What Do Gödel's Incompleteness Theorems Mean?
quantamagazine.org
What Do Gödel's Incompleteness Theorems Mean?
1–10 of 72 posts
Re: What Do Gödel's Incompleteness Theorems Mean?
#2Interesting 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.
Re: What Do Gödel's Incompleteness Theorems Mean?
#3Interesting 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…
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.
Re: What Do Gödel's Incompleteness Theorems Mean?
#4I don’t think we’ll ever entirely know what they mean.
Re: What Do Gödel's Incompleteness Theorems Mean?
#5It hints at something fundamental to how the universe works, in that there is always an adjacent possible.
Re: What Do Gödel's Incompleteness Theorems Mean?
#6> “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.'
There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic
Re: What Do Gödel's Incompleteness Theorems Mean?
#7Of 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.
Re: What Do Gödel's Incompleteness Theorems Mean?
#8> “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.' There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic
Right, you need to be able to construct numbers for Gödel's proof to apply.
Hilbert's incidence geometry, for instance, is consistent and complete. It's just rather small.
Re: What Do Gödel's Incompleteness Theorems Mean?
#9Of 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.
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.
Re: What Do Gödel's Incompleteness Theorems Mean?
#10It may mean our brains are not currently equipped to understand the universe.