
Math problems that challenge the world - ohjeez
https://www.hpe.com/us/en/insights/articles/the-toughest-math-problems-that-challenge-the-world-1805.html
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joe_the_user
_" 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)

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andrepd
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_.

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Avshalom
it's almost an antithetical non-sequitur at that, because Euclidean Geometry
is consistent.

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hinkley
“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.

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TeMPOraL
> _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.

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whatshisface
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.

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hgibbs
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.

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ape4
Yes I think it would be hard to build a functional model of the brain

~~~
clishem
That's not even a problem of mathematics.

