Topographic map of the subcontinent and surrounding regions
The Indian subcontinent is a physiographic region of Asia below the Himalayas which projects into the Indian Ocean between the Bay of Bengal to the east and the Arabian Sea to the west. It is now divided between Bangladesh, India, and Pakistan. Although the terms "Indian subcontinent" and "South Asia" are often
also used interchangeably to denote a wider region which includes, in
addition, Bhutan, the Maldives, Nepal and Sri Lanka, the "Indian subcontinent" is more of a geophysical term, whereas "South Asia" is more geopolitical. "South Asia" frequently also includes Afghanistan, which is not considered part of the subcontinent even in extended usage.
Name
Historically, the region surrounding and southeast of the Indus River was often simply referred to as India
in many historical sources. Even today, historians use this term to
denote the entire Indian subcontinent when discussing history up until
the era of the British Raj. Over time, however, "India" evolved to refer to a distinct political entity that eventually became a nation-state (today the Republic of India).
According to the Oxford English Dictionary, the term subcontinent
signifies a "subdivision of a continent which has a distinct
geographical, political, or cultural identity" and also a "large land
mass somewhat smaller than a continent". Its use to signify the Indian subcontinent is evidenced from the early
twentieth century when most of the territory was either part of the British Empire or allied with them. It was a convenient term to refer to the region comprising both British India and the princely states.
The term has been particularly common in the British Empire and its successors, while the term South Asia is the more common usage in Europe and North America as well as in most countries in South Asia itself sometimes. According to historians Sugata Bose and Ayesha Jalal, the Indian subcontinent has come to be known as South Asia "in more recent and neutral parlance". Indologist Ronald B. Inden argues that the usage of the term South Asia is becoming more widespread since it clearly distinguishes the region from East Asia. While South Asia, a more accurate term that reflects the region's contemporary political demarcations, is replacing the Indian subcontinent,
a term closely linked to the region's colonial heritage, as a cover
term, the latter is still widely used in typological studies.
Since the Partition of India,
citizens of Pakistan (which became independent of British India in
1947) and Bangladesh (which became independent of Pakistan in 1971)
often perceive the use of the Indian subcontinent as offensive and suspicious because of the dominant placement of India in the term. As such it is being increasingly less used in those countries. Meanwhile, many Indian analysts prefer to use the term because of the socio-cultural commonalities of the region. The region has also been called the "Asian subcontinent", the "South Asian subcontinent", as well as "India" or "Greater India" in the classical and pre-modern sense.
The sport of cricket, introduced to the region by the British, is notably popular in India, Pakistan, Sri Lanka, Nepal, Bangladesh and Afghanistan. Within a cricket context, these countries are sometimes referred to simply as the subcontinent e.g. "Australia's tour of the subcontinent". The term is also sometimes used adjectivally in cricket e.g. "subcontinental conditions".
Cimmeria,
having rifted from Gondwana shown drifting towards Eurasia, closing the
Paleo-Tethys Ocean above, opening the Neo-Tethys Ocean below, and
carrying parts of what is today the Tibetan PlateauThe accretions of the Karakoram, the Kohistan-Ladakh island arc, and the Gangdese belt to Eurasia preceded the final India-Eurasia collision. The stars mark the syntaxis-causing obtrustions.
After the Lhasa terrane had adjoined Eurasia, an active
continental margin opened along its southern flank, below which the
Neo-Tethys oceanic plate had begun to subduct. Magmatic activity along this flank produced the Gangdese batholith in what is today the Tibetan trans-Himalaya. Another subduction zone opened to the west, in the ocean basin above the Kohistan-Ladakh island arc.
This island arc—formed by one oceanic plate subducting beneath another,
its magma rising and creating continental crust—drifted north, closed
its ocean basin and collided with Eurasia. Ladakh is today in the Indian-administered region of Kashmir and Kohistan in the Khyber-Pakhtunkhwa province of Pakistan, both on the Indian subcontinent.
The collision of India with Eurasia closed the Neo-Tethys Ocean. The suture zone (in this instance, the remnants of the Neo-Tethys
subduction zone pinched between the two continental crusts), which marks
India's welding to Eurasia, is called the Indus-Yarlung suture zone. It lies north of the Himalayas. The headwaters of the Indus River and the Yarlung Tsangpo (later in its course, the Brahmaputra) flow along this suture zone. These two Eurasian rivers, whose courses were continually diverted by
the rising Himalayas, define the western and eastern limits,
respectively, of the Himalayan mountain range.
The origin of the RSDLP split was Lenin's support for a smaller
party of professional revolutionaries, as opposed to the Menshevik
desire for a broad party membership. The influence of the factions
fluctuated in the years up to 1912, when the RSDLP formally split in
two. The political philosophy of the Bolsheviks was based on the
Leninist principles of vanguardism and democratic centralism.
Lenin was also more willing to use illegal means such as robbery to
fund the party's activities. By 1917, influenced by the experiences of World War I,
he reached the conclusion that the chain of world capitalism could
"break at its weakest link" in Russia before it assumed the level of the
advanced countries, opposing theorists such as Georgi Plekhanov. Lenin had also come to view poorer peasants as potential allies of the relatively small Russian proletariat.
After the February Revolution of 1917, Lenin returned to Russia and issued his April Theses, which called for "no support for the Provisional Government" and "all power to the soviets." During the summer of 1917, which saw events including the July Days and Kornilov affair,
large numbers of radicalized workers joined the Bolsheviks, which
planned the October Revolution that overthrew the government. The
Bolsheviks initially governed in coalition with the Left Socialist-Revolutionaries, but increasingly centralized power and suppressed opposition during the Russian Civil War. After 1921, it became the sole legal party in Soviet Russia and the Soviet Union. Under Joseph Stalin's leadership, Bolshevism became linked to his policies of "socialism in one country," rapid industrialization, collectivized agriculture, and centralized state control.
Lenin's political pamphlet What Is to Be Done?, written in 1901, helped to precipitate the Bolsheviks' split from the Mensheviks. In Germany, the book was published in 1902, but in Russia, strict censorship outlawed its publication and distribution. One of the main points of Lenin's writing was that a revolution
can only be achieved by a strong, professional leadership with deep
dedication to Marxist theoretical principles and an organization that
spanned through the whole of Russia, abandoning what Lenin called
"artisanal work" towards a more organized revolutionary work. After the
proposed revolution had successfully overthrown the Russian autocracy,
this strong leadership would relinquish power and allow a socialist party to fully develop within the principles of democratic centralism. Lenin said that if professional revolutionaries
did not maintain influence over the fight of the workers, then that
fight would steer away from the party's objective and carry on under the
influence of opposing beliefs or even away from revolution entirely.
The pamphlet also showed that Lenin's view of a socialist intelligentsia was in line with Marxist theory. For example, Lenin agreed with the Marxist ideal of social classes ceasing to be and for the eventual "withering away of the state". Most party members considered unequal treatment of workers immoral and were loyal to the idea of a completely classless society. This pamphlet also showed that Lenin opposed another group of reformers, known as "Economists",
who were for economic reform while leaving the government relatively
unchanged and who, in Lenin's view, failed to recognize the importance
of uniting the working population behind the party's cause.
Second Party Congress
At the 2nd Congress of the RSDLP, which was held in Brussels and then London during August 1903, Lenin and Julius Martov disagreed over the party membership rules. Lenin, who was supported by Georgy Plekhanov,
wanted to limit membership to those who supported the party full-time
and worked in complete obedience to the elected party leadership. Martov
wanted to extend membership to anyone "who recognises the Party
Programme and supports it by material means and by regular personal
assistance under the direction of one of the party's organisations." Lenin believed his plan would develop a core group of professional
revolutionaries who would devote their full time and energy towards
developing the party into an organization capable of leading a
successful proletarian revolution against the Tsarist autocracy.
The base of active and experienced members would be the
recruiting ground for this professional core. Sympathizers would be left
outside and the party would be organised based on the concept of democratic centralism.
Martov, until then a close friend of Lenin, agreed with him that the
core of the party should consist of professional revolutionaries, but he
argued that party membership should be open to sympathizers,
revolutionary workers, and other fellow travellers. The two had
disagreed on the issue as early as March–May 1903, but it was not until
the Congress that their differences became irreconcilable and split the
party. At first, the disagreement appeared to be minor and inspired by
personal conflicts. For example, Lenin's insistence on dropping less
active editorial board members from Iskra
or Martov's support for the Organizing Committee of the Congress which
Lenin opposed. The differences grew and the split became irreparable.
Internal unrest also arose over the political structure that was best suited for Soviet power. As discussed in What Is To Be Done?,
Lenin firmly believed that a rigid political structure was needed to
effectively initiate a formal revolution. This idea was met with
opposition from once close allies, including Martov, Plekhanov, Vera Zasulich, Leon Trotsky, and Pavel Axelrod. Plekhanov and Lenin's major dispute arose addressing the topic of nationalizing land or leaving it for private use. Lenin wanted to nationalize to aid in collectivization, whereas Plekhanov
thought worker motivation would remain higher if individuals were able
to maintain their own property. Those who opposed Lenin and wanted to
continue on the socialist mode of production path towards complete socialism
and disagreed with his strict party membership guidelines became known
as "softs" while Lenin supporters became known as "hards".
Some of the factionalism could be attributed to Lenin's steadfast
belief in his own opinion and what was described by Plekhanov as
Lenin's inability to "bear opinions which were contrary to his own" and
loyalty to his own self-envisioned utopia. Lenin was seen even by fellow party members as being so narrow-minded
and unable to accept criticism that he believed that anyone who did not
follow him was his enemy. Trotsky, one of Lenin's fellow revolutionaries, compared Lenin in 1904 to the French revolutionary Maximilien Robespierre.
Etymology of Bolshevik and Menshevik
The two factions of the Russian Social Democratic Labour Party (RSDLP) were originally known as hard (Lenin supporters) and soft (Martov supporters). In the 2nd Congress vote, Lenin's faction won votes on the majority of important issues, and soon came to be known as Bolsheviks, from the Russian bolshinstvo, 'majority'. Likewise, Martov's group came to be known as Mensheviks, from menshinstvo, 'minority'. However, Martov's supporters won the vote concerning the question of
party membership, and neither Lenin nor Martov had a firm majority
throughout the Congress as delegates left or switched sides. In the end,
the Congress was evenly split between the two factions.
Starting in 1907, English-language articles sometimes used the term Maximalist for "Bolshevik" and Minimalist for "Menshevik", which proved to be confusing as there was also a "Maximalist" faction within the Russian Socialist Revolutionary Party in 1904–1906 (which, after 1906, formed a separate Union of Socialists-Revolutionaries Maximalists) and then again after 1917.
The average party member was very young: in 1907, 22% of Bolsheviks
were under 20 years of age; 37% were 20–24 years of age; and 16% were
25–29 years of age. By 1905, 62% of the members were industrial workers (3% of the population in 1897). Twenty-two percent of Bolsheviks were gentry (1.7% of the total population) and 38% were uprooted peasants; compared with 19% and 26% for the Mensheviks. In 1907, 78% of the Bolsheviks were Russian and 10% were Jewish;
compared to 34% and 20% for the Mensheviks. Total Bolshevik membership
was 8,400 in 1905, 13,000 in 1906, and 46,100 by 1907; compared to
8,400, 18,000 and 38,200 for the Mensheviks. By 1910, both factions
together had fewer than 100,000 members.
Beginning of the 1905 Revolution (1903–05)
Between 1903 and 1904, the two factions were in a state of flux, with
many members changing sides. Plekhanov, the founder of Russian Marxism,
who at first allied himself with Lenin and the Bolsheviks, had parted
ways with them by 1904. Trotsky at first supported the Mensheviks, but
left them in September 1904 over their insistence on an alliance with
Russian liberals and their opposition to a reconciliation with Lenin and
the Bolsheviks. He remained a self-described "non-factional social democrat" until August 1917, when he joined Lenin and the Bolsheviks, as their positions resembled
his and he came to believe that Lenin was correct on the issue of the
party.
All but one member of the RSDLP Central Committee were arrested
in Moscow in early 1905. The remaining member, with the power of
appointing a new committee, was won over by the Bolsheviks. The lines between the Bolsheviks and the Mensheviks hardened in April
1905 when the Bolsheviks held a Bolsheviks-only meeting in London, which
they called the 3rd Party Congress. The Mensheviks organised a rival conference and the split was thus finalized.
The Bolsheviks played a relatively minor role in the 1905 Revolution and were a minority in the Saint Petersburg Soviet of Workers' Deputies led by Trotsky. However, the less significant Moscow Soviet was dominated by the Bolsheviks. These Soviets became the model for those formed in 1917.
Mensheviks (1906–07)
As the Russian Revolution of 1905 progressed, Bolsheviks, Mensheviks,
and smaller non-Russian social democratic parties operating within the Russian Empire attempted to reunify at the 4th Congress of the RSDLP held in April 1906 at Folkets hus, Norra Bantorget, in Stockholm. When the Mensheviks made an alliance with the Jewish Bund, the Bolsheviks found themselves in a minority.
However, all factions retained their respective factional structure and the Bolsheviks formed the Bolshevik Centre, the de facto governing body of the Bolshevik faction within the RSDLP. At the 5th Congress
held in London in May 1907, the Bolsheviks were in the majority, but
the two factions continued functioning mostly independently of each
other.
Split between Lenin and Bogdanov (1908–10)
Tensions had existed between Lenin and Alexander Bogdanov from as early as 1904. Lenin had fallen out with Nikolai Valentinov after Valentinov had introduced him to Ernst Mach's Empiriocriticism, a viewpoint that Bogdanov had been exploring and developing as Empiriomonism. Having worked as co-editor with Plekhanov, on Zarya, Lenin had come to agree with the Valentinov's rejection of Bogdanov's Empiriomonism.
With the defeat of the revolution in mid-1907 and the adoption of
a new, highly restrictive election law, the Bolsheviks began debating
whether to boycott the new parliament known as the Third Duma. Lenin, Grigory Zinoviev, Lev Kamenev, and others argued for participating in the Duma while Bogdanov, Anatoly Lunacharsky, Mikhail Pokrovsky, and others argued that the social democratic faction in the Duma should be recalled. The latter became known as "recallists" (Russian: otzovists).
A smaller group within the Bolshevik faction demanded that the RSDLP
Central Committee should give its sometimes unruly Duma faction an
ultimatum, demanding complete subordination to all party decisions. This
group became known as "ultimatists" and was generally allied with the recallists.
With most Bolshevik leaders either supporting Bogdanov or
undecided by mid-1908 when the differences became irreconcilable, Lenin
concentrated on undermining Bogdanov's reputation as a philosopher. In
1909, he published a scathing book of criticism entitled Materialism and Empirio-criticism (1909), assaulting Bogdanov's position and accusing him of philosophical idealism. In June 1909, Bogdanov proposed the formation of Party Schools as
Proletarian Universities at a Bolshevik mini-conference in Paris
organised by the editorial board of the Bolshevik magazine Proletary. However, this proposal was not adopted and Lenin tried to expel Bogdanov from the Bolshevik faction. Bogdanov was then involved with setting up Vpered, which ran the Capri Party School from August to December 1909.
Final attempt at party unity (1910)
With both Bolsheviks and Mensheviks weakened by splits within their
ranks and by Tsarist repression, the two factions were tempted to try to
reunite the party. In January 1910, Leninists, recallists, and various
Menshevik factions held a meeting of the party's Central Committee in
Paris. Kamenev and Zinoviev were dubious about the idea; but under
pressure from conciliatory Bolsheviks like Victor Nogin, they were willing to give it a try.
One of the underlying reasons that prevented any reunification of
the party was the Russian police. The police were able to infiltrate
both parties' inner circles by sending in spies who then reported on the
opposing party's intentions and hostilities. This allowed the tensions to remain high between the Bolsheviks and Mensheviks and helped prevent their uniting.
Lenin was firmly opposed to any reunification but was outvoted
within the Bolshevik leadership. The meeting reached a tentative
agreement, and one of its provisions was to make Trotsky's Vienna-based Pravda,
a party-financed central organ. Kamenev, Trotsky's brother-in-law who
was with the Bolsheviks, was added to the editorial board; but the
unification attempts failed in August 1910 when Kamenev resigned from
the board amid mutual recriminations.
The factions permanently broke relations in January 1912 after the Bolsheviks organised a Bolsheviks-only Prague Party Conference
and formally expelled Mensheviks and recallists from the party. As a
result, they ceased to be a faction in the RSDLP and instead declared
themselves an independent party, called Russian Social Democratic Labour Party (Bolsheviks)
– or RSDLP(b). Unofficially, the party has been referred to as the
Bolshevik Party. Throughout the 20th century, the party adopted a number
of different names. In 1918, RSDLP(b) became All-Russian Communist
Party (Bolsheviks) and remained so until 1925. From 1925 to 1952, the
name was All-Union Communist Party (Bolsheviks) and from 1952 to 1991,
the Communist Party of the Soviet Union.
As the party split became permanent, further divisions became
evident. One of the most notable differences was how each faction
decided to fund its revolution. The Mensheviks decided to fund their
revolution through membership dues while Lenin often resorted to more
drastic measures since he required a higher budget. One of the common methods the Bolsheviks used was committing bank
robberies, one of which, in 1907, resulted in the party getting over
250,000 roubles, which is the equivalent of about $125,000. Bolsheviks were in constant need of money because Lenin practised his
beliefs, expressed in his writings, that revolutions must be led by
individuals who devote their entire lives to the cause. As compensation,
he rewarded them with salaries for their sacrifice and dedication. This
measure was taken to help ensure that the revolutionaries stayed
focused on their duties and motivated them to perform their jobs. Lenin
also used the party money to print and copy pamphlets which were
distributed in cities and at political rallies in an attempt to expand
their operations. Both factions received funds through donations from
wealthy supporters.
The elections to the Russian Constituent Assembly
took place in November 1917 in which the Bolsheviks came second with
23.9% of the vote and dissolved the Assembly in January 1918
Further differences in party agendas became evident as the beginning of World War I loomed near. Joseph Stalin was especially eager for the start of the war, hoping that it would turn into a war between classes or essentially a Russian Civil War. This desire for war was fuelled by Lenin's vision that the workers and
peasants would resist joining the war effort and therefore be more
compelled to join the socialist movement. Through the increase in
support, Russia would then be forced to withdraw from the Allied powers
in order to resolve her internal conflict. Unfortunately for the
Bolsheviks, Lenin's assumptions were incorrect. Despite his and the
party's attempts to push for a civil war through involvement in two
conferences in 1915 and 1916 in Switzerland, the Bolsheviks were in the
minority in calling for a ceasefire by the Imperial Russian Army in World War I.
Although the Bolshevik leadership had decided to form a separate
party, convincing pro-Bolshevik workers within Russia to follow suit
proved difficult. When the first meeting of the Fourth Duma was convened
in late 1912, only one out of six Bolshevik deputies, Matvei Muranov (another one, Roman Malinovsky, was later exposed as an Okhrana agent), voted on 15 December 1912 to break from the Menshevik faction within the Duma. The Bolshevik leadership eventually prevailed, and the Bolsheviks formed their own Duma faction in September 1913.
One final difference between the Bolsheviks and Mensheviks was
how ferocious and tenacious the Bolshevik party was in order to achieve
its goals, although Lenin was open minded to retreating from political
ideals if he saw the guarantee of long-term gains benefiting the party.
This practice was seen in the party's trying to recruit peasants and
uneducated workers by promising them how glorious life would be after
the revolution and granting them temporary concessions.
Bolshevik figures such as Anatoly Lunacharsky, Moisei Uritsky and Dmitry Manuilsky
considered that Lenin's influence on the Bolshevik party was decisive
but the October insurrection was carried out according to Trotsky's, not
to Lenin's plan.
In 1918, the party renamed itself the Russian Communist Party
(Bolsheviks) at Lenin's suggestion. In 1925, this was changed to
All-Union Communist Party (Bolsheviks). At the 19th Party Congress in 1952 the Party was renamed the Communist Party of the Soviet Union at Stalin's suggestion.
Non-Russian / Soviet political groups having used the name "Bolshevik"
During the Cold War in the United Kingdom, trade union leaders and other leftists were sometimes derisively described as Bolshies. The usage is roughly equivalent to the term "commie", "Red", or "pinko" in the United States during the same period. The term Bolshie later became a slang term for anyone who was rebellious, aggressive, or truculent.
According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves (sometimes called "the Russell set"). If R
is not a member of itself, then its definition entails that it is a
member of itself; yet, if it is a member of itself, then it is not a
member of itself, since it is the set of all sets that are not members
of themselves. The resulting contradiction is Russell's paradox. In
symbols:
Let Then
Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by the German philosopher and mathematician Gottlob Frege, hence undermining Frege's attempt to reduce mathematics to logic and calling into question the logicist programme. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory and the Zermelo set theory. In particular, Zermelo's axioms restricted the unlimited comprehension principle. With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice).
The main difference between Russell's and Zermelo's solution to the
paradox is that Zermelo modified the axioms of set theory while
maintaining a standard logical language, while Russell modified the
logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be that of first-order logic.
The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor
– considered the founder of modern set theory – had already realized
that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.
Informal presentation
Most sets commonly encountered are not members of themselves. Call a
set "normal" if it is not a member of itself, and "abnormal" if it is a
member of itself. Clearly every set must be either normal or abnormal.
For example, consider the set of all squares in a plane.
This set is not itself a square in the plane, thus it is not a member
of itself and is therefore normal. In contrast, the complementary set
that contains everything which is not a square in the plane is itself not a square in the plane, and so it is one of its own members and is therefore abnormal.
Consider the set of all normal sets, R, and try to determine whether R is normal or abnormal. If R were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if R
were abnormal, it would not be contained in the set of all normal sets
(itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox.
a contradiction. Therefore, this naive set theory is inconsistent.
Philosophical implications
Prior to Russell's paradox (and to other similar paradoxes discovered around the time, such as the Burali-Forti paradox), a common conception of the idea of set was the "extensional concept of set", as recounted by von Neumann and Morgenstern:
A set is an arbitrary collection of
objects, absolutely no restriction being placed on the nature and
number of these objects, the elements of the set in question. The
elements constitute and determine the set as such, without any ordering
or relationship of any kind between them.
In particular, there was no distinction between sets and proper
classes as collections of objects. Additionally, the existence of each
of the elements of a collection was seen as sufficient for the existence
of the set of said elements. However, paradoxes such as Russell's and
Burali-Forti's showed the impossibility of this conception of a set, by
examples of collections of objects that do not form sets, despite all
said objects being existent.
Set-theoretic responses
From the principle of explosion of classical logic, any proposition can be proved from a contradiction.
Therefore, the presence of contradictions like Russell's paradox in an
axiomatic set theory is disastrous; since if any formula can be proved
true it destroys the conventional meaning of truth and falsity.
Further, since set theory was seen as the basis for an axiomatic
development of all other branches of mathematics, Russell's paradox
threatened the foundations of mathematics as a whole. This motivated a
great deal of research around the turn of the 20th century to develop a
consistent (contradiction-free) set theory.
In 1908, Ernst Zermelo proposed an axiomatization
of set theory that avoided the paradoxes of naive set theory by
replacing arbitrary set comprehension with weaker existence axioms, such
as his axiom of separation (Aussonderung). (Avoiding paradox was not Zermelo's original intention, but instead to document which assumptions he used in proving the well-ordering theorem.) Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo's axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the modern day.
ZFC does not assume that, for every property, there is a set of
all things satisfying that property. Rather, it asserts that given any
set X, any subset of X definable using first-order logic exists. The object R defined by Russell's paradox above cannot be constructed as a subset of any set X, and is therefore not a set in ZFC. In some extensions of ZFC, like von Neumann–Bernays–Gödel set theory, objects like R are called proper classes.
ZFC is silent about types, although the cumulative hierarchy
has a notion of layers that resemble types. Zermelo himself never
accepted Skolem's formulation of ZFC using the language of first-order
logic. As José Ferreirós notes, Zermelo insisted instead that
"propositional functions (conditions or predicates) used for separating
off subsets, as well as the replacement functions, can be 'entirely arbitrary' [ganz beliebig]"; the modern interpretation given to this statement is that Zermelo wanted to include higher-order quantification in order to avoid Skolem's paradox. Around 1930, Zermelo also introduced (apparently independently of von Neumann), the axiom of foundation,
thus—as Ferreirós observes—"by forbidding 'circular' and 'ungrounded'
sets, it [ZFC] incorporated one of the crucial motivations of TT [type
theory]—the principle of the types of arguments". This 2nd order ZFC
preferred by Zermelo, including axiom of foundation, allowed a rich
cumulative hierarchy. Ferreirós writes that "Zermelo's 'layers' are
essentially the same as the types in the contemporary versions of simple
TT [type theory] offered by Gödel and Tarski. The cumulative hierarchy
into which Zermelo developed his models can be described as the universe
of a cumulative TT in which transfinite types are allowed. (Once an
impredicative standpoint is adopted, abandoning the idea that classes
are constructed, it is natural to accept transfinite types.) Thus,
simple TT and ZFC could be regarded as systems that 'talk' essentially
about the same intended objects. The main difference is that TT relies
on a strong higher-order logic, while Zermelo employed second-order
logic, and ZFC can also be given a first-order formulation. The
first-order 'description' of the cumulative hierarchy is much weaker, as
is shown by the existence of countable models (Skolem's paradox), but
it enjoys some important advantages."
In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.
Through the work of Zermelo and others, especially John von Neumann,
the structure of what some see as the "natural" objects described by
ZFC eventually became clear: they are the elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set
operation. It is thus possible again to reason about sets in a
non-axiomatic fashion without running afoul of Russell's paradox, namely
by reasoning about the elements of V. Whether it is appropriate to think of sets in this way is a point of contention among the rival points of view on the philosophy of mathematics.
Russell discovered the paradox in May or June 1901. By his own account in his 1919 Introduction to Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal". In a 1902 letter, he announced the discovery to Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function:
There is just one point where I
have encountered a difficulty. You state (p. 17 [p. 23 above]) that a
function too, can act as the indeterminate element. This I formerly
believed, but now this view seems doubtful to me because of the
following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w
is not a predicate. Likewise there is no class (as a totality) of those
classes which, each taken as a totality, do not belong to themselves.
From this I conclude that under certain circumstances a definable
collection [Menge] does not form a totality.
Russell would go on to cover it at length in his 1903 The Principles of Mathematics, where he repeated his first encounter with the paradox:
Before taking leave of fundamental
questions, it is necessary to examine more in detail the singular
contradiction, already mentioned, with regard to predicates not
predicable of themselves. ... I may mention that I was led to it in the
endeavour to reconcile Cantor's proof....
Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der Arithmetik. Frege responded to Russell very quickly; his letter dated 22 June 1902
appeared, with van Heijenoort's commentary in Heijenoort 1967:126–127.
Frege then wrote an appendix admitting to the paradox, and proposed a solution that Russell would endorse in his Principles of Mathematics, but was later considered by some to be unsatisfactory. For his part, Russell had his work at the printers and he added an appendix on the doctrine of types.
Ernst Zermelo in his (1908) A new proof of the possibility of a well-ordering (published at the same time he published "the first axiomatic set theory") laid claim to prior discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell9
gave to the set-theoretic antinomies could have persuaded them [J.
König, Jourdain, F. Bernstein] that the solution of these difficulties
is not to be sought in the surrender of well-ordering but only in a
suitable restriction of the notion of set". Footnote 9 is where he stakes his claim:
91903, pp.
366–368. I had, however, discovered this antinomy myself, independently
of Russell, and had communicated it prior to 1903 to Professor Hilbert
among others.
Frege sent a copy of his Grundgesetze der Arithmetik to
Hilbert; as noted above, Frege's last volume mentioned the paradox that
Russell had communicated to Frege. After receiving Frege's last volume,
on 7 November 1903, Hilbert wrote a letter to Frege in which he said,
referring to Russell's paradox, "I believe Dr. Zermelo discovered it
three or four years ago". A written account of Zermelo's actual
argument was discovered in the Nachlass of Edmund Husserl.
In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows:
The reason why a function cannot be its own argument is that the sign
for a function already contains the prototype of its argument, and it
cannot contain itself. For let us suppose that the function F(fx) could
be its own argument: in that case there would be a proposition F(F(fx)), in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(fx) and the outer one has the form Y(O(fx)).
Only the letter 'F' is common to the two functions, but the letter by
itself signifies nothing. This immediately becomes clear if instead of F(Fu) we write (do) : F(Ou) . Ou = Fu. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)
Russell and Alfred North Whitehead wrote their three-volume Principia Mathematica hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of naive set theory
by employing a theory of types they devised for this purpose. While
they succeeded in grounding arithmetic in a fashion, it is not at all
evident that they did so by purely logical means. While Principia Mathematica avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.
In any case, Kurt Gödel in 1930–31 proved that while the logic of much of Principia Mathematica, later known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent. This is very widely—though not universally—regarded as having shown the logicist program of Frege to be impossible to complete.
In 2001, A Centenary International Conference celebrating the
first hundred years of Russell's paradox was held in Munich and its
proceedings have been published.
Applied versions
Some versions of this paradox are closer to real-life and may be easier to understand for non-logicians. For example, the barber paradox
supposes a male barber who shaves all men who do not shave themselves
and only men who do not shave themselves. When one thinks about whether
the barber shaves himself or not, a similar paradox begins to emerge.
An exception may be the Grelling–Nelson paradox,
in which words and meaning are the elements of the scenario rather than
people and hair-cutting. Though it is easy to refute the barber's
paradox by saying that such a barber does not (and cannot) exist, it is impossible to say something similar about a meaningfully defined word.
One way that the paradox has been dramatised is: suppose that
every public library has to compile a catalogue of all its books. Since
the catalogue is itself one of the library's books, some librarians
include it in the catalogue for completeness; while others leave it out
as it being one of the library's books is self evident. Next, imagine
that all these catalogues are sent to the national library. Some of them
include themselves in their listings, others do not. The national
librarian compiles two master catalogues—one of all the catalogues that
list themselves, and one of all those that do not.
The question is: should these master catalogues list themselves?
The 'catalogue of all catalogues that list themselves' is no problem. If
the librarian does not include it in its own listing, it remains a true
catalogue of those catalogues that do include themselves. If he does
include it, it remains a true catalogue of those that list themselves.
However, just as the librarian cannot go wrong with the first master
catalogue, he is doomed to fail with the second. When it comes to the
'catalogue of all catalogues that do not list themselves', the librarian
cannot include it in its own listing, because then it would include
itself, and so belong in the other catalogue, that of catalogues that do
include themselves. However, if the librarian leaves it out, the
catalogue is incomplete. Either way, it can never be a true master
catalogue of catalogues that do not list themselves.
Applications and related topics
Russell-like paradoxes
As illustrated above for the barber paradox, Russell's paradox is not hard to extend. Take:
The original Russell's paradox with "contain": The container (Set)
that contains all (containers) that do not contain themselves.
The Grelling–Nelson paradox with "describer": The describer (word) that describes all words, that do not describe themselves.
Richard's paradox
with "denote": The denoter (number) that denotes all denoters (numbers)
that do not denote themselves. (In this paradox, all descriptions of
numbers get an assigned number. The term "that denotes all denoters
(numbers) that do not denote themselves" is here called Richardian.)
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