Poincarae's Philosophy: From Conventionalism to Phenomenology

By Elie Zahar | Go to book overview

APPENDIX I
Logical Independence

[A] A set Δ of sentences is said to be logically independent if no element of A logically follows from the remaining members of Δ; i.e. if Not [(Δ-{X})

X]for all X∈Δ.

[B] For any set Δ of formulas, < Δ> will denote the class of all logical consequences of Δ. Thus A⊆ <Δ>; <Δ> is moreover closed under the relation of logical consequence; i.e. « Δ» = < Δ>.

[C] A theory is any set of sentences closed under the relation of logical consequence; i.e. <Ω> = Ω. Ω will henceforth denote a first-order theory expressed in a denumerable language; and Ω() will be an arbitrary axiom system of Ω; e.g. Ω0 = Ω. More generally: Ω0⊆ < Ω0>= Ω.

[D] We propose to show that Ω possesses at least one logically independent set of axioms; i.e that there exists at least one logically independent subset Σ of Ω such that < Σ > = Ω.

[E] Since the underlying language is denumerable, we can write:

(1) Ω0 = { A0, A1, A2, …} = {Ai: i ∈ω)}: where, as usual, ω denotes the set of all natural numbers. Let us define the sequence of sentences B0, B1 … as follows:

(2) B0≡A0;Bn+1,≡(A0∧A1∧…∧An)⇒An+1).Put

(3) Σ0={B0,B1,…} = {Bi:i∈ω}.

[F] To show that Σ0 ⊆ Ω.

Proof: We have the tautology

Am ⇒ ((A0 ∧ … ∧ Am-1) ⇒ Am); i.e. Am Bm . But by definition of Ω0,. and of Ω: Am Ω0… ⊆Ω. Thus: Am ∈ Ω. Hence Ω Bm ; from which Bm ∈ Ω follows, for Ω. is closed under logical consequence. By (3): Σ0∈ Ω

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