## 1. The Paradox



Central to any theory of sets is a statement of the conditions under
which sets are formed. In addition to simply listing the members of a
set, it was initially assumed that any well-defined condition (or
precisely specified property) could be used to determine a set. For
example, if \(T\) is the property of being a teacup, then the
set, \(S\), of all teacups might be defined as \(S = \{x: T(x)\}\), the set of all individuals,
\(x\), such that \(x\) has the property of being \(T\).
Even a contradictory property might be used to determine a set. For
example, the property of being both \(T\) and not\(-T\)
would determine the empty set, the set having no members. 



More precisely, naive set theory assumes the so-called naive
or unrestricted Comprehension Axiom, the axiom that for any formula
\(\phi(x)\) containing \(x\) as a free variable, there will
exist the set \(\{x: \phi(x)\}\) whose members are
exactly those objects that satisfy \(\phi(x)\). Thus, if the
formula \(\phi(x)\) stands for "\(x\) is
prime", then \(\{x: \phi(x)\}\) will be the set of
prime numbers. If \(\phi(x)\) stands for "\({\sim}(x = x)\)", then \(\{x: \phi(x)\}\) will be the
empty set. 



But from the assumption of this axiom, Russell's contradiction
follows. For example, if we let \(\phi(x)\) stand for \(x \in x\) and
let \(R = \{x: {\sim}\phi(x)\}\), then \(R\) is the set whose members
are exactly those objects that are not members of themselves.



Is \(R\) a member of itself? If it is, then it must satisfy the
condition of not being a member of itself and so it is not. If it is
not, then it must not satisfy the condition of not being a member of
itself, and so it must be a member of itself. Since by classical logic
one case or the other must hold - either \(R\) is a member
of itself or it is not - it follows that the theory implies a
contradiction.



As Russell tells us, it was after he applied the same kind of
reasoning found in Cantor's diagonal argument to a
"supposed class of all imaginable objects" that he was led
to the contradiction:



> 
> The comprehensive class we are considering, which is to embrace
> everything, must embrace itself as one of its members. In other words,
> if there is such a thing as "everything," then,
> "everything" is something, and is a member of the class
> "everything." But normally a class is not a member of
> itself. Mankind, for example, is not a man. Form now the assemblage of
> all classes which are not members of themselves. This is a class: is
> it a member of itself or not? If it is, it is one of those classes
> that are not members of themselves, i.e., it is not a member of
> itself. If it is not, it is not one of those classes that are not
> members of themselves, i.e. it is a member of itself. Thus of the two
> hypotheses - that it is, and that it is not, a member of itself
> - each implies its contradictory. This is a contradiction.
> (1919, 136)
> 



Standard responses to the paradox attempt to limit in some way the
conditions under which sets are formed. The goal is usually both to
eliminate \(R\) (and similar contradictory sets) and, at the same
time, to retain all other sets needed for mathematics. This is often
done by replacing the unrestricted Comprehension Axiom with the more
restrictive Separation Axiom, namely the axiom that given any
(consistent) set \(S\) and any formula \(\phi(x)\) with \(x\) free,
there will be a set \(\{x \in S: \phi(x)\}\) whose members are exactly
those members of \(S\) that satisfy \(\phi(x)\). If we now let
\(\phi(x)\) stand for the formula \(x \not\in x\), it turns out that
the corresponding set, \(\{x \in S: x \not\in x\}\) will not be
contradictory since it consists only of those members found within S
that are not members of themselves. Hence the set fails to include
itself. 



A variety of related paradoxes is discussed in the second chapter of
the Introduction to Whitehead and Russell (1910, 2nd edn 60-65), as
well as in the entry on
 paradoxes and contemporary logic
 in this encyclopedia. 


## 2. History of the Paradox



Russell appears to have discovered his paradox in the late spring of
1901, while working on his *Principles of Mathematics* (1903).
Exactly when the discovery took place is not clear. Russell initially
states that he came across the paradox "in June 1901"
(1944, 13). Later he reports that the discovery took place "in
the spring of 1901" (1959, 75). Still later he reports that he
came across the paradox, not in June, but in May of that year (1969,
221). Cesare Burali-Forti, an assistant to Giuseppe Peano, had
discovered a similar antinomy in 1897 when he noticed that since the
set of ordinals is well-ordered, it too must have an ordinal. However,
this ordinal must be both an element of the set of all ordinals and
yet greater than every such element. 



Unlike Burali-Forti's paradox, Russell's paradox does not
involve either ordinals or cardinals, relying instead only on the
primitive notions of set and set inclusion. Zermelo noticed a similar
contradiction sometime between 1897 and 1902, possibly anticipating
Russell by some years (Ebbinghaus and Peckhaus 2007, 43-48;
Tappenden 2013, 336), although Kanamori concludes that the discovery
could easily have been as late as 1902 (Kanamori 2009, 411). As Linsky
points out, Zermelo's argument, while similar to
Russell's, is best understood as one of a cluster of arguments
by Zermelo, Schroder and Cantor that "indeed
anticipated" the mathematical argument Russell developed but
that turned out to be different in small but significant ways from
Russell's argument (Linsky 2013, 11). In any case, the arguments
were thought to be of minor importance until it was realized how
detrimental they were to Gottlob Frege's
 foundations for arithmetic.



Russell wrote to Frege with news of his paradox on June 16, 1902. (For
the relevant correspondence, see Russell (1902) and Frege (1902) in
van Heijenoort (1967).) The paradox was of significance to
Frege's logical work since, in effect, it showed that the axioms
Frege was using to formalize his logic were inconsistent.
Specifically, Frege's Axiom V requires that an expression such
as \(\phi(x)\) be considered both a function of the argument \(x\) and
a function of the argument \(\phi\). (More precisely, Frege's
Law states that the course-of-values of a concept \(f\) is identical
to the course-of-values of a concept \(g\) if and only if \(f\) and
\(g\) agree on the value of every argument, i.e., if and only if for
every object \(x, f(x) = g(x)\). See section 2.4.1 of the entry on
 Gottlob Frege
 in this encyclopedia for more discussion.) In effect, it was this
ambiguity that allowed Russell to construct \(R\) in such a way
that it could both be and not be a member of itself.



Russell's letter arrived just as the second volume of
Frege's  *Grundgesetze der Arithmetik* (*The Basic
Laws of Arithmetic*, 1893, 1903) was in press. Immediately
appreciating the difficulty the paradox posed, Frege added to the
*Grundgesetze* a hastily composed appendix discussing
Russell's discovery. In the appendix Frege observes that the
consequences of Russell's paradox are not immediately clear. For
example, "Is it always permissible to speak of the extension of
a concept, of a class? And if not, how do we recognize the exceptional
cases? Can we always infer from the extension of one concept's
coinciding with that of a second, that every object which falls under
the first concept also falls under the second? These are the
questions," Frege notes, "raised by Mr Russell's
communication" (1903, 127). Because of these worries, Frege
eventually felt forced to abandon many of his views about logic and
mathematics. 



Even so, as Russell points out, Frege met the news of the paradox with
remarkable fortitude:



> 
> As I think about acts of integrity and grace, I realise that there is
> nothing in my knowledge to compare with Frege's dedication to
> truth. His entire life's work was on the verge of completion,
> much of his work had been ignored to the benefit of men infinitely
> less capable, his second volume was about to be published, and upon
> finding that his fundamental assumption was in error, he responded
> with intellectual pleasure clearly submerging any feelings of personal
> disappointment. It was almost superhuman and a telling indication of
> that of which men are capable if their dedication is to creative work
> and knowledge instead of cruder efforts to dominate and be known.
> (Quoted in van Heijenoort (1967), 127)
> 



Of course, Russell too was concerned about the consequences of the
contradiction. Upon learning that Frege agreed with him about the
significance of the result, he immediately began writing an appendix
for his own soon-to-be-released *Principles of Mathematics.*
Entitled "Appendix B: The Doctrine of Types," the appendix
represents Russell's first attempt at providing a principled
method for avoiding what soon was to become known as
"Russell's paradox."


## 3. Early Responses to the Paradox



The significance of Russell's paradox can be seen once it is
realized that, using classical logic, all sentences follow from a
contradiction. For example, assuming both \(P\) and \({\sim}P,\)
any arbitrary proposition, \(Q\), can be proved as follows: from
\(P\) we obtain \(P \vee Q\) by the rule of
Addition; then from \(P \vee Q\) and \({\sim}P\) we
obtain \(Q\) by the rule of Disjunctive Syllogism. Because set
theory underlies all branches of mathematics, many people began to
worry that the inconsistency of set theory would mean that no
mathematical proof could be completely trustworthy. Only by
eliminating Russell's paradox could mathematics as a whole
regain its consistency.



Russell's paradox ultimately stems from the idea that any
condition or property may be used to determine a set. For example, the
property of being evenly divisible by only itself and the number one
distinguishes the set of prime numbers from the set of whole numbers.
The property of having mammary glands distinguishes the set of mammals
from reptiles, birds and other living organisms. The property of being
both square and not square (or any other conjunction of contradictory
properties) determines the empty set, and so on.



One early skeptic concerning an unrestricted Comprehension (or
Abstraction) axiom was the originator of modern set theory, Georg
Cantor. Even prior to Russell's discovery, Cantor had rejected
unrestricted Comprehension in favour of what was, in effect, a
distinction between sets and classes, recognizing that some properties
(such as the property of being an ordinal) produced collections that
were simply too large to be sets, and that any assumption to the
contrary would lead to inconsistency. (Details can be found in Moore
(1982), Hallett (1984) and Menzel (1984).) 



Russell's own response to the paradox came with his aptly named
 theory of types.
 Believing that self-application lay at the heart of the paradox,
Russell's basic idea was that we can avoid commitment to
\(R\) (the set of all sets that are not members of themselves) by
arranging all sentences (or, more precisely, all
 propositional functions,
 functions which give propositions as their values) into a hierarchy.
It is then possible to refer to all objects for which a given
condition (or predicate) holds only if they are all at the same level
or of the same "type." 



This solution to Russell's paradox is motivated in large part by
adoption of the so-called *vicious circle principle*. The
principle in effect states that no propositional function can be
defined prior to specifying the function's scope of application.
In other words, before a function can be defined, one must first
specify exactly those objects to which the function will apply (the
function's domain). For example, before defining the predicate
"is a prime number," one first needs to define the
collection of objects that might possibly satisfy this predicate,
namely the set, \(N\), of natural numbers.



As Whitehead and Russell explain,



> 
> 
> An analysis of the paradoxes to be avoided shows that they all result
> from a kind of vicious circle. The vicious circles in question arise
> from supposing that a collection of objects may contain members which
> can only be defined by means of the collection as a whole. Thus, for
> example, the collection of *propositions* will be supposed to
> contain a proposition stating that "all propositions are either
> true or false." It would seem, however, that such a statement
> could not be legitimate unless "all propositions" referred
> to some already definite collection, which it cannot do if new
> propositions are created by statements about "all
> propositions." We shall, therefore, have to say that statements
> about "all propositions" are meaningless. ... The
> principle which enables us to avoid illegitimate totalities may be
> stated as follows: "Whatever involves *all* of a
> collection must not be one of the collection"; or, conversely:
> "If, provided a certain collection had a total, it would have
> members only definable in terms of that total, then the said
> collection has no total." We shall call this the
> "vicious-circle principle," because it enables us to avoid
> the vicious circles involved in the assumption of illegitimate
> totalities. (1910, 2nd edn 37)
> 
> 
> 



If Whitehead and Russell are right, it follows that no
function's scope of application will ever be able to include any
object presupposing the function itself. As a result, propositional
functions (along with their corresponding propositions) will end up
being arranged in a hierarchy of the kind Russell proposes.



Although Russell first introduced his theory of types in his 1903
*Principles of Mathematics*, he recognized immediately that
more work needed to be done since his initial account seemed to
resolve some but not all of the paradoxes. Among the alternatives he
considered was a so-called substitutional theory (Galaugher 2013).
This in turn led to type theory's more mature expression five
years later in Russell's 1908 article, "Mathematical Logic
as Based on the Theory of Types," and in the monumental work he
co-authored with
 Alfred North Whitehead,
 *Principia Mathematica* (1910, 1912, 1913). Russell's type theory thus
appears in two versions: the "simple theory" of 1903 and
the "ramified theory" of 1908. Both versions have been
criticized for being too ad hoc to eliminate the paradox successfully.




In response to Russell's paradox,
 David Hilbert
 also expanded his program of building a consistent, axiomatic
foundation for mathematics so that it included an axiomatic foundation
for logic and set theory (Peckhaus 2004). Underlying this
*formalist* approach was the idea of allowing the use of only
finite, well-defined and constructible objects, together with rules of
inference deemed to be absolutely certain.



Finally,
 Luitzen Brouwer
 developed *intuitionism*, whose basic idea was that one cannot
assert the existence of a mathematical object unless one can define a
procedure for constructing it.



Together, all of these responses helped focus attention on the
connections between logic, language and mathematics. They also helped
logicians develop an explicit awareness of the nature of formal
systems and of the kinds of metalogical and metamathematical results
that have proved to be central to research in the foundations of logic
and mathematics over the past one hundred years.


## 4. Russell's Paradox in Contemporary Logic



Russell's paradox is sometimes seen as a negative development
- as bringing down Frege's *Grundgesetze* and as
one of the original conceptual sins leading to our expulsion from
Cantor's paradise.
 W.V. Quine
 describes the paradox as an "antinomy" that "packs
a surprise that can be accommodated by nothing less than a repudiation
of our conceptual heritage" (1966, 11). Quine is referring to
the Naive Comprehension principle mentioned earlier. In symbols,
the principle states that 



(NC)   \(\exists A \forall x (x \in A \equiv \phi),\)




where \(A\) is not free in the formula \(\phi\).This says,
"There is a set \(A\) such that for any object \(x, x\) is an
element of \(A\) if and only if the condition expressed by \(\phi\)
holds." Russell's paradox arises by taking \(\phi\) to be
the formula: \(x \not\in x\). 



Despite Quine's comment, it is possible to see Russell's
paradox in a more positive light. For one thing, although the matter
remains controversial, later research has revealed that the paradox
does not necessarily short circuit Frege's derivation of
arithmetic from logic alone. Frege's version of NC (his Axiom V)
can simply be abandoned. (For details, see the entry on
 Frege's Theorem.)
 For another, Church gives an elegant formulation of the simple theory
of types that has proven fruitful even in areas removed from the
foundations of mathematics. (For details, see the entry on
 Type Theory.)
 Finally, the development of axiomatic (as opposed to naive) set
theories which exhibit various ingenious and mathematically and
philosophically significant ways of dealing with Russell's
paradox paved the way for stunning results in the metamathematics of
set theory. These results have included
 Godel's
 and Cohen's theorems on the independence of the
 axiom of choice
 and Cantor's continuum hypothesis. So let us see, roughly, how
some of these methods - specifically, the so-called
"untyped" methods - deal with Russell's
paradox.



Zermelo replaces NC with the following axiom schema of Separation (or
*Aussonderungsaxiom*): 



(ZA)   \(\forall A \exists B \forall x (x \in B \equiv(x \in A \wedge \phi)).\)




Again, to avoid circularity, \(B\) cannot be free in \(\phi\). This
demands that in order to gain entry into \(B, x\) must be a member of
an existing set \(A\). As one might imagine, this requires a host of
additional set-existence axioms, none of which would be required if NC
had held up. 



How does ZA avoid Russell's paradox? One might think at first
that it doesn't. After all, if we let \(A\) be \(V\) - the
whole universe of sets - and \(\phi\) be \(x \not\in x\), a
contradiction again appears to arise. But in this case, all the
contradiction shows is that \(V\) is not a set. All the contradiction
shows is that "\(V\)" is an empty name (i.e., that it has no
reference, that \(V\) does not exist), since the ontology of
Zermelo's system consists solely of sets. 



This same point can be made in yet another way, involving a
relativized form of Russell's argument. Let \(B\) be any set. By
ZA, the set \(R\_B = \{x \in B: x \not\in x\}\) exists, but it cannot
be an element of \(B\). For if it is an element of \(B\), then we can
ask whether or not it is an element of \(R\_B\); and it is if and only
if it is not. Thus something, namely \(R\_B\), is "missing"
from each set \(B\). So again, \(V\) is not a set, since nothing can be
missing from \(V\). But notice the following subtlety: unlike the previous
argument involving the direct application of *Aussonderungs* to
V, the present argument hints at the idea that, while \(V\) is not a set,
"V" is not an empty name. The next strategy for dealing
with Russell's paradox capitalizes on this hint. 



John von Neumann's (1925) untyped method for dealing with
paradoxes, and with Russell's paradox in particular, is simple
and ingenious. Von Neumann introduces a distinction between membership
and non-membership and, on this basis, draws a distinction between
sets and classes. An object is a *member* (simpliciter) if it
is a member of some class; and it is a *non-member* if it is
not a member of any class. (Actually, von Neumann develops a theory of
functions, taken as primitive, rather than classes, wherein
corresponding to the member/non-member distinction one has a
distinction between an object that can be an argument of some function
and one that cannot. In its modern form, due to Bernays and
 Godel,
 it is a single-sorted theory of classes.) 



Sets are then defined as members, and non-members are labeled
"proper classes." So for example, the Russell class,
\(R\), cannot be a member of any class, and hence it must be a
proper class. If \(R\) is assumed to be an element of a class
\(A\), then it follows from one of von Neumann's axioms that
\(R\) is not equivalent to \(V\). But \(R\) is equivalent to
\(V\), and hence not an element of \(A\). Thus, von
Neumann's method is closely related to the result stated above
about the set \(R\_B\), for arbitrary \(B\). Von
Neumann's method, while admired by the likes of Godel and
Bernays, has been undervalued in recent years. 



Quine (1937) and (1967) similarly provide another untyped method (in
letter if not in spirit) of blocking Russell's paradox, and one
that is rife with interesting anomalies. Quine's basic idea is
to introduce a *stratified comprehension axiom*. In effect, the
axiom blocks circularity by introducing a hierarchy (or
stratification) that is similar to type theory in some ways, and
dissimilar in others. (Details can be found in the entry on
 Quine's New Foundations.)
 



In contrast to Zermelo's, von Neumann's and Quine's
strategies, which are in a sense purely set theoretic, there have also
been attempts to avoid Russell's paradox by altering the
underlying logic. There have been many such attempts and we shall not
review them all, but one stands out as being, at the moment, both
radical and somewhat popular (although not with set theorists *per
se*): this is the *paraconsistent* approach, which limits
the overall effect of an isolated contradiction on an entire theory.
Classical logic mandates that any contradiction trivializes a theory
by making every sentence of the theory provable. This is because, in
classical logic, the following is a theorem:



(*Ex Falso Quadlibet*)   \(A \supset({\sim}A \supset B).\)




Now, virtually the only way to avoid EFQ is to give up disjunctive
syllogism, that is, given the usual definitions of the connectives,
*modus ponens*! So altering basic sentential logic in this way
is radical indeed - but possible. Unfortunately, even giving up
EFQ is not enough to retain a semblance of NC. One also has to give up
the following additional theorem of basic sentential logic:



(Contraction)   \((A \supset (A \supset B)) \supset (A \supset B).\)




It can then be argued that NC leads directly, not merely to an
isolated contradiction, but to triviality. (For the argument that this
is so, see the entry on
 Curry's paradox,
 section 2.2. Note too that it is not enough merely to retain the name
"*modus ponens*"; it is the rule itself that
becomes modified within non-traditional logics.) Thus it seems that
the woes of NC are not confined to Russell's paradox but also
include a negation-free paradox due to Curry.



Another suggestion might be to conclude that the paradox depends upon
an instance of the principle of Excluded Middle, that either \(R\) is
a member of \(R\) or it is not. This is a principle that is rejected
by some non-classical approaches to logic, including
intuitionism. However it is possible to formulate the paradox without
appealing to Excluded Middle by relying instead upon the Law of
Non-contradiction. We do so as follows: Given the definition of \(R\)
it follows that \(R \in R \equiv{\sim}(R \in R)\). So \(R \in R
\supset{\sim}(R \in R)\). But we also know that \(R \in R \supset R
\in R\). So \(R \in R \supset(R \in R \wedge{\sim}(R \in R))\). But by
the Law of Non-contradiction we know that \({\sim}(R \in R
\wedge{\sim}(R \in R))\). So by *modus tollens* we conclude
that \({\sim}(R \in R)\). At the same time we also know that since \(R
\in R \equiv{\sim}(R \in R)\), it follows that \({\sim}(R \in R)
\supset R \in R\), and thus that \(R \in R\). So we can deduce both
\(R \in R\) and its negation using only intuitionistically acceptable
methods. 



It seems, therefore, that proponents of non-classical logics cannot
claim to have preserved NC in any significant sense, other than
preserving the purely syntactical form of the principle, and neither
intuitionism nor paraconsistency plus the abandonment of Contraction
will offer an advantage over the untyped solutions of Zermelo, von
Neumann, or Quine. (Further discussion can be found in Meyer, Routley
and Dunn (1979), Irvine (1992), Priest (2006, ch. 18), Weber (2010),
Weber (2012), and in the entries on
 Curry's paradox
 (sec. 2.2) and
 paraconsistent logic
 (sec. 2.3).) 



It is also worth noting that Russell's paradox was not the only
paradox that troubled Russell and, hence, not the only motivation for
the type restrictions one finds in *Principia Mathematica.* In
his earlier work, *The Principles of Mathematics*, Russell
devotes a chapter to "the Contradiction" (Russell's
paradox), presenting it in several forms and dismissing several
non-starter responses. He then signals that he will
"shortly" discuss the doctrine of types. This
doesn't happen for several hundred pages, until we reach the
very end of the book, in Appendix B! There Russell presents an
incipient, simple theory of types, not the theory of types we find in
*Principia Mathematica.* Why was the later theory needed? The
reason is that in Appendix B Russell also presents another paradox
which he thinks cannot be resolved by means of the simple theory of
types. This new paradox concerns propositions, not classes, and it,
together with the semantic paradoxes, led Russell to formulate his
ramified version of the theory of types. 



The new, propositional version of the paradox has not figured
prominently in the subsequent development of logic and set theory, but
it sorely puzzled Russell. For one thing, it seems to contradict
Cantor's theorem. Russell writes: "We cannot admit that
there are more ranges [classes of propositions] than
propositions" (1903, 527). The reason is that there seem to be
easy, one to one correlations between classes of propositions and
propositions. For example, the class \(m\) of propositions can be
correlated with the proposition that every proposition in \(m\)
is true. This, together with a fine-grained principle of individuation
for propositions (asserting, for one thing, that if the classes
\(m\) and \(n\) of propositions differ, then any proposition
about \(m\) will differ from any proposition about \(n)\)
leads to contradiction. 



There has been relatively little discussion of this paradox, although
it played a key role in the development of Church's logic of
sense and denotation. While we have several set theories to choose
from, we do not have anything like a well-developed theory of
Russellian propositions, although such propositions are central to the
views of Millians and direct-reference theorists. One would think that
such a theory would be required for the foundations of semantics, if
not for the foundations of mathematics. Thus, while one of
Russell's paradoxes has led to the fruitful development of the
foundations of mathematics, his "other" paradox has yet to
lead to anything remotely similar in the foundations of semantics. To
be sure, Church (1974a) and Anderson (1989) have attempted to develop
a Russellian intensional logic based on the ramified theory of types,
but an argument can be made that the ramified theory is too
restrictive to serve as a foundation for the semantics of natural
language. There have also been some recent attempts to obtain the
beginnings of a Russellian intensional logic based on untyped set
theories (Cantini 2004; Deutsch 2014). It is rather ironic that
although fine-grained Russellian propositions are favored in the
philosophy of language, the formal development of intensional logic is
dominated by Montague grammar, with its coarse-grained theory of
propositions. 



It is also worth noting that a number of seemingly purely
set-theoretical principles are actually (applied) instances of
theorems of pure logic (i.e., of first-order quantification theory
with identity)! There is a (partial) list of these in Kalish,
Montague, and Mar (2000). Russell's paradox is an instance of
T269 in this list:



(T269)   \({\sim}\exists y \forall x (Fxy \equiv{\sim} Fxx).\)




Reading the dyadic predicate letter "\(F\)" as
"is a member of", this says that it is not the case that
there is a \(y\) such that for any \(x, x\) is a
member of \(y\) if and only if \(x\) is not a member of
\(x\). Does this mean that Russell's paradox reduces to
T269? 



Certainly the proof of T269 distills the essence of Russell's
argument, its pattern of reasoning. But that pattern also underwrites
an endless list of seemingly frivolous "paradoxes" such as
the famous paradox of the barber who shaves all and only those who do
not shave themselves or, similarly, the paradox of the benevolent but
efficient God who helps all and only those who do not help themselves.




How do these "pseudo paradoxes," as they are sometimes
called, differ, if at all, from Russell's paradox? The pattern
of reasoning is the same and the conclusion - that there is no
such Barber, no such efficient God, no such set of non-self-membered
sets - is the same: such things simply don't exist.
(However, as von Neumann showed, it is not necessary to go quite this
far. Von Neumann's method instructs us not that such things as
\(R\) do not exist, but just that we cannot say much about them,
inasmuch as \(R\) and the like cannot fall into the extension of
any predicate that qualifies as a class.) 



The standard answer to this question is that the difference lies in
the subject matter. Quine asks, "why does it [Russell's
paradox] count as an antinomy and the barber paradox not?"; and
he answers, "The reason is that there has been in our habits of
thought an overwhelming presumption of there being such a class but no
presumption of there being such a barber" (1966, 14). Even so,
psychological talk of "habits of thought" is not
particularly illuminating. More to the point, Russell's paradox
sensibly gives rise to the question of what sets there are; but it is
nonsense to wonder, on such grounds as T269, what barbers or Gods
there are! 



This verdict, however, is not quite fair to fans of the Barber or of
T269 generally. They will insist that the question raised by T269 is
not what barbers or Gods there are, but rather what non-paradoxical
objects there are. This question is virtually the same as that raised
by Russell's paradox itself. Thus, from this perspective, the
relation between the Barber and Russell's paradox is much closer
than many (following Quine) have been willing to allow (Salmon 2013).




We note that there is a first-order logical formula that bears the
same relation to the principle about the
\(R\_B\)'s that T269 bears to Russell's
paradox. It is the following:



(T273)   
\(\forall z \forall y (\forall x [Fxy \equiv (Fxz \wedge{\sim} Fxx)] \supset{\sim} Fyz).\)




(We have taken the liberty of extending the numbering used in Kalish,
Montague and Mar (2000) to T273.) But not all set-theoretic paradoxes
are similarly related to first-order logical theorems. The
Burali-Forti paradox is an example, since the notion of a
well-ordering is not elementary; that is, it is not first-order
definable. 



Russell's paradox has never been passe, but recently
there has been an explosion of interest in it by scholars involved in
research in mathematical logic and in philosophical and historical
studies of modern logic. A glance at the contents of the 2004 volume
*One Hundred Years of Russell's Paradox* shows prominent
mathematical and philosophical logicians and historians of logic
poring over the paradox, proposing new ways back into Cantor's
paradise, or other ways of resolving the issue. Their investigations
include radically new ways out of the dilemma posed by the paradox,
new studies of the theories of types (simple and ramified, and
extensions thereof), new interpretations of Russell's paradox
and constructive theories, of Russell's paradox of propositions
and of his own attempt at an untyped theory (the substitution theory),
and so forth. 



All of this reminds us that fruitful work can arise from the most
unlikely of observations. As Dana Scott has put it, "It is to be
understood from the start that Russell's paradox is *not*
to be regarded as a disaster. It and the related paradoxes show that
the naive notion of all-inclusive collections is untenable. That
\(is\) an interesting result, no doubt about it" (1974,
207).