New Foundations

New Foundations
Name	New Foundations
Author	W. V. O. Quine
Country	United States
Language	English
Subject	Set theory
Publisher	Harvard University Press
Pub date	1937
Media type	Print
Pages	201

New Foundations

New Foundations is a set-theoretic system proposed in 1937 by W. V. O. Quine aiming to reconcile paradox-avoidance with an unrestricted comprehension principle. It contrasts with contemporaneous systems such as Zermelo–Fraenkel set theory and Type theory while engaging with problems explored by Bertrand Russell, Ernst Zermelo, and Abraham Fraenkel. New Foundations has influenced work by logicians including Alonzo Church, Dana Scott, and Paul Bernays and remains a topic of interest in studies connected to automorphic set theory, category theory, and foundational debates addressed at venues like the International Congress of Mathematicians.

Overview

New Foundations is notable for admitting a universal set while attempting to block the Russell's paradox via a syntactic stratification constraint on comprehension. The system's axioms include extensionality and a scheme that allows comprehension for formulas that can be assigned a stratified typing akin to constraints studied in simple type theory and by researchers in model theory such as Saharon Shelah. New Foundations inspired later variations like NFU (New Foundations with urelements) studied by Roland M. Solovay and William T. Tait and has been examined in relation to independence results pursued by scholars including Kurt Gödel and Gerhard Gentzen.

History and Development

Quine introduced New Foundations in a monograph published by Harvard University Press; his proposal responded to paradoxes identified by figures like Gottlob Frege and Bertrand Russell. Early reactions engaged philosophers and mathematicians at institutions such as Harvard University, Princeton University, and the University of Cambridge, producing critiques from proponents of Zermelo set theory and elaborations by contemporaries including Alfred North Whitehead and Norbert Wiener. Mid-20th century work on models and consistency was advanced by logicians at University of California, Berkeley and Princeton University with contributions from Dana Scott who produced relative consistency proofs for variants with urelements. Later development saw connections to permutation models investigated by H. Jerome Keisler and Paul J. Cohen techniques for forcing and independence applied to related contexts.

Formal Definition and Axioms

The formal language of New Foundations uses the first-order apparatus familiar from Alfred Tarski's semantics and shares extensionality with Zermelo–Fraenkel set theory. Its central axiom is a comprehension schema restricted to formulas that are stratified: variables can be assigned integer levels so that atomic membership assertions correspond to level differences as in constraints studied by Willard Van Orman Quine and conceptualized relative to early type theory of Russell and Whitehead. The system includes axioms for unordered pair constructions analogous to treatments found in Ernst Zermelo's work and permits ordered pairs often defined via encodings popularized by Kurt Gödel. Variants may add axioms such as Infinity and Choice comparable to those in Zermelo–Fraenkel set theory and interact with large cardinal hypotheses investigated by Robert M. Solovay and John von Neumann.

Models and Consistency Results

Models of New Foundations have been constructed using techniques from model theory and permutation models developed in the wake of research by C. C. Chang and H. Jerome Keisler. Relative consistency results for NFU—where urelements are allowed—were established via methods connected to Zermelo–Fraenkel set theory and inner model techniques attributable to Dana Scott and Roland Hinnion. Complete consistency of the original system without urelements remains unresolved, prompting model constructions in contexts like tangled type theories and relations to the Aczel's anti-foundation axiom studied by Peter Aczel. Independence and equiconsistency efforts have invoked combinatorial set-theoretic tools developed by Paul Erdős and consistency strength comparisons drawn against theories involving large cardinals considered by W. Hugh Woodin.

Mathematics within New Foundations

Mathematical development inside New Foundations includes reconstruction of arithmetic and parts of analysis analogous to work in Peano arithmetic and frameworks comparable to those in Zermelo–Fraenkel set theory. Analysts and algebraists have formalized vector spaces, ring theory, and measure theory in NFU-based settings influenced by algebraic formalism from Emmy Noether and integration theory related to Henri Lebesgue. Category-theoretic formulations connecting to Saunders Mac Lane's ideas have been explored since NF-style comprehension can permit large categories including a category of all sets, engaging debates tied to Grothendieck-style universes and structural approaches promoted by William Lawvere.

Criticisms and Alternatives

Criticisms of New Foundations focus on its opaque stratification restriction and the unresolved consistency of the urelement-free form, leading many to prefer Zermelo–Fraenkel set theory with Choice as presented in mainstream texts by authors like Serge Lang and Kurt Gödel. Alternative approaches include Type theory as developed by Per Martin-Löf, non-well-founded set theories advanced by Peter Aczel, and weak systems with urelements studied by J. Barkley Rosser. Philosophical objections have been voiced by scholars at institutions including Oxford University and University of Chicago where foundational programs influenced by Bertrand Russell and Ludwig Wittgenstein inform debates about ontological commitments and formal expressiveness.

Category:Set theory