A set of axioms for logic

Abstract

One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.

The Axiom Scheme of Acyclic Comprehension

“…The second author also conjectured that the scheme was very weak (meaning not equivalent to full stratified comprehension). The first author realized that one could attack this problem by attempting to prove all propositions in a finite axiomatization of stratified comprehension (that stratified comprehension is finitely axiomatizable was originally shown in [4], though the axiomatization given there is very unpleasant to work with). Undaunted by the skepticism of the second author, he proceeded to prove that each of the axioms of the finite axiomatization used in the second author's [5] (adapted to the Wiener ordered pair of [9]) follows from acyclic comprehension: a precis of his proof can be seen at [1].…”

“…We include the verification of this finite axiomatization by demonstrating that each of Hailperin's axioms in [4] follow from these axioms.…”

“…An indication of the proof is found in the second author's [5] (this is the version modified for use of the Kuratowski ordered pair). The theorem that stratified comprehension is finitely axiomatizable is due to Hailperin in [4]; details of the implementation (which is due to the second author) are inspired by the reduction of first-order logic to relation algebra in [8]. A complete verification that the axioms of Hailperin follow from the axioms given here is found below.…”

“…Define x ∈ k+1 y as (∃z.x ∈ z ∧ z ∈ k y). 4 We define [⊆] here as simply {(x, y) | (∀z.z ∈ x → z ∈ y)}, whereas in [5] the urelements are excluded from the domain and range of [⊆]: to see that this is harmless it is sufficient to note that we can define the class of urelements as U = (dom ([⊆] c )) c − {∅}, and then the restricted subset relation is realized as…”

The Usual Model Construction for NFU Preserves Information

“…Note that the Extensionality axiom of NF is an axiom, not an axiom scheme. Although this is less obvious, the same is true of stratified comprehension, because the stratified comprehension scheme is equivalent to the conjunction of finitely many of its instances (the canonical reference for this, though the specific implementation given is terrible, is [7]). …”