quote:Original post by Paradigm Shifter
Godel showed that even with axioms, maths is inconsistent (i.e. there exist statements which can't be proven or disproven; isn't the Riemmann hypothesis one of those?).

What he proved was not that math is inconsistent. He proved that if it is consistent, which we assume it is, there must exist true statements that cannot be proven, from a fix set of axioms.

Which statments that cannot be proven, using for example Peano's axioms, we can never know. This is easy to realize when you think about it for a while. If you would show that a statement is true and that it cannot be proven, you would have proved it already, since you showed that it was true -- contradiction. Therefor, it is not the case that you can prove that you can't prove a specific true statement. Uhm... that suddenly became complicated.

Many believed Fermat's last theorem was such an improvable, yet true, statement until it was proved a number of years ago.

[edited by - aleph on December 10, 2002 8:40:25 AM]

nice nickname aleph

I have to agree that especially linear algebra is not about memorizing - it''s always good to know the terminology if you
wantto describe problems, but it won''t help you much while trying solve them. Remember the fundamentals of each topic since
you will need them to identify linear algebra topics as such.

For example knowing the basics of matrix operations and vector spaces is useful for the image processing - filters, edge detection, color spaces, and color dithering are
simple matrix transformations.

Cheers,
Pat

quote:Original post by johnb
Fortunately the sorts of propositions that are undecidable have little effect on day-to-day mathematics.

Here is a good read for you.

http://www.ams.org/notices/200203/fea-knuth.pdf

There is a nice paragraph about Gödel''s theorem and its impact on day-to-day mathematics. I agree with Knuth in this (as in most other things).