[marojejian]

Interesting points in here.

e.g. that Godel didn't think this scrapped Hilbert's project totally:

>Gödel believed that it was possible to redefine what we mean by a formal mathematical framework, or allow for alternative frameworks. He often discussed an infinite sequence of acceptable logical systems, each more powerful than the last. Every well-formulated mathematical question might be answerable within one of them.

[lioeters]

That part you quoted was interesting to me too. I remember once re-reading the incompleteness theorems - where it talks about a "finite set of axioms", it seemed there may be a loophole if we can imagine a theoretically infinite set of axioms, as a way to approach completeness.

Overall I really enjoyed this article, short interviews with mathematicians and philosophers on a topic I've often thought about.

[AndrewOMartin]

Some people, when confronted with a Gödel's Incompleteness Theorems, think "I know, I’ll use a theoretically infinite set of axioms." Now they have aleph-nought problems.

[lioeters]

"..How about infinity plus one! Or infinite infinities!" -- Every child who learns about infinity for the first time, but also serious mathematicians in philosophical struggle with the truth. And Cantor, may the angels soothe his troubled soul.

[joe_the_user]

I think the combination of Godel's completeness theorem and Godel's incompleteness theorem stakes out a position in between "truth absolutism" (everything certainly knowable etc) and "truth nihilism" (nothing is truly knowable with any certainty). Which I think is great. Thing, however, is that a lot of philosophers and mathematicians fall into one of these views of truth and so you see people constantly fighting, chaffing at the bit against, this middle ground, claiming it "satisfies nobody" etc. Well, it satisfies me quite a bit.

[whattheheckheck]

I wonder if thats why agile is best. We can never fully "prove" the program as theory building