Timeline for Schemes (as in algebraic geometry) and first-order logic.
Current License: CC BY-SA 2.5
3 events
| when toggle format | what | by | license | comment | |
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| Nov 23, 2010 at 20:18 | comment | added | Neil Strickland | I don't really want to talk explicitly about open sets at all. Given a topos $\mathcal{E}$ and a sentence $\phi$ in a kind of first order language that refers to objects and morphisms of $\mathcal{E}$, there is a standard system of semantics that gives a subobject $[\phi]$ of the terminal object (or in other words, a truth value). If $\mathcal{E}$ is the category of sheaves on a scheme and $R$ is the structure sheaf and $\phi$ is the sentence $\forall a\in R (\exists b\in R ab=1)\vee (\exists b\in R (1-a)b=1)$ then $[\phi]$ is the whole terminal object (so $\phi$ is 'true'). | |
| Nov 23, 2010 at 19:58 | comment | added | David Feldman | Just so I understand, you mean this to speak to my final paragraph? One can't first-order capture "closure under arbitrary unions," so one can't presuppose first-order topology (even with the open sets rather than the points as primitives). | |
| Nov 23, 2010 at 17:06 | history | answered | Neil Strickland | CC BY-SA 2.5 |