> “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete.'
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.
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.
There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic
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.
Overall I really enjoyed this article, short interviews with mathematicians and philosophers on a topic I've often thought about.