New Foundations
A non-well-founded set theory conceived by Quine.
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New Foundations (NF) is a set theory in mathematical logic that is finitely axiomatizable and non-well-founded. It was devised by Willard Van Orman Quine as a streamlined version of the theory of types found in *Principia Mathematica*. The well-formed formulas of NF are the standard ones from propositional calculus, using just two primitive predicates: equality (=) and membership (∈).
The system can be presented with only two axiom schemata. The first is Extensionality: if two objects have exactly the same elements, they are the same object. The second is a restricted axiom schema of comprehension: the set {x | φ} exists for every stratified formula φ. A formula φ is stratified if there is a function assigning natural numbers to its syntactic parts so that for any atomic subformula x ∈ y, the number for y is one greater than the number for x, and for any atomic subformula x = y, the numbers for x and y are equal.
NF can also be given a finite axiomatization, which has the advantage of removing the concept of stratification. In this approach, the axioms correspond to natural basic constructions, whereas stratified comprehension is powerful but less intuitive. In his introductory book, Holmes took the finite axiomatization as fundamental and proved stratified comprehension as a theorem. The exact set of axioms can vary, but it typically includes most of the following, with the rest provable as theorems:
- Extensionality: If A and B are sets, and for every object x, x is in A exactly when x is in B, then A = B. (This can also serve as a definition of equality, but then another axiom is needed to justify substitution with that definition.) - Singleton: For every object x, the set ι(x) = {x} = {y | y = x} exists. - Cartesian Product: For any sets A and B, the set A × B = {(a, b) | a ∈ A and b ∈ B} exists. This can be restricted to just A × V or V × B. - Converse: For each relation R, the set R⁻¹ = {(x, y) | (y, x) ∈ R} exists. - Singleton Image: For any relation R, the set Rι = {({x}, {y}) | (x, y) ∈ R} exists. - Domain: If R is a relation, the set dom(R) = {x | ...} exists.
- field
- Mathematical logic
- known_for
- Non-well-founded, finitely axiomatizable set theory; simplification of the theory of types
Lore & Background
New Foundations was conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. The well-formed formulas of NF are the standard formulas of propositional calculus with two primitive predicates: equality and membership. NF can be presented with only two axiom schemata: Extensionality and a restricted axiom schema of comprehension, where the formula must be stratified. A formula is stratified if there exists a function from pieces of its syntax to natural numbers such that for any atomic subformula x ∈ y, f(y) = f(x) + 1, and for any atomic subformula x = y, f(x) = f(y).
Reader's Guide
New Foundations can be finitely axiomatized, which eliminates the notion of stratification. The axioms in a finite axiomatization correspond to natural basic constructions, whereas stratified comprehension is powerful but not necessarily intuitive. In his introductory book, Holmes opted to take the finite axiomatization as basic, and prove stratified comprehension as a theorem. The precise set of axioms can vary, but includes most of the following, with the others provable as theorems: Extensionality, Singleton, Cartesian Product, Converse, Singleton Image, Domain, Inclusion, Complement, Boolean Union, Universal Set, and Ordered Pair. The finite axiomatization provides a more intuitive foundation for the theory, making it accessible for further development in mathematical logic.
Did You Know?
- NF is a non-well-founded, finitely axiomatizable set theory.
- NF was conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica.
- NF can be presented with only two axiom schemata: Extensionality and a restricted axiom schema of comprehension.
- A formula in NF is stratified if there exists a function from pieces of its syntax to natural numbers satisfying specific conditions for atomic subformulas.
Frequently Asked Questions
Who created New Foundations and why?
Willard Van Orman Quine devised NF as a leaner alternative to the full theory of types in Principia Mathematica. His aim was to recover the essential typed structure of that system using far fewer axioms and a simpler logical base.
What makes New Foundations distinct from ZFC or other standard set theories?
NF is both non-well-founded and finitely axiomatizable, two properties that ZFC does not share. It also operates with only two primitive predicates—equality and membership—inside ordinary first-order logic, rather than requiring a richer language.
What axioms does New Foundations actually rest on?
The system is built from just two schemata: an Extensionality axiom saying that objects with exactly the same members are identical, and a restricted Comprehension schema that permits set-formation only for formulas satisfying a stratification condition. This compact axiom base is what earns NF the label 'finitely axiomatizable.'
How does New Foundations relate to Principia Mathematica?
Quine designed NF as a direct streamlining of the ramified type hierarchy that Whitehead and Russell constructed in Principia Mathematica. Instead of an infinite tower of types, NF enforces a single stratification constraint on well-formed formulas to control which comprehensions are allowed.
Why do set theorists and logicians still study New Foundations?
NF provides a concrete, finitely axiomatized setting where questions about the universal set, large cardinals, and circular membership can be investigated without the full apparatus of ZFC. Its non-well-founded character also makes it a natural laboratory for exploring self-referential and circular mathematical structures.
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