Identity (address persistence through change)
Identity, in the τ-framework, is NF-address persistence through a morphism sequence. An entity x and an entity y are 'the same entity' across change iff their NF-address a appears at the source of the first morphism and the target of the last. No substance is required — Heraclitean flow with persistent invariants.
τ-Definition
Identity, in the τ-framework, is NF-address persistence through a morphism sequence. An entity x and an entity y are 'the same entity' across change iff their NF-address a appears at the source of the first morphism and the target of the last. No substance is required — Heraclitean flow with persistent invariants.
Categorical invariant. x =_τ y across Φ = (f₁,…,fₙ) ⟺ ∃ a ∈ NF-Addr : a ∈ Addr(src(f₁)) ∧ a ∈ Addr(tgt(fₙ)).
Primary registry anchor:
VII.D34
τ-Derivation Chain
-
VII.D25— Internal Set Ontology — existence is NF-addressability. -
VII.D23— Relational primacy — identity is determined by relational position, not intrinsic substance. -
VII.D34— Identity as Address Persistence — same entity iff NF-address persists through morphism sequence. -
VII.P06— Temporal ordering from persistence — the persistence relation induces the temporal partial order.
Lean modules referenced:
TauLib.BookVII.Meta.Registers
Phenomenological Correlate
Identity is instantiated whenever we say 'this is the same X as before' across observable change. The Ship of Theseus, personal identity over biographical time, the same river one cannot step into twice — all become well-posed once 'sameness' is reduced to address persistence rather than substance retention. What persists is structural position, not stuff.
Examples:
- Personal identity: 'I am the same person I was at age ten' (NF-address of the biographical agent persists through morphism sequences encoding maturation).
- Ship of Theseus: the ship is the same iff its NF-address is preserved by the plank-replacement morphisms; substance change does not break identity.
- Heraclitean river: 'the same river' — the river's NF-address persists; the water is replaced; the kernel-level invariant is intact.
- Computational object: a process retains identity through state updates iff its address in the kernel survives the update morphisms.
Register codomain: Cross-register (identity claims appear in all four registers; the persistence test is uniform).
Manuscript reference: manuscript-sources/book-07/part02/ch26.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookVII.Meta.Registers
Lean kind: structure
Lean symbol: Tau.BookVII.Meta.Registers.IdentityAsAddressPersistence