I want to show that the law of the excluded middle do not hold in a bicartesian closed category (BCCC), interpreted as follows:

In general, there need not be a morphism $1 \to A + 0^A$ for $A \in \text{Obj}(\mathcal{C})$ in a BCCC $\mathcal{C}$.

Well, this is obviously true, because we can see Heyting algebras as BCCCs (in fact, I guess they are indeed equivalent as categories). And if there exists one such a morphism for each object $A$, then each Heyting algebras would be a Boolean algebra.

But is there a more direct proof of this statement?

  • 2
    $\begingroup$ this isn't indirect at all-to prove something need not hold, you just need to give an example. $\endgroup$ – Kevin Carlson Jun 23 '16 at 18:15

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