# New Foundations

In mathematical logic, **New Foundations** (**NF**) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of *Principia Mathematica*.

## Definition

The

- equalto
*B*. - A restricted axiom schema of comprehension: exists for eachstratified formula.

A formula is said to be *stratified* if there exists a function *f* from pieces of 's syntax to the natural numbers, such that for any atomic subformula of we have *f*(*y*) = *f*(*x*) + 1, while for any atomic subformula of , we have *f*(*x*) = *f*(*y*).

### Finite axiomatization

NF can be finitely axiomatized.^{[1]} One advantage of such a finite axiomatization is that it 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.^{[2]} The precise set of axioms can vary, but includes most of the following, with the others provable as theorems:^{[3]}^{[1]}

- Extensionality: If and are sets, and for each object , is an element of if and only if is an element of , then .
^{[4]}This can also be viewed as defining the equality symbol.^{[5]}^{[6]} - Singleton: For every object , the set exists, and is called the singleton of .
^{[7]}^{[8]} - Cartesian Product: For any sets , , the set , called the Cartesian product of and , exists.
^{[9]}This can be restricted to the existence of one of the cross products or .^{[10]}^{[11]} - Converse: For each relation , the set exists; observe that exactly if .
^{[12]}^{[13]}^{[14]} - Singleton Image: For any relation , the set , called the singleton image of , exists.
^{[15]}^{[16]}^{[17]} - Domain: If is a relation, the set , called the domain of , exists.
^{[12]}This can be defined using the operation of type lowering.^{[18]} - Inclusion: The set exists.
^{[19]}Equivalently, we may consider the set .^{[20]}^{[21]}

- Complement: For each set , the set , called the complement of , exists.
^{[22]} - (Boolean) Union: If and are sets, the set , called the (Boolean) union of and , exists.
^{[23]} - Universal Set: exists. It is straightforward that for any set , .
^{[22]} - Ordered Pair: For each , , the ordered pair of and , , exists; exactly if and . This and larger tuples can be a definition rather than an axiom if an ordered pair construction is used.
^{[24]} - Projections: The sets and exist (these are the relations which an ordered pair has to its first and second terms, which are technically referred to as its projections).
^{[25]} - Diagonal: The set exists, called the equality relation.
^{[25]} - Set Union: If is a set all of whose elements are sets, the set , called the (set) union of , exists.
^{[26]} - Relative Product: If , are relations, the set , called the relative product of and , exists.
^{[12]}

- Anti-intersection: exists. This is equivalent to complement and union together, with and .
^{[27]} - Cardinal one: The set of all singletons, , exists.
^{[28]}^{[29]} - Tuple Insertions: For a relation , the sets and exist.
^{[30]}^{[31]} - Type lowering: For any set , the set exists.
^{[32]}^{[33]}

### Typed Set Theory

New Foundations is closely related to **Russellian unramified typed set theory** (**TST**), a streamlined version of the theory of types of *Principia Mathematica* with a linear hierarchy of types. In this many-sorted theory, each variable and set is assigned a type. It is customary to write the *type indices* as superscripts: denotes a variable of type *n*. Type 0 consists of individuals otherwise undescribed. For each (meta-) natural number *n*, type *n*+1 objects are sets of type *n* objects; objects connected by identity have equal types and sets of type *n* have members of type *n*-1. The axioms of TST are extensionality, on sets of the same (positive) type, and comprehension, namely that if is a formula, then the set exists. In other words, given any formula , the formula is an axiom where represents the set and is not free in . This type theory is much less complicated than the one first set out in the *Principia Mathematica*, which included types for relations whose arguments were not necessarily all of the same types.

There is a correspondence between New Foundations and TST in terms of adding or erasing type annotations. In NF's comprehension schema, a formula is stratified exactly when the formula can be assigned types according to the rules of TST. This can be extended to map every NF formula to a set of corresponding TST formulas with various type index annotations. The mapping is one-to-many because TST has many similar formulas. For example, raising every type index in a TST formula by 1 results in a new, valid TST formula.

### Tangled Type Theory

Tangled Type Theory (TTT) is an extension of TST where each variable is typed by an ordinal rather than a natural number. The well-formed atomic formulas are and where . The axioms of TTT are those of TST where each variable of type is mapped to a variable where is an increasing function.

TTT is considered a "weird" theory because each type is related to *each* lower type in the same way. For example, type 2 sets have both type 1 members and type 0 members, and extensionality axioms assert that a type 2 set is determined uniquely by *either* its type 1 members or its type 0 members. Whereas TST has natural models where each type is the power set of type , in TTT each type is being interpreted as the power set of each lower type simultaneously. Regardless, a model of NF can be easily converted to a model of TTT, because in NF all the types are already one and the same. Conversely, with a more complicated argument, it can also be shown that the consistency of TTT implies the consistency of NF.^{[34]}

### NFU and other variants

**NF with urelements** (**NFU**) is an important variant of NF due to Jensen^{[35]} and clarified by Holmes.^{[3]} Urelements are objects that are not sets and do not contain any elements, but can be contained in sets. One of the simplest forms of axiomatization of NFU regards urelements as multiple, unequal empty sets, thus weakening the extensionality axiom of NF to:

- Weak extensionality: Two
*non-empty*objects with the same elements are the same object; formally,

In this axiomatization, the comprehension schema is unchanged, although the set will not be unique if it is empty (i.e. if is unsatisfiable).

However, for ease of use, it is more convenient to have a unique, "canonical" empty set. This can be done by introducing a sethood predicate to distinguish sets from atoms. The axioms are then:^{[36]}

- Sets: Only sets have members, i.e.
- Extensionality: Two
*sets*with the same elements are the same set, i.e. - Comprehension: The
*set*exists for each stratified formula , i.e.

NF_{3} is the fragment of NF with full extensionality (no urelements) and those instances of comprehension which can be stratified using at most three types. NF_{4} is the same theory as NF.

Mathematical Logic (ML) is an extension of NF that includes proper classes as well as sets. ML was proposed by Quine and revised by Hao Wang, who proved that NF and the revised ML are equiconsistent.

## Constructions

This section discusses some problematic constructions in NF. For a further development of mathematics in NFU, with a comparison to the development of the same in ZFC, see implementation of mathematics in set theory.

### Ordered pairs

Relations and functions are defined in TST (and in NF and NFU) as sets of ordered pairs in the usual way. For purposes of stratification, it is desirable that a relation or function is merely one type higher than the type of the members of its field. This requires defining the ordered pair so that its type is the same as that of its arguments (resulting in a **type-level** ordered pair). The usual definition of the ordered pair, namely , results in a type two higher than the type of its arguments *a* and *b*. Hence for purposes of determining stratification, a function is three types higher than the members of its field. NF and related theories usually employ Quine's set-theoretic definition of the ordered pair, which yields a type-level ordered pair. However, Quine's definition relies on set operations on each of the elements *a* and *b*, and therefore does not directly work in NFU.

As an alternative approach, Holmes^{[3]} takes the ordered pair *(a, b)* as a primitive notion, as well as its left and right projections and , i.e., functions such that and (in Holmes' axiomatization of NFU, the comprehension schema that asserts the existence of for any stratified formula is considered a theorem and only proved later, so expressions like are not considered proper definitions). Fortunately, whether the ordered pair is type-level by definition or by assumption (i.e., taken as primitive) usually does not matter.

### Natural numbers and the axiom of infinity

The usual form of the axiom of infinity is based on the von Neumann construction of the natural numbers, which is not suitable for NF, since the description of the successor operation (and many other aspects of von Neumann numerals) is necessarily unstratified. The usual form of natural numbers used in NF follows Frege's definition, i.e., the natural number *n* is represented by the set of all sets with *n* elements. Under this definition, 0 is easily defined as , and the successor operation can be defined in a stratified way: Under this definition, one can write down a statement analogous to the usual form of the axiom of infinity. However, that statement would be trivially true, since the universal set would be an inductive set.

Since inductive sets always exist, the set of natural numbers can be defined as the intersection of all inductive sets. This definition enables mathematical induction for stratified statements , because the set can be constructed, and when satisfies the conditions for mathematical induction, this set is an inductive set.

Finite sets can then be defined as sets that belong to a natural number. However, it is not trivial to prove that is not a "finite set", i.e., that the size of the universe is not a natural number. Suppose that . Then (it can be shown inductively that a finite set is not

**axiom of infinity**for NF:

^{[37]}

It may intuitively seem that one should be able to prove *Infinity* in NF(U) by constructing any "externally" infinite sequence of sets, such as . However, such a sequence could only be constructed through unstratified constructions (evidenced by the fact that TST itself has finite models), so such a proof could not be carried out in NF(U). In fact, *Infinity* is logically independent of NFU: There exists models of NFU where is a non-standard natural number. In such models, mathematical induction can prove statements about , making it impossible to "distinguish" from standard natural numbers.

However, there are some cases where *Infinity* can be proven (in which cases it may be referred to as the **theorem of infinity**):

- In NF (without urelements), Specker
^{[38]}has shown that the axiom of choice is false. Since it can be proved through induction that every finite set has a choice function (a stratified condition), it follows that is infinite. - In NFU with axioms asserting the existence of a type-level ordered pair, is equinumerous with its proper subset , which implies
*Infinity*.^{[37]}Conversely, NFU +*Infinity*+*Choice*proves the existence of a type-level ordered pair.^{[citation needed]}NFU +*Infinity*interprets NFU + "there is a type-level ordered pair" (they are not quite the same theory, but the differences are inessential).^{[citation needed]}

Stronger axioms of infinity exist, such as that the set of natural numbers is a strongly Cantorian set, or NFUM = NFU + *Infinity* + *Large Ordinals* + *Small Ordinals* which is equivalent to Morse–Kelley set theory plus a predicate on proper classes which is a *κ*-complete nonprincipal ultrafilter on the proper class ordinal *κ*.^{[39]}

### Large sets

NF (and NFU + *Infinity* + *Choice*, described below and known consistent) allow the construction of two kinds of sets that

*The universal set V*. Because is astratified formula, the universal setstructure.*V*= {*x*|*x=x*} exists by*Comprehension*. An immediate consequence is that all sets have complements, and the entire set-theoretic universe under NF has a Boolean- Frege's definition of the cardinal numbers works in NF and NFU: a cardinal number is an equivalence class of sets under the relation of equinumerosity: the setsbetween them, in which case we write . Likewise, an ordinal number is an
*A*and*B*are equinumerous if there exists a bijectionwell-orderedsets.

*
*

### Cartesian closure

The category whose objects are the sets of NF and whose arrows are the functions between those sets is not Cartesian closed;^{[40]} Since NF lacks Cartesian closure, not every function curries as one might intuitively expect, and NF is not a topos.

## Resolution of set-theoretic paradoxes

NF may seem to run afoul of problems similar to those in naive set theory, but this is not the case. For example, the existence of the impossible Russell class is not an axiom of NF, because cannot be stratified. NF steers clear of the three well-known paradoxes of set theory in drastically different ways than how those paradoxes are resolved in well-founded set theories such as ZFC. Many useful concepts that are unique to NF and its variants can be developed from the resolution of those paradoxes.

### Russell's paradox

The resolution of Russell's paradox is trivial: is not a stratified formula, so the existence of is not asserted by any instance of *Comprehension*. Quine said that he constructed NF with this paradox uppermost in mind.^{[41]}

### Cantor's paradox and Cantorian sets

Cantor's paradox boils down to the question of whether there exists a largest cardinal number, or equivalently, whether there exists a set with the largest cardinality. In NF, the universal set is obviously a set with the largest cardinality. However, Cantor's theorem says (given ZFC) that the power set of any set is larger than (there can be no injection (one-to-one map) from into ), which seems to imply a contradiction when .

Of course there is an injection from into since is the universal set, so it must be that Cantor's theorem (in its original form) does not hold in NF. Indeed, the proof of Cantor's theorem uses the

This failure is not surprising since makes no sense in TST: the type of is one higher than the type of . In NF, is a syntactical sentence due to the conflation of all the types, but any general proof involving *Comprehension* is unlikely to work.

The usual way to correct such a type problem is to replace with , the set of one-element subsets of . Indeed, the correctly typed version of Cantor's theorem is a theorem in TST (thanks to the diagonalization argument), and thus also a theorem in NF. In particular, : there are fewer one-element sets than sets (and so fewer one-element sets than general objects, if we are in NFU). The "obvious" bijection from the universe to the one-element sets is not a set; it is not a set because its definition is unstratified. Note that in all models of NFU + *Choice* it is the case that ; *Choice* allows one not only to prove that there are urelements but that there are many cardinals between and .

However, unlike in TST, is a syntactical sentence in NF(U), and as shown above one can talk about its truth value for specific values of (e.g. when it is false). A set which satisfies the intuitively appealing is said to be **Cantorian**: a Cantorian set satisfies the usual form of Cantor's theorem. A set which satisfies the further condition that , the restriction of the singleton map to *A*, is a set is not only Cantorian set but **strongly Cantorian**.^{[42]}

### Burali-Forti paradox and the T operation

The *Burali-Forti paradox* of the largest ordinal number is resolved in the opposite way: In NF, having access to the set of ordinals does not allow one to construct a "largest ordinal number". One can construct the ordinal that corresponds to the natural well-ordering of all ordinals, but that does not mean that is larger than all those ordinals.

To formalize the Burali-Forti paradox in NF, it is necessary to first formalize the concept of ordinal numbers. In NF, ordinals are defined (in the same way as in

*all*ordinals, not any proper initial segment of them.

However, the statement " is the order type of the natural order on the ordinals less than " is not stratified, so the transfinite induction argument does not work in NF. In fact, "the order type of the natural order on the ordinals less than " is at least *two* types higher than : The order relation is one type higher than assuming that is a type-level ordered pair, and the order type (equivalence class) is one type higher than . If is the usual Kuratowski ordered pair (two types higher than and ), then would be *four* types higher than .

To correct such a type problem, one needs the **T operation**, , that "raises the type" of an ordinal , just like how "raises the type" of the set . The T operation is defined as follows: If , then is the order type of the order . Now the lemma on order types may be restated in a stratified manner:

- The order type of the natural order on the ordinals is or , depending on which ordered pair is used.

Both versions of this statement can be proven by transfinite induction; we assume the type level pair hereinafter. This means that is always less than , the order type of *all* ordinals. In particular, .

Another (stratified) statement that can be proven by transfinite induction is that T is a strictly monotone (order-preserving) operation on the ordinals, i.e., iff . Hence the T operation is not a function: The collection of ordinals cannot have a least member, and thus cannot be a set. More concretely, the monotonicity of T implies , a "descending sequence" in the ordinals which also cannot be a set.

One might assert that this result shows that no model of NF(U) is "

## Consistency

Some mathematicians have questioned the

^{[43]}

Although NFU resolves the paradoxes similarly to NF, it has a much simpler consistency proof. The proof can be formalized within

^{[35]}

### Models of NFU

Jensen's proof gives a fairly simple method for producing models of NFU in bulk. Using well-known techniques of

^{}[note 1] of the cumulative hierarchy of sets. We may suppose without loss of generality that .

The domain of the model of NFU will be the nonstandard rank . The basic idea is that the automorphism *j* codes the "power set" of our "universe" into its externally isomorphic copy inside our "universe." The remaining objects not coding subsets of the universe are treated as urelements. Formally, the membership relation of the model of NFU will be

It may now be proved that this actually is a model of NFU. Let be a stratified formula in the language of NFU. Choose an assignment of types to all variables in the formula which witnesses the fact that it is stratified. Choose a natural number *N* greater than all types assigned to variables by this stratification. Expand the formula into a formula in the language of the nonstandard model of Zermelo set theory with automorphism *j* using the definition of membership in the model of NFU. Application of any power of *j* to both sides of an equation or membership statement preserves its truth value because *j* is an automorphism. Make such an application to each atomic formula in in such a way that each variable *x* assigned type *i* occurs with exactly applications of *j*. This is possible thanks to the form of the atomic membership statements derived from NFU membership statements, and to the formula being stratified. Each quantified sentence can be converted to the form (and similarly for

*j*is never applied to a bound variable. Choose any free variable

*y*in assigned type

*i*. Apply uniformly to the entire formula to obtain a formula in which

*y*appears without any application of

*j*. Now exists (because

*j*appears applied only to free variables and constants), belongs to , and contains exactly those

*y*which satisfy the original formula in the model of NFU. has this extension in the model of NFU (the application of

*j*corrects for the different definition of membership in the model of NFU). This establishes that

*Stratified Comprehension*holds in the model of NFU.

To see that weak *Extensionality* holds is straightforward: each nonempty element of inherits a unique extension from the nonstandard model, the empty set inherits its usual extension as well, and all other objects are urelements.

If is a natural number *n*, one gets a model of NFU which claims that the universe is finite (it is externally infinite, of course). If is infinite and the

*Choice*

*Infinity*+

*Choice*.

### Self-sufficiency of mathematical foundations in NFU

For philosophical reasons, it is important to note that it is not necessary to work in

Note that the construction of such sequences of sets is limited by the size of the type in which they are being constructed; this prevents TSTU from proving its own consistency (TSTU + *Infinity* can prove the consistency of TSTU; to prove the consistency of TSTU+*Infinity* one needs a type containing a set of cardinality , which cannot be proved to exist in TSTU+*Infinity* without stronger assumptions). Now the same results of model theory can be used to build a model of NFU and verify that it is a model of NFU in much the same way, with the 's being used in place of in the usual construction. The final move is to observe that since NFU is consistent, we can drop the use of absolute types in our metatheory, bootstrapping the metatheory from TSTU to NFU.

### Facts about the automorphism *j*

The

*W*has type α in NFU, then

*j*(

*W*) will be a well-ordering of type

*T*(α) in NFU.

In fact, *j* is coded by a function in the model of NFU. The function in the nonstandard model which sends the singleton of any element of to its sole element, becomes in NFU a function which sends each singleton {*x*}, where *x* is any object in the universe, to *j*(*x*). Call this function *Endo* and let it have the following properties: *Endo* is an injection from the set of singletons into the set of sets, with the property that *Endo*( {*x*} ) = {*Endo*( {*y*} ) | *y*∈*x*} for each set *x*. This function can define a type level "membership" relation on the universe, one reproducing the membership relation of the original nonstandard model.

## History

In 1914,

^{[44]}Hao Wang showed how to amend Quine's axioms for ML so as to avoid this problem.

^{[45]}

In 1944, Theodore Hailperin showed that *Comprehension* is equivalent to a finite conjunction of its instances,^{[1]} In 1953, Ernst Specker showed that the axiom of choice is false in NF (without urelements).^{[38]} In 1969, Jensen showed that adding urelements to NF yields a theory (NFU) that is provably consistent.^{[35]} That same year, Grishin proved NF_{3} consistent.^{[46]} Specker additionally showed that NF is equiconsistent with TST plus the axiom scheme of "typical ambiguity".^{[citation needed]} NF is also equiconsistent with TST augmented with a "type shifting automorphism", an operation (external to the theory) which raises type by one, mapping each type onto the next higher type, and preserves equality and membership relations.^{[citation needed]}

In 1983, Marcel Crabbé proved consistent a system he called NFI, whose axioms are unrestricted extensionality and those instances of comprehension in which no variable is assigned a type higher than that of the set asserted to exist. This is a

^{[according to whom?]}Holmes has

^{[date missing]}shown that NFP has the same consistency strength as the predicative theory of types of

*Principia Mathematica*without the axiom of reducibility

The Metamath database implemented Hailperin's finite axiomatization for New Foundations.^{[47]} Since 2015, several candidate proofs by Randall Holmes of the consistency of NF relative to ZF were available both on arXiv and on the logician's home page. His proofs were based on demonstrating the equiconsistency of a "weird" variant of TST, "tangled type theory with λ-types" (TTT_{λ}), with NF, and then showing that TTT_{λ} is consistent relative to ZF with atoms but without choice (ZFA) by constructing a class model of ZFA which includes "tangled webs of cardinals" in ZF with atoms and choice (ZFA+C). These proofs were "difficult to read, insanely involved, and involve the sort of elaborate bookkeeping which makes it easy to introduce errors". In 2024, Sky Wilshaw formalized a version of Holmes' proof using the proof assistant Lean, finally resolving the question of NF's consistency.^{[48]} Timothy Chow characterized Wilshaw's work as showing that the reluctance of peer reviewers to engage with a difficult to understand proof can be addressed with the help of proof assistants.^{[49]}

## See also

- Alternative set theory
- Axiomatic set theory
- Implementation of mathematics in set theory
- Positive set theory
- Set-theoretic definition of natural numbers

## Notes

- ^
^{a}^{b}^{c}Hailperin 1944. **^**Holmes 1998, chpt. 8.- ^
^{a}^{b}^{c}Holmes 1998. **^**Holmes 1998, p. 16.**^**Hailperin 1944, Definition 1.02 and Axiom PId.- W. V. O. Quine,
*Mathematical Logic*(1981) uses "three primitive notational devices: membership, joint denial, and quantification", then defines = in this fashion (pp.134–136) **^**Holmes 1998, p. 25.**^**Fenton 2015, ax-sn.**^**Holmes 1998, p. 27.**^**Hailperin 1944, p. 10, Axiom P5.**^**Fenton 2015, ax-xp.- ^
^{a}^{b}^{c}Holmes 1998, p. 31. **^**Hailperin 1944, p. 10, Axiom P7.**^**Fenton 2015, ax-cnv.**^**Holmes 1998, p. 32.**^**Hailperin 1944, p. 10, Axiom P2.**^**Fenton 2015, ax-si.**^**Hailperin 1944, p. 10.**^**Holmes 1998, p. 44.**^**Hailperin 1944, p. 10, Axiom P9.**^**Fenton 2015, ax-sset.- ^
^{a}^{b}Holmes 1998, p. 19. **^**Holmes 1998, p. 20.**^**Holmes 1998, pp. 26–27.- ^
^{a}^{b}Holmes 1998, p. 30. **^**Holmes 1998, p. 24.**^**Fenton 2015, ax-nin.**^**Hailperin 1944, p. 10, Axiom P8.**^**Fenton 2015, ax-1c.**^**Hailperin 1944, p. 10, Axioms P3,P4.**^**Fenton 2015, ax-ins2,ax-ins3.**^**Hailperin 1944, p. 10, Axiom P6.**^**Fenton 2015, ax-typlower.**^**Holmes & Wilshaw 2024.- ^
^{a}^{b}^{c}Jensen 1969. **^**Holmes 2001.- ^
^{a}^{b}Holmes 1998, sec. 12.1. - ^
^{a}^{b}Specker 1953. - JSTOR 2694912.
**^**Forster 2007.**^**Quine 1987.**^**Holmes 1998, sec. 17.5.**^**Smith, Peter (21 April 2024). "NF really is consistent".*Logic Matters*. Retrieved 21 April 2024.**^**Rosser 1942.**^**Wang 1950.**^**Grishin 1969.**^**Fenton 2015.**^**Smith, Peter (21 April 2024). "NF really is consistent".*Logic Matters*. Retrieved 21 April 2024.**^**Chow, Timothy (3 May 2024). "Timothy Chow on the NF consistency proof and Lean".*Logic Matters*. Retrieved 3 May 2024.

**
****^**We talk about the automorphism moving the rank rather than the ordinal because we do not want to assume that every ordinal in the model is the index of a rank.

## References

- Crabbé, Marcel (1982). "On the consistency of an impredicative fragment of Quine's NF".
*The Journal of Symbolic Logic*.**47**(1): 131–136.S2CID 42174966. - Fenton, Scott (2015). "New Foundations Explorer Home Page".
*Metamath*. Retrieved 25 April 2024. - Forster, Thomas (October 14, 2007). "Why the Sets of NF do not form a Cartesian-closed Category" (PDF).
*www.dpmms.cam.ac.uk*. - Forster, T. E. (2008). "The iterative conception of set" (PDF).
*The Review of Symbolic Logic*.**1**: 97–110.S2CID 15231169. - Forster, T. E. (1992),
*Set theory with a universal set. Exploring an untyped universe*, Oxford Science Publications, Oxford Logic Guides, vol. 20, New York: The Clarendon Press, Oxford University Press, - Forster, T. E. (2018). "Quine's New Foundations".
*Stanford Encyclopedia of Philosophy*. - Grishin, V. N. (1969). "Consistency of a fragment of Quine's NF system".
*Dokl. Akad. Nauk SSSR*.**189**(2): 41–243. - Hailperin, T (1944). "A set of axioms for logic". S2CID 39672836.
- Holmes, M. Randall (1998),
*Elementary set theory with a universal set*(PDF), Cahiers du Centre de Logique, vol. 10, Louvain-la-Neuve: Université Catholique de Louvain, Département de Philosophie,MR 1759289 - Holmes, M. Randall (2008). "Symmetry as a Criterion for Comprehension Motivating Quine's 'New Foundations'".
*Studia Logica*.**88**(2): 195–213. . - Holmes, M. Randall; Wilshaw, Sky (2024). "New Foundations is consistent" (PDF).
- S2CID 46960777With discussion by Quine.
- Quine, W. V. (1937), "New Foundations for Mathematical Logic",
*The American Mathematical Monthly*,**44**(2), Mathematical Association of America: 70–80, - Quine, Willard Van Orman (1940),
*Mathematical Logic*(first ed.), New York: W. W. Norton & Co., Inc., - Quine, Willard Van Orman (1951),
*Mathematical logic*(Revised ed.), Cambridge, Mass.: Harvard University Press, - Quine, W. V., 1980, "New Foundations for Mathematical Logic" in
*From a Logical Point of View*, 2nd ed., revised. Harvard Univ. Press: 80–101. The definitive version of where it all began, namely Quine's 1937 paper in the*American Mathematical Monthly*. - Quine, Willard Van Orman (1987). "The Inception of "New Foundations"".
*Selected Logic Papers - Enlarged Edition*. Harvard University Press.ISBN 9780674798373. - Rosser, Barkley (1942), "The Burali-Forti paradox",
*Journal of Symbolic Logic*,**7**(1): 1–17, - .
- Wang, Hao (1950), "A formal system of logic",
*Journal of Symbolic Logic*,**15**(1): 25–32,

*
*## External links

- "Enriched Stratified systems for the Foundations of Category Theory" by Solomon Feferman (2011)
- Stanford Encyclopedia of Philosophy:
- Quine's New Foundations — by Thomas Forster.
- Alternative axiomatic set theories — by Randall Holmes.

- Randall Holmes: New Foundations Home Page.
- Randall Holmes: Bibliography of Set Theory with a Universal Set.
- Randall Holmes: A new pass at the NF consistency proof