point-surjectivity

Compiled with AI assistance and selectively reviewed. Please consult the cited passages; the books remain the authoritative source. About this companion

Editorial synthesis · provisional

Definition

Point-surjectivity is the representability condition used in the manuscript’s Lawvere argument: every relevant map can be represented by an index in an evaluation scheme. The supplement explicitly defines a weak form by agreement at all global points. This differs from merely calling an arrow epimorphic, and its philosophical use requires justifying both the ambient category and the identification of exhaustive self-representation with that condition.

Explanation

For an evaluation map with two inputs in A and output in Y, weak point-surjectivity says that each map from A to Y has an index a₀ whose evaluation agrees with it on every global point of A. Surjectivity on global points of the curried map into the exponential is a stronger sufficient formulation. The proposed entry keeps those distinctions.

The diagonal proof applies an endomorphism to self-evaluation and selects an index representing the resulting map. Evaluating at that index gives a fixed point. A fixed-point-free endomorphism therefore prevents weak point-surjectivity in the stated setting; the conclusion is not a general slogan that every self-description must fail in every category.

The application requires enough relevant points and an output object admitting the needed fixed-point-free map. PB distinguishes a suitable configuration category from the Hilbert-space setting, where only zero global elements make this version vacuous. Identifying a world’s self-account and its openness with the required mathematical objects remains a contestable interpretation, not a consequence of the word exhaustive.

Scope and attribution

Preserve the pointwise definition, stronger sufficient formulation and ambient-category caveat.

PB — The Possible Book

The supplement defines weak point-surjectivity and explicitly limits its ambient and interpretive use.

Occurrence locators: 157, 182, 183, 184, 185.

Background used in the editorial explanation

Named theories and mechanisms are distinguished from neighboring background, homonyms and the author’s own synthesis. Reproduced AI passages retain their separate voice.