Quine's New Foundations and the concept of a Universal Set

[madness]

Has anyone come across Quine's New Foundations?

https://plato.stanford.edu/entries/quine-nf/
https://en.wikipedia.org/wiki/New_Foundations

I'm not very knowledgeable about set theory, mathematical logic, or the foundations of mathematics, but I found what I read interesting. The basic idea (as I understand it) is to modify the axioms of set theory to allow for the existence of a Universal Set without succumbing to Russell's Paradox. It doesn't seem to have made a big impact in mathematics as far as I can tell (and perhaps not in philosophy either). Why not? Is this not considered interesting by mathematicians? Or am I mistaken about its impact in the field?

 

[jedishrfu]

This article from Sciam might shed some light here:

https://www.scientificamerican.com/article/what-is-russells-paradox/

and some discussion on Universal sets:

https://en.wikipedia.org/wiki/Universal_set

I think though it may hinge on Godel's two theorems:

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

which basically say that in any system of logic there will statements which are undecidable that is we can't prove them true or false.

The original goal was to cast all of mathematics as derived from set theory but Russell's paradox and later Godel's theorems flattened that goal and so I guess that's what made it less interesting to mathematicians as shown in this wikipedia discussion:

https://en.wikipedia.org/wiki/Set_theory

Objections to set theory as a foundation for mathematics
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop's influential book Foundations of Constructive Analysis.[13]

A different objection put forth by Henri Poincaré is that defining sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo-Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".[14]

Ludwig Wittgenstein condemned set theory philosophically for its connotations of Mathematical platonism.[15] He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers".[16] Wittgenstein identified mathematics with algorithmic human deduction;[17] the need for a secure foundation for mathematics seemed, to him, nonsensical.[18] Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism. Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics.[19] Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in Remarks on the Foundations of Mathematics: Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel, Bernays, Dummett, and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments.[20]

Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.[21][22] Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces.[23]

An active area of research is the univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.[24][25]

Perhaps @fresh_42 or @Mark44 can provide a better answer.

 

[fresh_42]

I never had heard about it. However, after I read "NF cannot prove $\mathbb{N}$ is a set" and "NF is incompatible with AC" I think I haven't missed something important. It seems to be just another play toy for logicians and totally irrelevant for anybody else.

 

[madness]

This article from Sciam might shed some light here:

https://www.scientificamerican.com/article/what-is-russells-paradox/

and some discussion on Universal sets:

https://en.wikipedia.org/wiki/Universal_set

I think though it may hinge on Godel's two theorems:

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

which basically say that in any system of logic there will statements which are undecidable that is we can't prove them true or false.

The original goal was to cast all of mathematics as derived from set theory but Russell's paradox and later Godel's theorems flattened that goal and so I guess that's what made it less interesting to mathematicians as shown in this wikipedia discussion:

https://en.wikipedia.org/wiki/Set_theory
Perhaps @fresh_42 or @Mark44 can provide a better answer.

Quine's New Foundations came after Godel's incompleteness theorems and was motivated at least partly as a way to avoid Russell's paradox.

I never had heard about it. However, after I read "NF cannot prove $\mathbb{N}$ is a set" and "NF is incompatible with AC" I think I haven't missed something important. It seems to be just another play toy for logicians and totally irrelevant for anybody else.

I must have missed that. Can you show where you read that? There seems to be a few variants - NF, NFU, and ML. I read that NF but not NFU is incompatible with AC. The below thread also states that "One can give stratified definitions for individual Frege-Russell natural numbers, and then so too for the set N of all Frege-Russell naturals, so that exists in NF"

https://mathoverflow.net/questions/108488/a-question-about-quines-set-theory-nf

 

[fresh_42]

Can you show where you read that?

The axiom of choice cannot be added as an axiom to NF either - Ernst Specker proved in 1949 that it was incompatible with the other axioms of NF. Furthermore, one cannot prove the existence of the set of natural numbers.

https://de.wikipedia.org/wiki/New_Foundations

 

[madness]

https://de.wikipedia.org/wiki/New_Foundations

NFU, which is NF plus "urelements", is consistent with axiom of choice. And one can define the set of natural numbers in terms of equivalence classes of all sets with n elements in both NF and NFU. I'm not sure whether there is a difference between proving that the set of natural numbers exists and providing a definition of the set of natural number within the theory.

 

[TeethWhitener]

Is this not considered interesting by mathematicians? Or am I mistaken about its impact in the field?

It seems to be just another play toy for logicians and totally irrelevant for anybody else.

This seems about right. It’s important for philosophers of mathematics exploring the extension/intension distinction, and it’s really good for highlighting a lot of the foundational issues that were bothering the mathematics community at the turn of the 20th century (e.g., why was the axiom of choice so controversial?).

There are a lot of things in NF that are unfamiliar to most practicing mathematicians, which is why it probably doesn't see too much use nowadays. NF doesn't use the von Neumann definition of ordinals, since that would lead to the Burali-Forti paradox. Instead, it uses the Frege-Russell definition. It also uses the Quine definition for ordered pairs, rather than the Kuratowski definition.

NF is incompatible with AC: it can be shown that the universal set that shows up in the theory cannot be well-ordered (and AC is equivalent to the well-ordering theorem). But apparently whether NF is compatible with various useful weakenings of AC (dependent choice, countable choice, etc.) is still an open problem. So get to work!

NFU, which is NF plus "urelements", is consistent with axiom of choice.

Urelements are weird. For instance, in theories with a universal set and urelements, the powerset of the universal set can actually be smaller than the universal set itself (it can't be bigger: that's Cantor's paradox), since the universal set can contain urelements but the powerset can't (since the powerset is by definition the set of all subsets, and urelements aren't subsets).