Godel's Incompleteness Theorem

[BaalChatzaf]

I have noticed that from time to time the Godel Incompleteness Theorem is used, abused, misinterpreted and stretched out of shape to support whatever philosophical principle is at hand. Please read the following and learn something correct:

http://www.ams.org/notices/200604/fea-franzen.pdf

This paper was written by the late Torkel Franzen (1950-2006), who jumped on anyone saying something nonsensical about Godel's Incompleteness Theorem on usenet. He was Godel's bulldog.

Ba'al Chatzaf

[Robert Campbell]

This paper was written by the late Torkel Franzen (1950-2006), who jumped on anyone saying something nonsensical about Godel's Incompleteness Theorem on usenet. He was Godel's bulldog.

Bob K,

Thanks for the reference.

Did Franzén ever jump on Leonard Peikoff, perchance?

Robert C

[Robert Campbell]

Bob K,

I was able to follow Franzén's piece without too much trouble.

He uses the consistency and completeness of Ayn Rand's philosophy as a foil, more than once.

Who put him up to that?

Robert C

[Christopher]

I've often simplified the application of this theorem to situations related to the following instances:

Any assertion which can be made within a system that contradicts the axioms of the system is representative of inconsistency.

Any system has axioms that cannot be proved. If those axioms are proved, it requires knowledge from outside the system.

This is so much simpler, isnt' it. Generally when we are dealing with an issue that relates to these topics, we can apply Godel's Theorem. In other words, it doesn't take an expert to recognize the application of the theorem, although it does take some heavy cognitive work to list a step-by-step procedure of how the theorem explicitly pertains to the situation.

Edited October 3, 2009 by Christopher

[BaalChatzaf]

I've often simplified the application of this theorem to situations related to the following instances:

Any assertion which can be made within a system that contradicts the axioms of the system is representative of inconsistency.

Any system has axioms that cannot be proved. If those axioms are proved, it requires knowledge from outside the system.

This is so much simpler, isnt' it. Generally when we are dealing with an issue that relates to these topics, we can apply Godel's Theorem. In other words, it doesn't take an expert to recognize the application of the theorem, although it does take some heavy cognitive work to list a step-by-step procedure of how the theorem explicitly pertains to the situation.

That is not what Godel proved. Why don't you read the theorem instead of producing a word salad.

Ba'al Chatzaf

[Robert Campbell]

Christopher,

Bob K is right.

Such points as you made did not require Kurt Gödel's effort or ingenuity to arrive at.

Robert C

[Philip Coates]

Asking a layman to 'just read' a theorem tied to instances from theoretical mathematics is not likely to be a productive endeavor. You will probably not know enough to have examples of formal systems to know what is being talked about (perhaps not even in fully systematizing arithmetic). From my grad school days: In topology you can -define- formal concepts such as "connectedness" and "compactness". [Topology was my very favorite subject in grad school. Elegant and beautiful!] In the branch of higher math called algebra (not the same as what you studied in high school) you can define the operations necessary for some entities to constitute a (closed) 'group' or a 'ring' or a 'field'. And then prove theorems.

These areas or branches of mathematics (and you don't need Godel for any of this) rest, for starting points especially, on both definitions (true because you use them to stipulate the kinds of things you are working with and - just as important - axioms Often particular to that area or branch. But they can be broader.

The axioms of a field, however, are not arbitrary just because their validation rests outside. They are valid, else they wouldn't be axioms. See Aristotle. Think of geometry and its axioms (valid in a Euclidean, non-curved surface context.)

Edited October 4, 2009 by Philip Coates

[Christopher]

Christopher,

Bob K is right.

Such points as you made did not require Kurt Gödel's effort or ingenuity to arrive at.

Robert C

Yes, Ba'al forced me to layout my knowledge explicitly and I find that I don't know it as well as I thought.