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I was driving and just happened to wonder if there existed some concepts that are simple to grasp, yet are not provable via current mathematical techniques. Does anyone know of concepts that fit this criteria?

I imagine the level of simple could vary considerably from person to person, myself being on the very low end of things.

agweber
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  • Please define "not mathematically provable." Are you talking about simple statements in math that are unprovable given the usual axiomatic setup, or about something else? –  Nov 07 '13 at 20:05
  • @T.Bongers I'm not restricting to any certain types of statements. I can't come up with anything which is why I'm asking. – agweber Nov 07 '13 at 20:09
  • I want to re-ask T. Bongers' question. "Unprovable" is kind of an ambiguous term, because it implies that something has been proven: if something is "unprovable" that kind of implies it's been proven to not be able to be proven. Otherwise, how do we know the difference between "unprovable" and "hasn't been proven yet"? One way of interpreting your question is that you are asking for statements that have been proven to not be provable from the usual axioms of set theory. Is this what you mean, or something else? – Ben Blum-Smith Nov 12 '13 at 18:25
  • @BenBlum-Smith Ah, I understand why you're asking now. I'm never certain what I was thinking a few days prior, but I believe I was after items that we cannot prove via current mathematical techniques. I'll see if I can reword the question. – agweber Nov 12 '13 at 18:36

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The simplest that I know of, I don't know if you consider this simple to grasp, is the Continuum Hypothesis.

$\textbf{The statement}$: there is no infinity between the cardinality of $\mathbb{Z}$ ($|\mathbb{Z}|=\aleph_0$) and the cardinality of $\mathbb{R}$.

It has been proven that this can neither be proven nor disproven with Zermelo-Fraenkel set theory with the axiom of choice.

Tim Ratigan
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we know that $e$ is transcendental, $\pi$ is transcendental. but still we dont know that whether $e + \pi$ and $e - \pi$ is transcendental or not. we know that atleast one of them is transcendental which follows from simple calculation.

GA316
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    I don't think this really counts as "unprovable." It's difficult and challenging, but I really doubt this is independent of the usual axioms. –  Nov 07 '13 at 20:19