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Volume 2 — Collapse Sets, Logic, and Predicate Paths

Logic and Set Theory in φ-Constrained Space​

This volume constructs set theory and logic entirely within the golden-base binary framework. Sets become clusters of collapse paths, membership is reachability, and logical operations respect the no-11 constraint throughout.

Chapter Index​

Chapter 032: SetBundle​

Collapse Path Clusters as φ-Structural Sets

Defines sets as bundles of related collapse paths, creating a new foundation for set theory.

Chapter 033: ReachIn​

Membership via Collapse Reachability Constraints

Membership is redefined as whether one trace can reach another through valid transformations.

Chapter 034: SetOverlay​

Union and Intersection via Path Bundle Superposition

Set operations through overlaying and intersecting trace path bundles.

Chapter 035: TraceMap​

Function as Collapse-Preserving Tensor Routing

Functions become trace-to-trace mappings that preserve structural constraints.

Chapter 036: CollapseCompose​

Compositional Mapping via Trace Chain Propagation

Function composition through chaining collapse transformations.

Chapter 037: PredTrace​

Logical Predicate as φ-Constrained Structural Selector

Predicates select traces based on structural properties.

Chapter 038: TruthEval​

Observer-Relative Evaluation over Collapse Structures

Truth becomes observer-dependent evaluation of trace configurations.

Chapter 039: QuantCollapse​

∀ / ∃ Quantification over Collapse Path Spaces

Universal and existential quantifiers range over trace structures.

Chapter 040: LogicTensor​

Structural Logic Connectives on Tensor Combinators

Logical operations (AND, OR, NOT) as tensor transformations.

Chapter 041: ImplCollapse​

Conditional Implication in Collapse Trace Systems

Implication as structural entailment between trace states.

Chapter 042: ConsistentTrace​

Logical Consistency via φ-Coherent Structure Composition

Consistency means maintaining φ-constraint across logical operations.

Chapter 043: CollapseDeduce​

Deductive Path Expansion under Constraint Entailment

Deduction as controlled expansion of trace paths.

Chapter 044: TruthNet​

Collapse-Aware Tensor Truth Table Generator

Truth tables in tensor form respecting structural constraints.

Chapter 045: LogicCircuit​

Constructing φ-Binary Circuits from Trace Primitives

Logic circuits built from golden-base components.

Chapter 046: CollapseSAT​

Trace-Constrained Structural Satisfiability

SAT problems in the context of φ-constrained structures.

Chapter 047: ModalCollapse​

Trace Modalities over Structure Observer Frames

Modal logic with necessity and possibility defined by trace reachability.

Key Concepts Introduced​

  1. Path Bundle Sets: Sets as collections of collapse paths
  2. Reachability Membership: ∈ as trace reachability
  3. Tensor Logic: Logical operations as tensor transformations
  4. Observer Truth: Truth relative to observer perspective
  5. Structural Consistency: φ-constraint preservation
  6. Modal Traces: Necessity and possibility in trace space

Dependencies​

  • Volume 0: Core language and trace grammar
  • Volume 1: Trace arithmetic for quantification

Next Steps​

  • Volume 3: Build algebraic structures on this logic
  • Volume 6: Extend to meta-logic and self-reference
  • Volume 7: Connect to observer systems

"In traces we trust, in reachability we prove."