# Quine's New Foundations

*First published Wed Jan 4, 2006; substantive revision Mon May 22, 2006*

Quine's system of axiomatic set theory, NF, takes its name from the title (“New Foundations for Mathematical Logic”) of the 1937 article which introduced it (Quine [1937a]). The axioms of NF are extensionality:

∀x∀y[x=y↔ ∀z(z∈x↔z∈y)]

together with *stratified comprehension*, which is to say all
universal closures of formulæ like

∃x∀y(y∈x↔ Φ)

where ‘*x*’ is not free in Φ and Φ is
(weakly) stratified. This last condition requires that there should be
a function σ (a “stratification”) from the variables
in Φ to an initial segment of the natural numbers such that if
‘*u* ∈ *v*’ is a subformula of Φ
then σ(‘*v*’) = σ
(‘*u*’) + 1 and if ‘*u* =
*v*’ is a subformula of Φ then σ
(‘*v*’) = σ(‘*u*’). The
origins of this constraint will be explained below.

Some illustrations may help: *x* ∈ *x* is not
stratified. *x* ∈ *y* is. Thus not every
substitution-instance of a stratified formula is stratified.
*y* =
(*x*)
is stratified, with the variable *y* being given a type one
higher than the type given to *x*. To check this we have to
write out ‘*y* =
(*x*)’
in primitive notation. In general it is always necessary to write a
formula out in primitive notation—at least until one gets the
hang of it.

The observant reader will have spotted the appearance above of the
expression *weakly stratified*. What is this? The instance of
the comprehension that says that *x* ∪ {*y*} always
exists is stratified, and is therefore an axiom. So *x* ∪
{*y*} always exists. So *x* ∪ {*x*} always
exists, by substitution. However the instance of the comprehension
scheme alleging its existence is not stratified. The term
‘*x* ∪ {*x*}’ is said to be *weakly
stratified* by which it is meant that it can be stratified if we
are allowed to give different types to distinct occurrences of
*free* variables. Weak stratification is what is needed to
admit those instances of the comprehension scheme that give us
substitution instances of stratified instances.

- 1. The Roots of Set Theory
- 2. Wellfoundedness and ZF
- 3. Type Theory and the Axiomatisation of NF
- 4. History
- 5. Other Current Research Areas
- 6. Implementation of Mathematical Concepts in NF
- 7. Coda
- Bibliography
- Other Internet Resources
- Related Entries

## 1. The Roots of Set Theory

There are various forces that gave rise to set theory, and various
forces that sustain it and drive it on. The following two conceptions
of *set* overlap but do not entirely coincide.

- The notion of
*arbitrary*set grew in the late 19th century from developments in analysis, somewhat in the way that the concept of arbitrary real-valued function had grown earlier in the century. To this day there are people who feel that the centre ground of set theory lies in Analysis. - A second idea is that of sets as one extensional (in the sense of the intension/extension distinction) limb (the others being lists and multisets) of a body whose intensional limbs are function-in-intension, properties, algorithm. (Sets as properties-in-extension is not a completely straightforward story: the point has been made that much of the heartsearching occasioned by the emergence of the Axiom of Choice as a principle of set existence arose precisely because sometimes the sets it postulated seemed not to be the extension of any property. But this is to make the same point in a back-handed way: it shows that people were expecting sets to be extensions of known intensions.)

If we think of sets as extensions, what are the corresponding intensions? The answer must surely be: one-place predicates. (Solace for the distractible: “So what are the extensions corresponding to many-place predicates?” The answer is not to worry; whatever you think they ought to be, you can fake them with ordered tuples and sets).

From our point of view, this is an interesting analysis, because on this account of set, we clearly ought to believe in an axiom of complementation for sets. If you think that a negation of a proposition is a proposition (so that in particular a negation of an atomic proposition is a proposition) then you also believe that a negation (or complement or whatever you want to call it) of a predicate is another predicate. Notice that this doesn't tell you that your complementation has to be classical, but it does tell you that a complement of a set should be a set. This will turn out to be one of the axioms of NF.

Sets are just extensional entities with particular multiplicity
properties (that mark them off from multisets) and certain
functionalities (that mark them off from functions-in-extension, graphs
etc). Among the variety of objects in the extensional canon, sets are
distinguished by their almost complete absence of structure. This
points to one of the most perplexing but prominent rôles of set
theory in modern mathematics. Almost any (indeed many people believe
*literally* any) fact in mathematics can be construed as a fact
about sets.

The idea that sets are extensions manifests itself in the idea (to which we will return) that set theory is the theory of one binary extensional relation (parallel to graph theory as the theory of a single symmetric irreflexive relation). Useful though this idea will turn out to be, it does not suggest any particular axioms for Set theory.

Those axioms have arisen largely out of our understanding of the
paradoxes. The most striking feature of the paradoxes is something
vulgarly called “self-reference” and ideas about how to
deal with it have been a useful constraint on our invention of axioms
for set theory. The search for a spell to ward off the evils attendant
upon self-reference has fed into the axiomatisation of set theory in
two ways, corresponding to the two ways in which self-reference has
been (mis)-understood. Is it circularity in the *specification*
of sets that is to be eschewed? Or do we locate the problem in
circularity in the *membership relation between sets*? At all
events, we have to be careful to keep to a minimum the number of
sinews we cut, for there are important theorems (Gödel's
Incompleteness theorem among others) whose proofs rely on the same
“self-reference” that we find in the paradoxes and which
we are loath to abandon.

## 2. Wellfoundedness and ZF

The axioms of ZF are motivated by the second point of view—in
fact by a very specific school of it: the view that the key is not
mere *noncircularity* of ∈ but *wellfoundedness*. A
set *x* is *wellfounded* if there are no infinite
sequences …*x*_{n} ∈
*x*_{n-1} ∈
…*x*_{3} ∈ *x*_{2} ∈
*x*_{1}. The axioms of ZF can usefully be thought of as
arising from an attempt to axiomatise the theory of wellfounded sets.
This project has been so successful that nowadays many people believe
that wellfounded sets are the only kind of set there is—if it
isn't wellfounded it isn't a set but a *hyperset* perhaps. This
idea is a modern rewriting of history: Set theory did not
*start* with the idea that all sets should be wellfounded (had
it done so, the axiom scheme of naïve comprehension would never
have been entertained!) The idea that we should admit a candidate set
as a genuine set only if promises to be wellfounded is an overreaction
to the uncontroversial observation that the suspect collections that
feature in the paradoxes (the universe, the Russell class) are all
flagrantly illfounded, or at least would be if they existed. It is
easy to jump from this insight to the conclusion that wellfoundedness
of ∈ is the only thing that matters. The result of this jump is
ZF set theory.

But historically-wrong and philosophically-hasty does not imply unproductive. There is no doubt that restricting attention to wellfounded sets (in the sense that it is them alone that one tries to axiomatise) concentrated the minds of 20th century mathematicians to great effect. ZF does seem to avoid the paradoxes, and the conception of set it formalises is one that apparently allows one to interpret the whole of mathematics. But how can an historically wrong and philosophically hasty story have such a happy ending? The answer is that, although it may be a mistake to think that all sets are wellfounded, it is not a mistake to think that the concept of wellfounded set is worth axiomatising.

## 3. Type Theory and the Axiomatisation of NF

The conception of set underlying NF arises from a related—indeed
complementary—oversimplification. ZF arises from the belief that
if we enforce a kind of wellfoundedness in the constitution of set
then all will be well. NF arises ultimately from the idea that all
will be well if we enforce a kind of wellfoundedness (or at least
noncircularity) in the *description* of sets. This concern can
take two forms: One might feel uneasy about conjuring up a set by
means of a spell that invokes other sets that are in some sense
“not yet” available (sets of which the candidate will turn
out to be a member, for example). This is a mathematically fruitful
concern (keyword “predicativity”) and it will reappear
later in the study of NF (and in ZF too, of course) but it is not this
first form that gives rise to the axiomatisation of NF. In the second
view, the key to neutralising the dangers of self-reference lies in
segregating sets into pairwise disjoint levels called
*types*. (That this concern is orthogonal to the first kind of
concern can be shown by consideration of the axiom of sumset. This is
a legitimate axiom from the second point of view, even though the
spell that conjures
∪*X* does it by invoking
the set *X*, which is of higher type.)

This view of set theory gave rise to the Theory of Types. In the
Theory of Types (as simplified successively by Ramsey, Carnap,
Gödel and Quine) the bottom type is a collection of atoms. Atoms
are non-sets (dormice, teapots, anything you please) with no internal
structure visible to the set theory. Thereafter type *n* + 1 is
simply the power set of type *n*, and there is a type for each
natural number. The process cannot be extended to give transfinite
types, since the types (being formally disjoint) are not cumulative
and there is no sensible way to define a ωth type. NF arises
from a consideration of the syntax used to describe structures of this
kind, and the syntax requires more careful handling. A first-order
language for describing structures of this sort has infinitely many
suites of variables, one suite for each type, and a membership symbol
(‘∈’ as usual) which can be placed only between
variables of consecutive types: ‘*x*_{n}
∈ *y*_{n+1}’ is wellformed but
‘*x*_{n} ∈
*y*_{n}’ is not. The equality sign can
be placed only between variables of the *same* type.

This type theory has the following axioms. Extensionality at each type:

∀x_{n}∀y_{n}[x_{n}=y_{n}↔ ∀z_{n+1}(x_{n}∈z_{n+1}↔y_{n}∈z_{n+1})]

and an axiom scheme of comprehension at each type. Notice that the
formula displayed above is not, strictly speaking, an axiom of
extensionality, since the variables do not have concrete type
subscripts, so it is not officially a formula at all. It's a kind of
axiom scheme. Annoyingly, this subtlety matters. The fact that
variables have type subscripts precludes the more usual exploitations
of subscripting for purposes such as indicating position in a
sequence. This makes the language of type theory infeasible for most
mathematical purposes, so one looks for excuses to omit the type
subscripts. Why might this be a safe manœuvre? Let us define
Φ^{+} to be the result of raising all the type subscripts
on all variables in Φ. It's safe because the theory has the same
axioms at each type, so it has the same theorems at each type, so
whenever one has a proof of a formula Φ one also has a proof of
Φ^{+}. One omits the type subscripts because even though,
once has omitted them, there are infinitely many ways of restoring
them, these infinitely many differently restored versions do not
appear to differ significantly. We shall take up later the theme of
the significance of this difference, but for the moment we note only
that it appears—as it appeared to Russell and his successors up
to and including Quine—that the differences could be safely
overlooked. This strategy of overlooking is called *typical
ambiguity*.

This version of Type theory, as simplified by Ramsey, Gödel, Carnap and Quine. is the point of departure for Quine [1937a] (which is still a good place to read the background). NF is obtained from this type theory by taking its axioms and stripping the type subscripts off all the variables. This is subtly different from Type Theory used with a policy of typical ambiguity. Under that scheme a formula like

(A) ∃x∀y(y∈x)

isn't really a formula at all, but a formula scheme, or shorthand for
any of the genuine formulæ to be obtained from it by adding
subscripts to ‘*x*’ and
‘*y*’. Which subscript? In every setting it will
either (i) become clear which subscript, or (ii) become clear that it
doesn't matter. At all events, (A) is not an axiom of Type theory: the
theory does not have a universal set, what it has is a universal set
at each type.

However it is not just the universal set that appears at each type.
We also get the naturals, the reals and the rationals etc. etc. at
each type, and this is clearly not what we want. Quine's move in
[1937a] is the natural one of taking the pseudoformulæ like (A)
and thinking of them as genuine formulæ in a new, one-sorted,
language. The new theory does not have a universal set at each type,
for it has no types: it has a single universal set. And a single
implementation of the reals, and so on. Formulæ of the
one-sorted language obtained in this way from formulæ of type
theory are said to be *stratified*. The axioms of NF are now
extensionality plus those instances of naïve comprehension that
happen to be stratified.

Interestingly, natural and well-motivated though this progression sounds, it does not provide a consistency proof for the system it ends with. We will discuss this later.

The hard part is to fully understand stratification. There is an easy
rule of thumb with formulæ that are in primitive notation, for
one can just ask oneself whether the formula could become a wff of
type theory by adding type indices. It's harder when one has
formulæ no longer in primitive notation, and the reader
encounters these difficulties very early on, since the ordered pair is
not a set-theoretic notion. How does one determine whether or not a
formula is stratified when it contains subformulæ like
ƒ(*x*) = *y*? The technical/notational difficulty
here lands on top of—as so often—a conceptual
difficulty. The answer is that of course one has to fix an
implementation of ordered pair and stick to it. Does that mean
that—for formulæ involving ordered pairs—whether or
not the given formula is stratified depends on how one implements
ordered pairs? The answer is ‘yes' but the situation is not as
grave as this suggests. There are various standard definitions of
ordered pair, and they are all legitimate in NF, and all satisfactory
in the sense that they are “level” or
*homogeneous*. All of them make the formula
“<*x*, *y*> = *z*” stratified
and give the variables *x* and *y* the same type;
*z* takes a higher type in most cases (never
lower). *How* much higher depends on the version of ordered
pair being used, but there are very few formulæ that come out
stratified on one version of ordered pair but unstratified on another,
and they are all pathological in ways reminiscent of the paradoxes.

The best way to illustrate this is by considering ordinals (=
isomorphism classes of wellorderings) in NF. For any ordinal α
the order type of the set (and it is a set) of the ordinals below
α is wellordered. In ZF one can prove that the wellordering of
the ordinals below α is of length α. In NF one cannot
prove this equation for arbitrary α since the formula in the set
abstract whose extension is the graph of the isomorphism is not
stratified for any implementation of ordered pair. Now any
wellordering *R* of a set *A* to length α gives
rise to a wellordering of {{{*a*}} : *a* ∈
*A*}, and if instead one tries to prove (in NF) that the
ordinals below α are isomorphic to the wellordering of length
α decorated with curly brackets, one finds that the very
assertion that there is an isomorphism between these two wellorderings
comes out stratified or unstratified depending on one's choice of
implementation of ordered pair! This is because, in some sense, the
applications of the pairing function are *two* deep in
wellordering of the ordinals below α, but only *one* deep
in the wellordering of the set of double singletons. If we use Quine
ordered pairs, the assertion is stratified—and provable. If one
uses Wiener-Kuratowski ordered pairs then the assertion is
unstratified and refutable. However if one uses Wiener-Kuratowki
ordered pairs there is instead the assertion that the ordinals below
α are isomorphic to the obvious wellordering of
{{{{{*a*}}}} : *a* ∈ *A*}, which comes out
stratified (and provable). In general for each implementation of
ordered pair there is a depth of nesting of curly brackets which will
make a version of this equality come out stratified and true. This
does not work with deviant implementations of ordered pair under
which “<*x*, *y*> = *z*” is
unstratified or even with those which are stratified but give the
variables *x* and *y* different types. Use of such
implementations of ordered pairs result in certain sets not being the
same size as themselves!

Perhaps a concrete example would help. Let us try to prove Cantor's
theorem. The key step in showing there is no surjection *f* :
*X*
(*X*) by *reductio ad
absurdum* is the construction of the diagonal set {*x*
∈ *X* : *x* ∉ ƒ(*x*)}. The proof
relies on this object being a set, which it will be a set if
“*x* ∈ *X* & *x* ∉
ƒ(*x*) & ƒ : *X* →
(*X*)”
is stratified. This in turn depends on
“∃*y*[*y* ∈
(*X*) &
<*y*, *x*>
∈ ƒ & ƒ : *X* →
(*X*)]” being
stratified. And it *isn't* stratified, because
“<*y*, *x*> ∈ ƒ” compels
‘*x*’ and ‘*y*’ to be given the
same type, while “ƒ : *X* →
(*X*)” will compel
‘*y*’ to be given a type one higher than
‘*x*’. This is because we have subformulæ
‘*x* ∈ *X*’ and ‘*y*
⊆ *x*’. Notice that we can draw this melancholy
conclusion without knowing whether the type of ‘ƒ’ is
one higher than that type of its argument, or two, or three
…. We cannot prove Cantor's theorem.

However if we try instead to prove that
{{*x*} : *x* ∈ *X*} is not the
same size as
(*X*)
we find that the diagonal set is defined by a stratified condition
and exists, so the proof succeeds. This tells us that we cannot prove
that |*X*| = |{{*x*} : *x* ∈ *X*}|
for arbitrary *X*: graphs of restrictions of the singleton
function tend not to exist. If they did, we would be able to prove
Cantor's theorem in full generality. This gives rise to an
endomorphism *T* on cardinals, where *T*|*X*|
=_{df} |{{*x*} : *x* ∈
*X*}|. *T* misbehaves in connection with the sets that
in NF studies we call *big* (as opposed to *large*, as
in *large cardinals* (in ZF)). These are the collections like
the universal set, and the set of all cardinals and the set of all
ordinals: collections denoted by expressions which in ZF-like theories
will pick out proper classes. If |*X*| = |{{*x*} :
*x* ∈ *X*}| we say that *X* is
*cantorian*. If the singleton function restricted to *X*
exists, we say that *X* is *strongly cantorian*. Sets
whose sizes are concrete natural numbers are strongly cantorian. IN
(the set of Frege natural numbers) is cantorian, but the assertion
that it is strongly cantorian implies the consistency of NF.

It can be plausibly argued that the bizarre behaviour of the
*T* function (and it is at its most bizarre in connection with
the big sets) is just the paradoxes bubbling up again. Assertions to
the effect that *T* is well-behaved have strong consistency
strength (Orey [1955], [1956]) and play in NF studies the kind of
rôle that large cardinal axioms play in ZF-like systems. Indeed
there are some exact equivalents. The *T*-assertions may look
odd, but their oddness is only skin-deep. They are actually expressing
the same mathematics that happens in ZF through an unfamiliar
formalisation.

## 4. History

For a long time (1937-1952) it was not known whether or not NF proved the axiom of infinity. Specker showed that it did, though he never published his proof: the (quite different) proof in Specker [1953] is of his stronger result that NF proves that the universe cannot be wellordered (and is therefore infinite) and the first published proof is in Henson [1973a])

So NF refutes AC. However—so far—it seems that (apart from
some weird weak formulations that make no sense outside an NF context
and which all arise from refinements of Specker's proof) it is only
*full* AC that fails, so that some strong consequences of AC
remain on the cards for the moment. In particular, no-one has refuted
the prime ideal theorem or countable choice or DC. Further, it is
striking that all known failures of AC involve big sets. An axiom
saying that all wellfounded sets are wellordered might—for all
we know—be consistent with NF. In particular there is no reason
to suppose that NF refutes the weak forms of AC needed in the study of
the reals and complexes (“Analysis”). Indeed as far as we
know we can safely add to NF some axioms to say that the wellfounded
sets form a model of ZFC. According to this view, NF does not so much
contradict ZFC as cover extra topics, such as the universal set
— phenomena about which ZF has nothing to say. This underpins
the view held by both Church and Quine that a synthesis of ZFC and NF
could be obtained.. Both Church and Quine, albeit in different ways,
were of the view that a synthesis of ZFC and NF could be obtained
along these lines.

Quite early on Quine added classes to NF to obtain a theory of
sets-with-classes known as ML. In the ZF context, adding classes is a
natural thing to do, for it enables one to reduce the infinite
replacement scheme to a single set-existence axiom: “the image
of a set in a class is a set” if augmented with suitable
class-existence axioms. However, none of the axioms of NF refer to
classes in the way the replacement scheme of ZF does, so there is
nothing for the class existence axioms to do. For this reason ML is
nowadays regarded as a pointless syntactic complication of NF with no
new mathematics and is not the subject of any research. At best the
classes can be a harmless halo of epiphenomena (Wang [1950]). However,
at worst—if the set existence axioms are allowed to have bound
class variables (as in the 1940 first edition of Quine
[1951a])—then the class of total orders that are genuinely
(“externally”) wellordered (no sub*classes* without
least members) becomes a set and we have the Burali-Forti paradox
(Rosser [1942]). The rumours still occasionally surfacing among
mathematicians of the inconsistency of NF probably have their origin
in a misreporting of this result but the result itself concerns ML
only so it has no bearing on the consistency question for NF. If NF is
inconsistent it is for totally different reasons.

In [1944] Hailperin gave the first of a number of finite
axiomatisations of NF now known. Many of them exploit the function
*x*
{*y* : *x* ∈ *y*} which is injective and
total and is an ∈-isomorphism. This function was known to
Whitehead, who suggested to Quine that {*y* : *x* ∈
*y*} should be called the “essence” of *x*
(a terminology clearly suggested by a view of sets as
properties-in-extension).

Another early result was the observation of Rosser [1952] that NF admitted a type-level ordered pair iff the axiom of infinity was a theorem of NF. (see Forster [1994])

The second major development was Scott's [1962] application of
Rieger-Bernays independence methods to NF. If
<*X*, *E*> is a binary structure with *E* an
extensional relation on *X* (so that
<*X*, *E*> is a model of a set theory of some kind)
and π is a permutation of *X*, then
{<*x*, π(*y*)> : <*x*, *y*>
∈ *E*} is also an extensional relation on *X* and
gives us a new model of some kind of set theory. If certain
constraints are respected, this method outputs models of ZF when given
models of ZF (and was first invented in order to prove the
independence of the axiom of foundation from the other axioms of ZF)
but it works even better with NF, since the construction preserves
stratified formulæ.

The third major development came when Specker (in [1958] and [1962])
placed the narrative in Quine [1937a] recounted above on a firm
mathematical footing. (For an illustrated translation, with
commentary, of Specker [1958], see Forster [2000], in the Other
Internet Resources section below.) We've seen that Φ^{+}
is the result of raising all the type subscripts on all variables in
Φ by one. Then Φ ↔ Φ^{+} is an *axiom of
typical ambiguity*. Specker established that the consistency of NF
is equivalent to the consistency of TST + the scheme of all axioms of
typical ambiguity. This reduction of the consistency question of NF to
the consistency of TST + the axiom scheme of typical ambiguity
clarifies matters but doesn't actually solve anything, since the
axioms of typical ambiguity turn out to be quite strong—as are
their analogues in other languages (such as geometry) that admit
endomorphisms. Notice that the mere fact that if φ is provable
then so is φ^{+} is not enough to ensure that φ →
φ^{+} is provable, so the axioms of typical ambiguity are
*prima facie* nontrivial and we should not be surprised that
they turn out to be strong.

Kaye [1991] generalised Specker's method to show how any typed theory
of sets, garnished with the axioms of typical ambiguity, turns out to
be equiconsistent with a one-sorted theory. Various weak fragments of
NF can be shown consistent by this kind of analysis. Oswald ([1976],
[1981] and [1982]) introduced a notation according to which
*NF*_{n} is the system axiomatised by the
*n*-stratified axioms of *NF*, and
*N*_{n}*F* is extensionality plus the
existence of {*x* : φ(*x*)} where φ is
*n*-stratified (parameters are allowed). Oswald showed that
*N*_{1}*F* ⊆ *NF*_{2} ⊆
*N*_{2}*F* ⊆ *NF*_{3} ⊆
*N*_{3}*F* = *NF* and that the inclusions
are all proper. The consistency of NF_{3} is a result of
Grishin [1969].

Crabbé [1982a] proved the consistency of two predicative
fragments of NF: NFP and NFI. Both are extensionality plus existence
of {*y* :
φ(*y*, *x*_{1},…,*x*_{n})}
where φ is stratified and there are extra restrictions on the
variables in φ. All variables (bound or free) are restricted to
type no higher than that of the set being defined; bound variable are
not to be of any type exceeding the type of *y* (NFP) or (type
of *y*) + 1 (NFI). NFP + Axiom of sumset = NF. They are
distinguished from the other subsystems of NF known to be consistent
by the circumstance that they allow a proof of the axiom of infinity
which makes essential use of Specker's proof of the axiom of
infinity. In NFP we argue: either the axiom of sumset holds, in which
case we have NF and can run Specker's proof of the Axiom of infinity,
or (ii) it doesn't, in which case—since finite sets have
sumsets—not every set is finite.

The most dramatic result obtained by a
type-theory-plus-ambiguity-axiom analysis came when Ronald Jensen
[1969] noticed that if we weaken TST to a theory called TSTU by
allowing the existence of *urelemente* (distinct empty sets)
then it is easy to prove the consistency of the axioms of typical
ambiguity. The result is a model of NFU, which is NF with
extensionality weakened to allow *urelemente*. Jensen referred
to NFU as “slight(?) modification of Quine's NF”,
but—as he showed—the modification has huge effects:
Specker's proof of the axiom of infinity no longer works, and AC is
not refutable either.

Jensen's consistency proof for TSTU (and thereby NFU) combines Ramsey theory with a technique that had been developed originally by Abraham Robinson [1939] to obtain a proof of the independence of extensionality from the other axioms of ZF.

Boffa [1988] noticed another consistency proof of NFU. If
<*V*, ∈> is a model of *Z* (Zermelo set
theory) with an external automorphism σ and an ordinal κ
such that κ > σ(κ), then
*V*_{κ} is the domain of a model of *NFU*
whose membership relation is

{<x,y> : σ(x) ∈y& ρ(y) ≤ σ(κ) + 1}.

(ρ is set-theoretic rank.) Remarkably, as Holmes showed, one can
even recover from this model of NFU the nonstandard model of ZF that
gave rise to it. He does this by exploiting the idea of set pictures.
We saw earlier the idea that set theory is the theory of one
extensional binary relation (with equality of course: extensionality
cannot be defined in a language without equality). Thus one could say
that a set (or, better, a *set picture*) is a isomorphism class
of binary structures consisting of a domain, an extensional binary
relation, and a designated element of the carrier set. Relational
structures of this kind look like transitive closures of sets and the
designated element of the domain indicates the set whose transitive
closure is being depicted. (Consequently there is an accessibility
condition for the designated element: if one thinks of the extensional
relation as a graph, there must be a directed path from any element to
the designated element).

Although it was Aczel's [1988] book that made this idea common
currency, the idea goes back at least to Hinnion's Ph.D. thesis [1974]
and in a less developed form to Specker's (unpublished)
*Habilitationsschrift* and quite possibly earlier. Aczel used
this technique to provide models of Forti-Honsell's antifoundation
axiom [1983]; Hinnion's idea was to use isomorphism classes of
*wellfounded* pointed extensional relational types to interpret
theories of wellfounded sets in NF. With wellfounded extensional
relations one does not need to make the pointing explicit, since the
wellfoundedness ensures that at most one element of the carrier set
can satisfy the accessibility condition. (This is not so in general:
the two sets *x* = {*y*}; *y* =
{*x*, Ø} have set pictures that have the same binary
relation but different pointing.) In NFU each set picture is a set (a
big set, as it happens) and Holmes showed that in a model arising
*à la Boffa* the set pictures reproduce the sets of the
original nonstandard model. This striking phenomenon—which has
no parallel in the study of NF—justifies Holmes' dictum that
“NFU can be understood as the theory of certain nonstandard
models of Zermelo-style set theory with automorphisms”

There is a widespread misapprehension that Cantor's theorem shows
that there cannot be a consistent set theory with a universal set. That
error can be refuted by Jensen's construction of his models for NFU,
but by far the most appealing demonstration is a later construction of
a model of a set theory with a universal set by Church and Oswald
simultaneously and independently in 1974. It has the advantage over NFU
of respecting extensionality: it is a genuine model of a theory of pure
sets. The construction is a refinement of Rieger-Bernays methods. It
goes as follows: start with a model of something like ZF, and a
bijection *k* between *V* and *V* × {0, 1}. We
now define a new membership relation *E* so that *xEy*
iff either the second component of *k*(*y*) is 1 and
*x* ∈ first component, or the second component of
*k*(*y*) is 0 and *x* ∉ first component.
*k*^{−1}(Ø) is now the universal set. In fact
*k*^{−1}(<*x*,0>) and
*k*^{−1}(<*x*,1>) are
set-complements in this new structure.

Perhaps surprisingly, given that constructions like this provide
complements for all sets, they give a central rôle to the concept
of *wellfounded set* in that the wellfounded sets of the new
model can be made to be an isomorphic copy of the wellfounded sets in
the structure one starts with. Further, these models satisfy a scheme
of replacement for wellfounded sets: anything the same size as a
wellfounded set is a set. (Replacement for, say, wellordered sets
doesn't come free in the same way)

## 5. Other Current Research Areas

ZF arose as an attempt to axiomatise the theory of wellfounded sets,
and this principle is kept constantly in mind by the ZF-istes as they
struggle to find new and more informative axioms. In contrast, the
syntactic version of the noncircularity insight (which is the one that
underlies NF) has been a much less fertile source of axioms for its
offspring. Most new axioms for NF are either versions of AC or
postulates of smooth behaviour by the *T* functions.

### 5.1 NF and Proof Theory

The proof theory of stratified formulæ is intriguing but frustratingly little work has been done on it. The sentence

∀x[∀y(y∈x→ ∀z(z∈y→ ⊥)) → ∀z(z∈x→ ⊥)]

is an example of a stratified theorem of first-order logic that has stratified proofs but no stratified cut-free proofs. Crabbé [1991], [1994] has given two proofs of the fact that SF (NF with extensionality dropped altogether) admits cut-elimination. The connections with type theory make constructive versions of NF an obvious target for investigation, but nothing publishable has emerged so far. The only known proof (Specker's) of the axiom of infinity in NF has too little constructive content to allow a demonstration that INF (NF with a constructive instead of classical ambient logic) admits an implementation of Heyting arithmetic, but opinion is divided on whether or not this tells us that INF is significantly weaker than NF.

## 6. Implementation of Mathematical Concepts in NF

To the extent that mathematical concepts can be implemented in the
theory of wellfounded sets (AKA ZF) they can also be interpreted in NF
since NF has a theory of wellfounded sets. However this is not the way
in which we routinely interpret mathematics in NF since the
interpreting formulæ are unstratified and unwieldy: extra axioms
are needed. For example, the Von Neumann implementation of natural
numbers gives us a collection of natural numbers that is not the
extension of a stratified formula, so it cannot be expected to be a
set. By the same token, ordinary functions from the naturals to the
naturals can fail to have extensions that are sets. Instead we trade
on the fact that NF (and NFU too) admits a natural implementation of
various kinds of mathematical entities as global isomorphism
classes—which will of course be big sets (qv). The natural
number *n* is implemented as the set of all *n*-membered
sets: this is the Frege implementation used also in TST. Rosser
[1953a] uses NF instead of ZF as a foundation for presenting ordinary
mathematics (and to this day remains the only book to do so, though
Holmes [1998] uses NFU for this purpose).

As we have seen, implementing mathematical objects (ZF-style) as
wellfounded sets and then reasoning about those well-founded sets) is
not a very satisfactory way to proceed, since NF does not have
sufficient strength for us to manipulate the implemented objects with
the freedom we want. The situation with the endogenous interpretation
(mathematical objects as global equivalence classes) is potentially
more satisfactory. In this scenario cardinals are implemented as
equivalence classes under equipollence and are (big) sets. The
collection of all cardinals is a set. Ordinals can be handled
similarly: ordinals are (big) sets and the collection of all ordinals
is a (big) set; the collection of all groups is a set, the collection
of all rings is a set; …. Being big, these sets cannot be
manipulated in the freewheeling way that small sets can be. (The axiom
of scheme of (full, unstratified) separation fails for big sets: if it
didn't then the Russell class would be {*x* ∈ *V*:
*x*
*x*})
and their manipulation is complicated by the appearance on the scene
of various analogues of the *T* function (that we saw in
connection with the cardinals) that are needed to work round the
stratification constraints. However, it is most striking that this
doesn't turn out to be much of a problem. On the whole, ordinary
mathematics seems to be what the Computer Scientists call *strongly
typed*, and to a large extent the stratification constraints of NF
coincide with the endogenous strong typing of the mathematical objects
being implemented. Sadly there is no space here for a proper
discussion of this deep and interesting question.

However, the expression “Ordinary mathematics” is disputed
turf. On another front there are logicians that are fond of making
the point that some of the further reaches of large cardinal theory,
routinely disregarded—and even disparaged—by many
mathematicians nevertheless have implications for a kind of low-level
mathematics whose core significance is not in dispute. Inevitably the
question of the adequacy of NF for “Ordinary mathematics”
will overlap with this debate. However there is a further
complication to be considered. All these big collections that
implement the collection of all cardinals (or of all ordinals etc.)
are nevertheless sets, so there are questions arising from their
sethood that we can ask about them, and some of these questions have
no parallel in ZF. What is the cofinality of the wellordering of the
ordinals? Can we map onto *V* the set of those sets that do
not map onto *V*? No doubt there will be people who say that
these questions are not proper mathematical questions at all, but are
mere pathological artefacts of a misconceived system of axioms.
However, there is the intriguing possibility that answers to these
questions could have implications for the study of the kind of small
sets that appear in “ordinary mathematics”. No such
implications have been found so far, but then no-one has looked very
hard. We know of subclasses of the naturals definable by formulæ
that have big sets as parameters, so we know where to start looking.
It is tempting to speculate what the consequences might be for NF
studies were any to be discovered.

## 7. Coda

The current status of NF studies is odd. In the 1970's Boffa and Coret
proved some interesting results about stratified formulæ in
theories of wellfounded sets, and work continues. However there is
little traffic in the other direction and it seems unlikely that NF
will attract the attention of a critical mass of scholars until
someone succeeds in proving it consistent relative to ZF or one of its
kin, (and even this is uncertain, for NFU is known to be consistent
and still finds few friends) and this consistency question seems to be
very hard. There are older unresolved issues in Set Theory—CH
for one—but the status of NF does seem to be the oldest
outstanding *consistency* question. The fact that NF refutes
the axiom of choice sharpens the consistency problem greatly, and
ought to sharpen public interest, but the widely-held view that sets
are, of their essence, wellfounded represents a huge rhetorical
obstacle to NF attracting the attention that its foundational interest
merits. That idea is now (post Aczel [1988]) less suffocating than it
was, but is still casts a shadow. The shadow it casts falls over NFU
as well, which is unfortunate, since NFU is known to be consistent,
and has a wealth of interesting model theory. It also distracts us
from the most basic feature of NF: as a theory of a universal set, it
does at least represent an earnest endeavour to take the paradoxical
classes seriously, and not just erase them. “Cowards, you
notice” says Higgins in Pygmalion “are always shrieking to
have troublesome people killed”.

## Bibliography

- Aczel, Peter [1988]
*Non-Well-Founded Sets*. Lecture notes, no. 14. CSLI: Stanford, California. - Boffa, M. [1988]
*ZFJ and the Consistency Problem for NF*. Jahrbuch der Kurt Gödel Gesellschaft (Wien), pp. 102-106 - Crabbé, M. [1982a] “On the Consistency of an
Impredicative Subsystem of Quine's NF.”
*Journal of Symbolic Logic*47, pp. 131-136. - -----. [1991] “Stratification and Cut-Elimination.”
*Journal of Symbolic Logic*56, pp. 213-226 - -----. [1994] “The Hauptsatz for Stratified Comprehension:
a Semantic Proof.”
*Mathematical Logic Quarterly*40, pp, 481-489. - Forster, Thomas. (1994) “Why Set Theory Without the Axiom of
Foundation?”,
*Journal of Logic and Computation*4, (August) pp. 333-335. [Preprint available online (in Postscript)] - Forti, M. and Honsell, F. [1983] “Set Theory with Free
Construction cprinciples.”
*Annali della Scuola Normale Superiore di Pisa, Scienze fisiche e matematiche*10, pp. 493-522. - Grishin, V.N. [1969] “Consistency of a Fragment of Quine's NF
System.”
*Soviet Mathematics Doklady*10, pp. 1387-1390 - Henson, C.W. [1973a] “Type-Raising Operations in NF.”
*Journal of Symbolic Logic*38, pp. 59-68. - Jensen, R.B. [1969] “On the Consistency of a Slight(?)
Modification of Quine's NF.”
*Synthese*19, pp. 250-263. - Kaye, R.W. [1991] “A Generalisation of Specker's Theorem on
Typical Ambiguity.”
*Journal of Symbolic Logic*56, pp 458-466. - Orey, S. [1955] “Formal Development of Ordinal Number
Theory.”
*Journal of Symbolic Logic*20, pp. 95-104. - -----. [1956] “On the Relative Consistency of Set
Theory.”
*Journal of Symbolic Logic*21, pp. 280-290. - -----. [1964] “New Foundations and the Axiom of
Counting.”
*Duke Mathematical Journal*31, pp. 655-660. - Oswald, U. [1976] “Fragmente von ‘New Foundations’ und Typentheorie.” Ph.D. thesis, ETH Zürich. 46 pp.
- -----. [1981] “Inequivalence of the Fragments of New
Foundations.”
*Archiv für mathematische Logik und Grundlagenforschung*21, pp. 77-82. - -----. [1982] “A Decision Method for the Existential
Theorems of NF2.”
*Cahiers du Centre de Logique (Louvain-la-neuve)*4, pp. 23-43. - Quine, W.V. [1937a] “New Foundations for Mathematical
Logic.”
*American Mathematical Monthly*44, pp. 70-80. Reprinted in Quine [1953a] - -----. [1951a]
*Mathematical Logic*, revised ed. Harvard University Press. - Robinson, A. [1939] “On the Independence of the Axioms of
Definiteness.”
*Journal of Symbolic Logic*4 (1939): 69-72. - Rosser, J.B. [1942] “The Burali-Forti paradox.”
*Journal of Symbolic Logic*7, pp. 11-17. - -----. [1952] “The Axiom of Infinity in Quine's New
Foundations.”
*Journal of Symbolic Logic*17, pp. 238-242. - Specker, E.P. [1953] “The Axiom of Choice in Quine's New
Foundations for Mathematical clogic.”
*Proceedings of the National Academy of Sciences of the USA*39, pp. 972-975. - -----. [1958] “Dualität.”
*Dialectica*12, pp. 451-465. [Translation available online—see Other Internet Resources] - -----. [1962] “Typical ambiguity.”
*Logic, Methodology and Philosophy of cscience*, ed. E. Nagel, Stanford University Press, pp. 116-123. - Scott, D.S. [1962] “Quine's Individuals.”
*Logic, Methodology and Philosophy of Science*, ed. E. Nagel, Stanford University Press, pp. 111-115. - Wang, H. [1950] “A formal system of logic.”
*Journal of Symbolic Logic*15, pp. 25-32.

## Other Internet Resources

- Specker, E.P., 1958, "Duality" (Postscript file). This is a translation with commentary by T. Forster, 2000.
- New Foundations home page, maintained by Randall Holmes (Boise State University)