New Foundations
Introduction
"New Foundations" (NF) is a system of set theory developed by American mathematician Willard Van Orman Quine in 1937. It is an alternative to the more commonly known Zermelo-Fraenkel set theory (ZF) and is notable for its unique approach to avoiding paradoxes such as Russell's Paradox. NF employs a stratification criterion for formulas, which ensures that sets are constructed in a way that avoids self-reference and other problematic constructions.
Historical Context
The development of NF occurred during a period of intense scrutiny and development in the foundations of mathematics. The early 20th century saw the emergence of several paradoxes that challenged the consistency of naive set theory. Russell's Paradox, discovered by Bertrand Russell in 1901, was particularly influential, leading to the development of more robust set theories, including Zermelo-Fraenkel set theory and Quine's New Foundations.
Stratification and Axioms
The central feature of NF is its stratification criterion. A formula is said to be stratified if it is possible to assign a type to each variable in such a way that the formula respects the types. Specifically, if a variable appears in a set membership relation \( x \in y \), then the type of \( x \) must be one less than the type of \( y \). This stratification ensures that sets cannot contain themselves, directly or indirectly, thus avoiding paradoxes.
The axioms of NF include:
- **Extensionality**: Two sets are equal if and only if they have the same elements.
- **Stratified Comprehension**: For any stratified formula \( \phi(x) \), there exists a set \( \{ x \mid \phi(x) \} \).
Comparison with Zermelo-Fraenkel Set Theory
While Zermelo-Fraenkel set theory (ZF) is based on a cumulative hierarchy of sets, NF does not require such a hierarchy. Instead, NF relies on the stratification criterion to ensure consistency. This difference leads to several interesting consequences:
- **Universal Set**: NF allows the existence of a universal set, a set that contains all sets, which is not possible in ZF due to the Burali-Forti paradox.
- **Cardinality**: The concept of cardinality in NF differs from that in ZF. In NF, the cardinality of the universal set is not well-defined in the same way as in ZF.
Extensions and Variations
Several extensions and variations of NF have been proposed to address its limitations and explore its implications further. These include:
- **NFU**: New Foundations with urelements, introduced by Jensen in 1969, which allows for the existence of objects that are not sets.
- **TST**: Typographical Set Theory, a precursor to NF, which explicitly assigns types to variables.
Philosophical Implications
NF has significant philosophical implications for the foundations of mathematics. It challenges the necessity of the cumulative hierarchy and offers an alternative perspective on the nature of sets and their construction. The existence of a universal set in NF also raises questions about the nature of mathematical infinity and the concept of totality.
Applications and Current Research
While NF is not as widely used as ZF, it has found applications in certain areas of mathematical logic and theoretical computer science. Current research in NF includes exploring its consistency relative to other set theories, developing models of NF, and investigating its implications for the philosophy of mathematics.
Criticisms and Controversies
NF has faced several criticisms and controversies since its inception. One major criticism is the lack of a widely accepted proof of its consistency. While Jensen's NFU is known to be consistent relative to Zermelo set theory with urelements, the consistency of NF itself remains an open question. Additionally, the stratification criterion has been seen as somewhat artificial and less intuitive compared to the cumulative hierarchy of ZF.
Conclusion
New Foundations offers a unique and intriguing approach to set theory, distinguished by its stratification criterion and allowance for a universal set. While it has not achieved the same level of acceptance as Zermelo-Fraenkel set theory, it remains an important area of study in the foundations of mathematics. Its philosophical implications and potential applications continue to inspire research and debate.