Math problems that challenge the world(hpe.com)
hpe.com
Math problems that challenge the world
https://www.hpe.com/us/en/insights/articles/the-toughest-math-problems-that-challenge-the-world-1805.html
20 comments
This as well:
>"Can it be proven that the axioms of logic are consistent?" Gödel also answered this problem with his "incompleteness theorem," which states that all consistent axiomatic formulations include some undecidable propositions. For more, see the short history of Euclidean and non-Euclidean geometries.
This is imprecise, and the last sentence is a non sequitur.
>"Can it be proven that the axioms of logic are consistent?" Gödel also answered this problem with his "incompleteness theorem," which states that all consistent axiomatic formulations include some undecidable propositions. For more, see the short history of Euclidean and non-Euclidean geometries.
This is imprecise, and the last sentence is a non sequitur.
Yeah, that statement is just as bad. Godel's second incompleteness theorem [1] does prove that an axiomatic system cannot be proven consistent within itself but the restatement of the first incompleteness theorem is not a sufficient summary of this:
[1] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...
[1] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...
I think the author tried to give an example of an incomplete theory. Euclid's geometry minus the parallel postulate is an instance of that as the two different models show.
it's almost an antithetical non-sequitur at that, because Euclidean Geometry is consistent.
“and non Euclidean geometry”. You are reading that as two statements instead of a compound statement.
Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse.
In a less extreme example, it also doesn’t hold for real world problems of the Traveling Salesman sort (one way roads and steep hills).
It’s really good for flat surfaces. But we don’t live on a flat surface.
Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse.
In a less extreme example, it also doesn’t hold for real world problems of the Traveling Salesman sort (one way roads and steep hills).
It’s really good for flat surfaces. But we don’t live on a flat surface.
> Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse.
What has one to do with the other? Am I missing some important context here, because the statement "euclidean geometry is not consistent with general relativity" doesn't make any sense to me. Euclidean geometry is mathematical abstraction built out of axioms that were chosen to sorta match our intuitive understanding of space. Relativity is a physics theory that uses some different mathematical geometry built out of different axioms, chosen so that resulting geometry would better fit observational data. And Gödelian consistency has nothing to do with physics at all, it considers mathematical constructs.
What has one to do with the other? Am I missing some important context here, because the statement "euclidean geometry is not consistent with general relativity" doesn't make any sense to me. Euclidean geometry is mathematical abstraction built out of axioms that were chosen to sorta match our intuitive understanding of space. Relativity is a physics theory that uses some different mathematical geometry built out of different axioms, chosen so that resulting geometry would better fit observational data. And Gödelian consistency has nothing to do with physics at all, it considers mathematical constructs.
I think that consistency can matter to physicsts, there are some theories that can be written as axiomatic postulates that may be shown to contradict, not contradict, or be equivalent to ZFC as far as that question is concerned.
It's mathematics. Not physics. You're mixing things on a very bad level.
Mathematics exists in the mind, so to speak. There is no requirement of consistency with the physical world, even though some mathematics may be a useful language for modelling our physical theories of reality.
Euclidean geometry is self-consistent, and so is non-euclidean geometry. The point I think they were trying to make is that the parallel postulate is independent of the other 4, something that quite flustered mathematicians for millennia, until it was discovered that changing the parallel postulate led to different geometries, also very rich and interesting. So the parallel postulate is an example of an unprovable proposition in geometry with the other 4 axioms.
Mathematics exists in the mind, so to speak. There is no requirement of consistency with the physical world, even though some mathematics may be a useful language for modelling our physical theories of reality.
Euclidean geometry is self-consistent, and so is non-euclidean geometry. The point I think they were trying to make is that the parallel postulate is independent of the other 4, something that quite flustered mathematicians for millennia, until it was discovered that changing the parallel postulate led to different geometries, also very rich and interesting. So the parallel postulate is an example of an unprovable proposition in geometry with the other 4 axioms.
Depending on what you mean, non-euclidean geometry may not be self-consistent (or it may! We don't know, unless we appeal to stronger axioms). In a large number of cases, the set of axioms which produce the geometries in question are powerful enough to embed natural-number-like structures, which would imply that the consistency of the systems cannot be proven from within the systems, themselves.
Godel's theorem doesn't apply to Euclidean geometry. Tarski proved it is complete, consistent, and decidable.
Yes, that's fine. That Tarski's axioms for Euclidean geometry are consistent is independent of that claim—the above is a statement about the "usual" non-euclidean geometries (i.e., Riemannian).
I think there are some crossed wires here. "Consistency" in this sense does not talk about how two systems are related to each other[0]—it is a question of whether you can prove that a system of axioms, call it S, is not absurd (i.e., cannot produce the statement, say, 0=1).
Godel's incompleteness answers the question, "can we prove S is consistent, assuming only the axioms given in S?" with a negative, assuming S is a complicated-enough set of axioms.[1]
----
[0] The usual disclaimer holds, we can talk about proving the consistency of some axiomatic system A, if some other axiomatic system B is consistent. E.g., we can show that the integers with the usual ring structure is consistent if ZFC is consistent, because we can construct integer arithmetic from ZFC. This, though, is a point I avoid for clarity since this is distinct from what I believe is being mentioned.
[1] I.e., if it can embed the rules of natural-number arithmetic in some way.
Godel's incompleteness answers the question, "can we prove S is consistent, assuming only the axioms given in S?" with a negative, assuming S is a complicated-enough set of axioms.[1]
----
[0] The usual disclaimer holds, we can talk about proving the consistency of some axiomatic system A, if some other axiomatic system B is consistent. E.g., we can show that the integers with the usual ring structure is consistent if ZFC is consistent, because we can construct integer arithmetic from ZFC. This, though, is a point I avoid for clarity since this is distinct from what I believe is being mentioned.
[1] I.e., if it can embed the rules of natural-number arithmetic in some way.
Per Wikipedia:
> Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals)
so if you don't go into the different advanced axioms of set theory (way beyond the target audience of the article), the article is close enough.
so if you don't go into the different advanced axioms of set theory (way beyond the target audience of the article), the article is close enough.
Joe was arguing that the restatement of the CH was incorrect, not that CH Has/hasn't been solved.
And on your point, Z=L is not a widely accepted axiom. Per wiki,
>Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms.
And on your point, Z=L is not a widely accepted axiom. Per wiki,
>Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms.
[deleted]
Given how loosely defined sets are, this is defining sets more than numbers or any continuum.
This article contains some glaring mistakes e.g. the statement of the Riemann hypothesis is wrong. I suspect the author tried to edit the mathematics for the sake of comprehensibility and, due to their unfamiliarity with the area, failed.
This is pretty confused statement of the continuum hypothesis; "There is no set whose cardinality is strictly between that of the integers and the real numbers."
Kurt Godel proved that the CH is consistent with standard set theory and Paul Cohen proved that the negation of the CH is consistent with standard set theory, the results together showing the CH is independent of standard set theory.
I'm all for simplification as long as you don't garble ideas beyond recognition.
[1] https://en.wikipedia.org/wiki/Continuum_hypothesis