Of all the incompleteness-style theorems, I find the Halting problem to be the most approachable and also the most interesting. Maybe it's because I'm a software dev that dabbles in math rather than the other way around. But that makes me wonder if all of Gödel's theorems can be stated if 'software form', so to speak.
basically generalized the halting problem to arbitrary semantic properties.
lacewing 15 minutes ago [-]
Right, if you're a software engineer, the realization that the two theorems are nearly-equivalent really takes the air out of a lot of the existential philosophizing around Gödel's incompleteness.
Gödel's argument basically says that any system of mathematics powerful enough to implement basic arithmetic is a computer. This shouldn't be surprising to software engineers because the equivalency between Boolean logic and arithmetic is easy to show. And if you have a computer, you can build algorithms whose outcome can't be programmatically decided by other algorithms.
marojejian 3 days ago [-]
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 2 days ago [-]
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.
MrDrDr 35 minutes ago [-]
> “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.'
basically generalized the halting problem to arbitrary semantic properties.
Gödel's argument basically says that any system of mathematics powerful enough to implement basic arithmetic is a computer. This shouldn't be surprising to software engineers because the equivalency between Boolean logic and arithmetic is easy to show. And if you have a computer, you can build algorithms whose outcome can't be programmatically decided by other algorithms.
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.
There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic
Hilbert's incidence geometry, for instance, is consistent and complete. It's just rather small.