Fixed-Point Multiplicity and Actuality

Abstract

Powerset endomorphisms realize every fixed-point multiplicity, while a unique fixed point need not belong to a nonempty actuality predicate.

Theorem 1.1 (Self-consistency neither forces uniqueness nor selects actuality).

$$\begin{array}{c}(\neg \exists S:\mathrm{Set}(Unit),\mathrm{compl}(S)=S)\wedge \\ (\exists !S:\mathrm{Set}(Unit),\varnothing =S)\wedge \\ (\exists S:\mathrm{Set}(Bool),T:\mathrm{Set}(Bool),S\ne T\wedge \mathrm{inter}(S,\{false\})=S\wedge \mathrm{inter}(T,\{false\})=T)\wedge \\ (\forall S:\mathrm{Set}(Unit),\mathrm{union}(S,\varnothing )=S)\wedge \\ ((\exists S:\mathrm{Set}(Bool),\mathrm{inter}(S,\{false\})=S)\wedge (\neg \exists !S:\mathrm{Set}(Bool),\mathrm{inter}(S,\{false\})=S))\wedge \\ (\exists \mathcal{A}:\mathrm{Set}(\mathrm{Set}(Unit)),\mathrm{Nonempty}(\mathcal{A})\wedge (\exists !S:\mathrm{Set}(Unit),\varnothing =S)\wedge \forall S:\mathrm{Set}(Unit),\varnothing =S\Rightarrow \neg S\in \mathcal{A}).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Interpretation/FixedPointMultiplicity.fixed_point_multiplicity_and_actuality_gap (✓ std3). ∎

Source. Repository-derived.

Commentary.

Complement on subsets of a singleton has no fixed point; a constant-empty map has exactly one; intersection with the Boolean singleton has distinct fixed points; and union with the empty set fixes every subset of the singleton.

The same multiple-fixed-point construction directly refutes uniqueness. For actuality, the theorem supplies a nonempty predicate on singleton subsets that excludes every fixed point of the uniquely fixing constant-empty map. The source’s selector list is qualitative guidance without in-scope predicates, so no selector semantics are invented.

References

• Truth anchor: D5/S3/ConceptDynamics/Interpretation/FixedPointMultiplicity.fixed_point_multiplicity_and_actuality_gap