As far as I can tell Godel proved that if your system is as strong as Peano number theory then it’s not complete, i.e there are assertions that are true but not provable.
But then its negation must be false but not refutable.
My question is how can we know that something is true if there’s no proof in the system we believe to be consistent for the statement how can we know that it’s indeed true in that system?
Someone once said to me that GOdel decoded a statement of the sort of “this statement is not provable”, if it’s not provable then it’s true and if it’s not true then it’s provable which is impossible.
This very assertion seems to me out of context of theorems and conjectures that I had seen in maths. Sure you cannot use first order logic to formalize this statement, so it’s provalbe by definition, but it’s a self referenced statement, which doesn’t even appear in the mathematics literature.
I mean what statement are there in number theory that are not preovable but true, how can we know that it’s true if we can’t prove it?
seems to me like a theorem that just repeats the Liar paradox, that mathematicians don’t even bother to look at.
I understand the demathematized version of Godel’s theorems. It’s simply that consistency XOR completeness. Some meta-stuff going on in the proof as the Godel sentence itself is proved from outside the system in which the sentence is formulated. It is true and unprovable (in the system in which it is a formula) i.e. in fact it is provable, that’s why it’s true, but not within the system but from outside it.
That’s true of the Gödel sentence, sure, but there’s the strengthened finite Ramsey theorem, Kruskal’s tree theorem, and Goodstein’s theorem which are all true in Peano arithmetic but not provable in Peano artithmetic, each requiring either second-order arithmetic or set theory to prove.
Didn’t know. Thanks. My understanding is that any system which has math worth doing is either incomplete or inconsistent. Perhaps I’m watching the wrong channels but I rarely get mathematicians talking about Godel’s incompleteness theorems on my feed.
The problem for me is that Godel’s results can’t really be appreciated by someone who doesn’t know advanced math or logic and I’m that.
I read (in wiki) that second order logic doesn’t have a well established deductive systems like first order logic.
What books do you recommend on higher-order logics?
My bigger question is: what conjectures in ordinary maths can we tell that are unprovable but true?
I mean how to prove for example that a statement is true but unprovable in that system?
I mean if you use another system with more axioms you make your system a differnet one than the original, a stronger one.
Take for exzample Riemann hypothesis, surely it contains Peano Airthmetic axioms (and also it’s a subject in complex analysis, so it contains all the field axioms etc); how can we tell if it’s provable or not in this system (I guess we use ZF or ZFC set theories)?
I have two books to read on Godel theorems and more, but unfortuanetly something always halts my reading the books.
As a long retired mathematician sometimes I can speak for my colleagues. In this regard I can recall virtually no significant discussion of the topic amongst those in my subject area in classical complex analysis. Relatively few mathematicians in general pay any attention to the topic, since most work in areas apart from mathematical logic, and have not stumbled across a situation that might reach that far into that quirky domain.
Gödel’s proof shows that for a system competent enough the to do arithmetic, that is consistent, and the sentences of which we can talk about in order to assign truth values, one can write a sentence sentence that in effect says “This sentence is not provable in this system”.
Now if that sentence is provable, then it is false.
Hence the sentence must be unprovable, and so we must assign to it the value “true”.
So the answer to your question is that because it is unprovable, it is true.
Godel’s theorems are described as profound in mathematics. To me it does undercut the Euclidean view of mathematics as an axiomatic system where we can start off with a few propositions and then derive all true theorems that can be formulated in that system.
Professional mathematicians must surely keep Godel in the back of their minds when they’re working on proofs. What if that which they’re working on is unprovable?
If I recall it is independent of the axioms. The axiom of choice is an add-on as well. If you assume it true, then you turn math into a stranger practice that it normally is.
Godel’s theorems matter to MUH (Mathematical Universe Hypothesis) according to which the universe is a mathematical object within which other mathematical objects are embedded. Mathematical laws of the universe would be theorems within the MU (?) and so physics is math. Isn’t that neat?
As I recall, the CH can’t be proved in set theory. Its negation cannot also not be proved. So set theory as a formal system leaves the issue undecided. Like a place in the video game that isn’t “filled in” by the code.
A “constructivist type” can only find this amusing. The CH is like a “theological” statement about how many angels fit into the bellybutton of a clown made of shark teeth and vaseline.
