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More on decidable proposition: a terminology change, and categorial interpretation
In terms of the internal logic of a topos, false is the bottom element in the poset of subobjects of the terminal object.
Isn’t that more in terms of the external logic of the topos? Wouldn’t the internal meaning of false be the bottom element of the subobject classifier regarded as an internal poset? Maybe this is splitting hairs, but we (or at least I) often seem to get confused on points like that.
I basically copied false from true, mutatis mutandis. So everything (well, almost everything) to be said about one should apply to the other. Anyway, I’ll work through this.
In the internal logic of a given category, the objects are contexts, and the propositions in a given context $X$ are the subobjects of the object $X$. So the propositions in the global context, which are the propositions with no free variables at all, are the subterminal objects. Then $false$ and $true$ are two of these.
However, you can speak of $false$ and $true$ in any context, so perhaps we should discuss the bottom/top element in any subobject poset.
Assuming that we’re in a topos (which after all is what was written), subterminal objects correspond to global elements of the subobject classifier $\Omega$, and subobjects of $X$ correspond to morphisms from $X$ to $\Omega$. If we regard $\Omega$ as an internal poset, then the elements of that in the context $X$ are precisely the morphisms from $X$ to the object $\Omega$. Which is what I just said.
So you’re saying the same as I did in my ‘However, […]’ paragraph. OK, I’ll change it.
While doing this, I had cause to write empty category and poset of subobjects.
Okay, sure.
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