Source
Stone (1937), LIT-358, Thms 1, 2, 4, 7 (NOTE-311). Stone (1936), LIT-353, Thms 12, 49, 63–64, 67, 70 (NOTE-272). Kochen & Specker (1967), LIT-298, Thms 0 and 1 (NOTE-279). Isham & Butterfield (1998), LIT-325, §2.3 (NOTE-271).
What was actually shown
Stone 1936 recasts a Boolean algebra as a ring in which every element is idempotent. A prime ideal is the kernel of a homomorphism onto {0, 1} (Thm 49). Sending a to the set of prime ideals that omit it is an isomorphism onto an algebra of sets (Thm 67). The step that needs choice is that prime ideals exist and separate elements (Thms 63–64). Theorem 70 shows, without choice, that representability of every Boolean ring is equivalent to the Fundamental Proposition of ideal arithmetic. A finite Boolean ring is the power set of its atoms (Thm 12), and its prime ideals correspond one-to-one with the atoms.
Stone 1937 topologises the prime ideals, taking those sets as a basis. The space is totally disconnected, locally compact and Hausdorff, and compact exactly when the ring has a unit. Its compact open sets are exactly the ring’s elements (Thm 1), and every such space is the prime-ideal space of its own compact open sets (Thm 2). Isomorphism corresponds to homeomorphism, ideals to open sets, quotients to closed subsets (Thm 4), and subrings to continuous images (Thm 7). With a unit, compact open means clopen.
Kochen and Specker’s Theorem 0 reads the representation as a test. A partial Boolean algebra embeds in a Boolean algebra iff its homomorphisms to Z₂ separate points. Their Theorem 1 gives a 117-direction partial Boolean algebra in E³ with no such homomorphism at all. Isham and Butterfield restate this as the absence of global sections of the presheaf Hom(W, {0,1}) over Boolean subalgebras W. Stone’s theorem guarantees that this presheaf is non-empty and separating at every single stage.
The record’s join: every context, taken alone, is a Boolean algebra and so has all the two-valued valuations Stone promises. The Kochen–Specker failure therefore cannot sit in any one context. It sits in the requirement that the valuations agree where contexts share elements. NOTE-272 and NOTE-279 both draw this pairing; neither paper states it.
What this does not say
- That Stone wrote this in the vocabulary it is now cited in. Neither paper uses “ultrafilter”, “filter” or “Stone space”. The single use of “duality” (LIT-358, pp. 392–393) is for a different relation. The contravariant correspondence of morphisms has to be assembled from Thms 4 and 7; no one theorem states it.
- That Boolean algebras without a unit behave the same way. They correspond to locally compact spaces (LIT-358 Thm 8).
- That the Kochen–Specker obstruction holds in every dimension ≥ 3. LIT-298 proves it for a three-dimensional space only.
- That ultrafilters are topes. The atoms of the Boolean algebra generated by a hyperplane arrangement’s half-spaces are all of its faces, not only the full-dimensional regions (NOTE-272). Nothing in these sources covers arrangements.
- Anything probabilistic. That is THEORY-012’s criterion.
Connections
- THEORY-012: the probabilistic form of “no global section”. This theory is the Boolean-algebra fact underneath its deterministic, possibilistic case.
- LIT-313 (Gelfand–Naimark, Lemma 1): the same pattern for commutative C*-algebras, which are C of their maximal-ideal space. For an algebra generated by projections the two coincide (NOTE-303). The theorem is filed as THEORY-592, and the characters leg, Pontryagin duality, as THEORY-645.
- LIT-299 (Döring–Isham III, NOTE-290): the points of a context’s Gelfand spectrum are the ultrafilters of its projection lattice.
- LIT-145 (NOTE-177): maximal, prime and ultra filters coincide, in propositional-theory language.