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Math problems that challenge the world

98 points by ohjeez 9y ago

ape4 9y ago
Yes I think it would be hard to build a functional model of the brain
- clishem 9y ago
  That's not even a problem of mathematics.
joe_the_user 9y ago
"Is there a number, which is larger than any finite number, between that of a countable set of numbers and the numbers of the continuum?" To think of a continuum, think of a number line—and all the numbers on it—without any gaps. This problem was answered by Kurt Gödel. 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 https://en.wikipedia.org/wiki/Continuum_hypothesis
- andrepd 9y ago
  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.
  - Avshalom 9y ago
    it's almost an antithetical non-sequitur at that, because Euclidean Geometry is consistent.
    - hinkley 9y ago
      “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.
      - TeMPOraL 9y ago
        > 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.
        whatshisface 9y ago
        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.
      - andrepd 9y ago
        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.
- aikol 9y ago
  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.
  - kss238 9y ago
    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.
    - deleted 9y ago
      [deleted]
- posterboy 9y ago
  Given how loosely defined sets are, this is defining sets more than numbers or any continuum.
hgibbs 9y ago
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.

98 points by ohjeez 9y ago

ape4 9y ago

Yes I think it would be hard to build a functional model of the brain

clishem 9y ago
That's not even a problem of mathematics.

joe_the_user 9y ago

"Is there a number, which is larger than any finite number, between that of a countable set of numbers and the numbers of the continuum?" To think of a continuum, think of a number line—and all the numbers on it—without any gaps. This problem was answered by Kurt Gödel. 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 https://en.wikipedia.org/wiki/Continuum_hypothesis

andrepd 9y ago
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.
- Avshalom 9y ago
  it's almost an antithetical non-sequitur at that, because Euclidean Geometry is consistent.
  - hinkley 9y ago
    “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.
    - TeMPOraL 9y ago
      > 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.
      - whatshisface 9y ago
        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.
    - andrepd 9y ago
      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.
aikol 9y ago
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.
- kss238 9y ago
  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.
  - deleted 9y ago
    [deleted]
posterboy 9y ago
Given how loosely defined sets are, this is defining sets more than numbers or any continuum.

hgibbs 9y ago

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.