3 ms·
Not quite; I think it was actually Kurt Gödel. Jokes aside, I assume they are referring to Gödel's incompleteness theorems, which might well reasonably be char
by jeffparsons 3y ago
Not quite; I think it was actually Kurt Gödel.
Jokes aside, I assume they are referring to Gödel's incompleteness theorems, which might well reasonably be characterised in that way.
- eru 3y agoThough it only applies to some mathematical systems. Some are simple enough to be decidable.
- throwoutway 3y agoWhich? Are they consistent? Are they useful? From wikipedia: > The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent and effectively axiomatized.
- eigenket 3y agoTarski's axioms for geometry are the standard example of a system which is both complete and consistent. It is even decidable - there is an algorithm which you can feed a statement and it outputs whether it is true or not.
- throwoutway 3y agoOk, how relevant is Tarski's though? Seems academically, they are weak > We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. https://sciendo.com/article/10.2478/forma-2014-0017 https://sciendo.com/article/10.2478/forma-2014-0017 Also, thats for geometric space but I dont see how that's particularly relevant to this. Elsewhere I find that it doesn't contradict Gödel
- eigenket 3y agoI don't understand your questions. They are relevant for geometry, they aren't really relevant to anything else but they're axioms for geometry so being relevant for geometry is what you'd expect.
- throwoutway 3y agoIIRC, incompleteness theorems applies to arithmetic not geometry, the difference being subtle right?
- eigenket 3y agoThe original incompleteness theorems apply to any logical system which can formulate and prove a certain chunk of arithmetic, yes. It turns out that the amount of arithmetic you actually need is incredibly small, so these results apply to many systems including some formulations of geometry.
- eigenket 3y agoOh, by the way, it matters here that "weak" in the source you quote had a very specific technical meaning. If axioms are weak it means they apply to many things. Stronger axioms make more restrictions and apply to more specific things. So (arguably) axioms being weak is a good thing. Generally you want to work with axioms that are as weak as you can get away with.
- throwoutway 3y agoGood to know thanks
- eru 3y agoYes, they are consistent. Yes, there are useful systems. See https://en.wikipedia.org/wiki/Agda_(programming_language) https://en.wikipedia.org/wiki/Agda_(programming_language) for a reasonably practical example: > Agda is a total language, i.e., each program in it must terminate and all possible patterns must be matched.
- nateburke 3y agoCan the theorems of Kirt Gödel be classified, though, is my question. And if so, would their domains of applicability within the universe of formal systems actually be a classification system of that universe in and of itself?
- bigdict 3y agoWho is Kirt Gödel?
- xeonmc 3y agoActually, I believe it was rather his distant cousin, Kirk Girdle, with his Incompetence Theorem.