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I'm wondering if we have ever obtained interesting results or theories by refuting the continuum hypothesis.

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    The generalized continuum hypothesis is a very strong statement - it implies the axiom of choice, for example. So it's negation is pretty weak; it says that there is *some* cardinality $\mathfrak c$ such that there is a cardinal strictly in between $\mathfrak c$ and $2^{\mathfrak c}$, but it doesn't tell you anything about how large this counterexample to the generalized continuum hypothesis may be; it could be so large that it doesn't really have an impact on mathematics at the level of most real analysis and algebra, for example. – 2017-02-16
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    Just refuting the continuum hypothesis is enough. And we know of some interesting theories to do that (many forcing axioms, for example). – 2017-02-16

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You might find Freiling's axiom of symmetry to be of interest,

Freiling's axiom of symmetry

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    Could you elaborate a bit more? As it stands, your "answer" is little more than a link to Wikipedia (which incidentally is broken). – 2017-02-16
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    Seeing how this is your sixth or seventh answer in the span of a few days, maybe it's time to register? – 2017-02-16
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    Note that this only applies to CH - the failure of the GCH doesn't imply anything about the symmetry axiom. (Of course this isn't meant as criticism - the OP mentions *both* the CH and GCH.) – 2017-02-16