(This is an old post I never got around to finishing. It was originally going to have a second half about pointless topology; the interested reader can consult Vickers’ Topology via Logic on this theme.)
Standard presentations of propositional logic treat the Boolean operators “and,” “or,” and “not” as fundamental (e.g. these are the operators axiomatized by Boolean algebras). But from the point of view of category theory, arguably the most fundamental Boolean operator is “implies,” because it gives a collection of propositions the structure of a category, or more precisely a poset. We can endow the set of propositions with a morphism 
is a familiar inference rule (hypothetical syllogism). Since it is possible to define “and,” “or,” and “not” in terms of “implies” in the Boolean setting, we might want to see what happens when we start from the perspective that propositional logic ought to be about certain posets and figure out how to recover the familiar operations from propositional logic by thinking about what their universal properties should be.
It turns out that when we do this, we don’t get ordinary propositional logic back in the sense that the posets we end up identifying are not just the Boolean algebras: instead we’ll get Heyting algebras, and the corresponding notion of logic we’ll get is intuitionistic logic.
is a non-empty subset 
such that
, there is some 
such that 
.
, if 
then 
.
. (This holds in particular if 
.) A filter is an ultrafilter if in addition it is maximal under inclusion among (proper) filters. For Boolean rings, an equivalent condition is that for every 
either 
, but not both. Recall that this condition tells us that ultrafilters are precisely complements of maximal ideals, and can be identified with morphisms in 
. If 
for some set 
, then we will sometimes call an ultrafilter on 
an ultrafilter on 
“).
is a compact Hausdorff space and 
is its ring of functions. From this definition it follows that 
is a 
of compact Hausdorff spaces to the category 
of 
-algebras (which we are assuming have identities). Explicitly, a continuous function
is sent to a continuous function 
. The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that 
is an 
with the restriction of 
to the image of 
whose kernel is the set of functions which are zero on 
, so it exhibits 
?
we’d like to know if there’s a natural way to decompose it into its “components” 
. These aren’t its connected components in the topological sense, but in any reasonable sense the two parts are unrelated except possibly where they intersect. It turns out that the Noetherian condition is a natural way to answer this question. In fact, we will see that the Noetherian condition allows us to write 
uniquely as a union of a finite number of “components” which have a natural property that is stronger than connectedness.![\mathbb{C}[x, x^{ \frac{1}{2} }, x^{ \frac{1}{3} }, ... ]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BC%7D%5Bx%2C+x%5E%7B+%5Cfrac%7B1%7D%7B2%7D+%7D%2C+x%5E%7B+%5Cfrac%7B1%7D%7B3%7D+%7D%2C+...+%5D&bg=ffffff&fg=333333&s=0&c=20201002)
, the ring of fractional polynomials. 
for any 
, the ascending chain of ideals 
never terminates. In any reasonable theory of factorization writing 
and then comparing the ideals 
, then repeating this process to obtain a chain of ideals 
, should eventually stabilize at a prime. This leads to the following definition.
is Noetherian if every ascending chain of ideals stabilizes.![R[x]](https://s0.wp.com/latex.php?latex=R%5Bx%5D&bg=ffffff&fg=333333&s=0&c=20201002)
.
. The answer turns out to be quite nice.![\mathbb{C}[ t]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BC%7D%5B+t%5D&bg=ffffff&fg=333333&s=0&c=20201002)
of polynomials as a useful tool because, on intervals, it is dense in the continuous functions 
in the uniform topology. If we want to understand the relationship between 
and polynomial rings in a more general context, it might pay off to expand our scope from polynomial rings to more general types of well-behaved rings.
of real-valued continuous functions 
on a topological space 
Annoying Precision 