Gödel (1940): The continuum hypothesis is not false. Cohen (1964): The continuum hypothesis is not true.
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Pedantic note: logicians don't say "true" here. These theorems aren't about what's "true". They're about what can be derived from the Zermelo-Fraenkel axioms (with choice, if you like). Not all truth is contained in these axioms.
Would it be more accurate to say then The Continuum hypothesis “cannot be shown” to be true or false?
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It's more accurate... but it's even more accurate if you stop saying "true" and "false", which are irrelevant here, and say "cannot be proved from the ZF axioms". The ZF axioms might be true, they might be false: that's irrelevant to the results we're talking about.
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