach (also known by its shortname

*GEB*).

**Update April-25-2008:**It is also important to emphasize what Gödel

**does not prove**, i.e. Gödel

**does not**prove the romantic and anti-rationalist notion that "There are true things which cannot be proved".

*in his book Gödel, Escher, Bach: an Eternal Golden Braid

**Emphasis mine.

## 9 comments:

Great summary. I saw Hofstadter's GED at Booksale but it was missing the first 60 or so pages. I have a poorly scanned soft-copy which is a pain to read.

GED? I meant GEB -- Godel Escher Bach.

Thanks Jeg. It's good that understanding

whatGodel proved is easier than understandinghowhe went about proving it.I have a copy of GEB but haven't gotten around to reading it in depth.

What the common tao has always known, Godel has taken to the scientific mainstream.

His method, known as Godel numbering, is a breakthrough.

CVJ, the great thing about mathematicians and philosophers is that unlike novelist, you do not have to read the original.

Hi Brian, i agree. Mathematical and philosophical (as well as scientific) concepts once introduced, break free of its author.

Hi,

we are a group of secondary school students working on a project related to Godel's first theorem. We found that your diagrams are of great help to us and would like to obtain your permission to use these diagrams for our project. Regarding figure 4, we are not clear what the assertion of the man means. Could you further explain on it?

Thank you.

Hi, yes please go ahead and use the diagrams. 'Assertion' is just a synonym for 'statement'. In diagram #4, it is drawn as free floating i.e. not connected to any axiom or theorem to show that its truth is not derived from any axiom or theorem. (In that way, the assertion is similar to an axiom, rather than a theorem.)

Thank you for allowing us to use your diagrams! As our project would be submitted into the Singapore Mathematics Festival competition, we need to know your name for acknowledgement. Please feel free to drop us an email at mathproject389@gmail.com if there are any inquiries. Please reply ASAP! Thanks:)

You're welcome. My name is Carlos Villamil Jugo. Good luck on the competition.

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