Openness to experience

From Wikipedia, the free encyclopedia

Openness to experience is one of the domains which are used to describe human personality in the Five Factor Model. Openness involves six facets, or dimensions: active imagination (fantasy), aesthetic sensitivity, attentiveness to inner feelings, preference for variety (adventurousness), intellectual curiosity, and challenging authority (psychological liberalism). A great deal of psychometric research has demonstrated that these facets or qualities are significantly correlated. Thus, openness can be viewed as a global personality trait consisting of a set of specific traits, habits, and tendencies that cluster together.

Openness tends to be normally distributed with a small number of individuals scoring extremely high or low on the trait, and most people scoring moderately. People who score low on openness are considered to be closed to experience. They tend to be conventional and traditional in their outlook and behavior. They prefer familiar routines to new experiences, and generally have a narrower range of interests. Openness has moderate positive relationships with creativity, intelligence and knowledge. Openness is related to the psychological trait of absorption, and like absorption has a modest relationship to individual differences in hypnotic susceptibility.

Openness has more modest relationships with aspects of subjective well-being than other Five Factor Model personality traits. On the whole, openness appears to be largely unrelated to symptoms of mental disorders.

Measurement

Openness to experience is usually assessed with self-report measures, although peer-reports and third-party observation are also used. Self-report measures are either lexical or based on statements. Which measure of either type is used is determined by an assessment of psychometric properties and the time and space constraints of the research being undertaken.

• Lexical measures use individual adjectives that reflect openness to experience traits, such as creative, intellectual, artistic, philosophical, deep. Goldberg (1992) developed a 20-word measure as part of his 100-word Big Five markers. Saucier (1994) developed a briefer 8-word measure as part of his 40-word mini-markers. However, the psychometric properties of Saucier’s original mini-markers have been found suboptimal with samples outside of North America. As a result, a systematically revised measure, the International English Mini-Markers, was developed and has proven good psychometric validity for assessing openness to experience and other five factor personality model dimensions, both within and, especially, without American populations. Internal consistency reliability of the openness to experience measure is .84 for both native and non-native English-speakers.
• Statement measures tend to comprise more words, and hence take up more research instrument space, than lexical measures. For example, the openness (intellect) scale of Goldberg's International Personality Item Pool is 45 words compared Saucier or Thompson’s (2008) 8-word lexical scale for Openness. Examples of statement measure items used are the NEO PI-R, based on the Five Factor Model, and the HEXACO-PI-R based on the HEXACO model of personality are "Love to think up new ways of doing things" and "Have difficulty understanding abstract ideas". In these tests, openness to experience is one of the five/six measured personality dimensions. In both tests openness to experience has a number of facets. The NEO PI-R assesses six facets called openness to ideas, feelings, values, fantasy, aesthetics, and actions respectively. The HEXACO-PI-R assesses four facets called inquisitiveness, creativity, aesthetic appreciation, and unconventionality.

A number of studies have found that openness to experience has two major subcomponents, one related to intellectual dispositions, the other related to the experiential aspects of openness, such as aesthetic appreciation and openness to sensory experiences. These subcomponents have been referred to as intellect and experiencing openness respectively, and have a strong positive correlation (r = .55) with each other.

According to research by Sam Gosling, it is possible to assess openness by examining people's homes and work spaces. Individuals who are highly open to experience tend to have distinctive and unconventional decorations. They are also likely to have books on a wide variety of topics, a diverse music collection, and works of art on display.

Psychological aspects

Openness to experience has both motivational and structural components. People high in openness are motivated to seek new experiences and to engage in self-examination. Structurally, they have a fluid style of consciousness that allows them to make novel associations between remotely connected ideas. Closed people by contrast are more comfortable with familiar and traditional experiences.

Creativity

Openness to experience correlates with creativity, as measured by tests of divergent thinking. Openness has been linked to both artistic and scientific creativity as professional artists, musicians, and scientists have been found to score higher in openness compared to members of the general population.

Intelligence and knowledge

Openness to experience correlates with intelligence, correlation coefficients ranging from about r = .30 to r = .45. Openness to experience is moderately associated with crystallized intelligence, but only weakly with fluid intelligence. A study examining the facets of openness found that the Ideas and Actions facets had modest positive correlations with fluid intelligence (r=.20 and r=.07 respectively). These mental abilities may come more easily when people are dispositionally curious and open to learning. Several studies have found positive associations between openness to experience and general knowledge. People high in openness may be more motivated to engage in intellectual pursuits that increase their knowledge. Openness to experience, especially the Ideas facet, is related to need for cognition, a motivational tendency to think about ideas, scrutinize information, and enjoy solving puzzles, and to typical intellectual engagement (a similar construct to need for cognition).

Absorption and hypnotisability

Openness to experience is strongly related to the psychological construct of absorption defined as "a disposition for having episodes of 'total' attention that fully engage one's representational (i.e. perceptual, enactive, imaginative, and ideational) resources.” The construct of absorption was developed in order to relate individual differences in hypnotisability to broader aspects of personality. The construct of absorption influenced Costa and McCrae's development of the concept of openness to experience in their original NEO model due to the independence of absorption from extraversion and neuroticism. A person's openness to becoming absorbed in experiences seems to require a more general openness to new and unusual experiences. Openness to experience, like absorption has modest positive correlations with individual differences in hypnotisability. Factor analysis has shown that the fantasy, aesthetics, and feelings facets of openness are closely related to absorption and predict hypnotisability, whereas the remaining three facets of ideas, actions, and values are largely unrelated to these constructs. This finding suggests that openness to experience may have two distinct yet related subdimensions: one related to aspects of attention and consciousness assessed by the facets of fantasy, aesthetics, and feelings; the other related to intellectual curiosity and social/political liberalism as assessed by the remaining three facets. However, all of these have a common theme of ‘openness’ in some sense. This two-dimensional view of openness to experience is particularly pertinent to hypnotisability. However, when considering external criteria other than hypnotisability, it is possible that a different dimensional structure may be apparent, e.g. intellectual curiosity may be unrelated to social/political liberalism in certain contexts.

Relationship to other personality traits

Although the factors in the Big Five model are assumed to be independent, openness to experience and extraversion as assessed in the NEO-PI-R have a substantial positive correlation. Openness to experience also has a moderate positive correlation with sensation-seeking, particularly, the experience seeking facet. In spite of this, it has been argued that openness to experience is still an independent personality dimension from these other traits because most of the variance in the trait cannot be explained by its overlap with these other constructs. A study comparing the Temperament and Character Inventory with the Five Factor model found that Openness to experience had a substantial positive correlation with self-transcendence (a "spiritual" trait) and to a lesser extent novelty seeking (conceptually similar to sensation seeking). It also had a moderate negative correlation with harm avoidance. The Myers–Briggs Type Indicator (MBTI) measures the preference of "intuition," which is related to openness to experience. Robert McCrae pointed out that the MBTI sensation versus intuition scale "contrasts a preference for the factual, simple and conventional with a preference for the possible, complex, and original," and is therefore similar to measures of openness.

Social and political attitudes

There are social and political implications to this personality trait. People who are highly open to experience tend to be liberal and tolerant of diversity. As a consequence, they are generally more open to different cultures and lifestyles. They are lower in ethnocentrism, right-wing authoritarianism, social dominance orientation, and prejudice. Openness has a stronger (negative) relationship with right-wing authoritarianism than the other five-factor model traits (conscientiousness has a modest positive association, and the other traits have negligible associations). Openness has a somewhat smaller (negative) association with social dominance orientation than (low) agreeableness (the other traits have negligible associations). Openness has a stronger (negative) relationship with prejudice than the other five-factor model traits (agreeableness has a more modest negative association, and the other traits have negligible associations). However, right-wing authoritarianism and social dominance orientation are each more strongly (positively) associated with prejudice than openness or any of the other five-factor model traits. Recent research has argued that the relationship between openness and prejudice may be more complex, as the prejudice examined was prejudice against conventional minority groups (for example sexual and ethnic minorities) and that people who are high in openness can still be intolerant of those with conflicting worldviews.

Regarding conservatism, studies have found that cultural conservatism was related to low openness and all its facets, but economic conservatism was unrelated to total openness, and only weakly negatively related to the Aesthetics and values facets. The strongest personality predictor of economic conservatism was low agreeableness (r= -.23). Economic conservatism is based more on ideology whereas cultural conservatism seems to be more psychological than ideological and may reflect a preference for simple, stable and familiar mores. Some research indicates that within-person changes in levels of openness do not predict changes in conservatism.

Subjective well-being and mental health

Openness to experience has been found to have modest yet significant associations with happiness, positive affect, and quality of life and to be unrelated to life satisfaction, negative affect, and overall affect in people in general. These relationships with aspects of subjective well-being tend to be weaker compared to those of other five-factor model traits, that is, extraversion, neuroticism, conscientiousness, and agreeableness. Openness to experience was found to be associated with life satisfaction in older adults after controlling for confounding factors. Openness appears to be generally unrelated to the presence of mental disorders. A meta-analysis of the relationships between five-factor model traits and symptoms of psychological disorders found that none of the diagnostic groups examined differed from healthy controls on openness to experience.

In addition, openness to experience may contribute to graceful aging, facilitating healthy memory and verbal abilities as well as a number of other significant cognitive features in older adults.

Personality disorders

Main article: Personality disorders

At least three aspects of openness are relevant to understanding personality disorders: cognitive distortions, lack of insight and impulsivity. Problems related to high openness that can cause issues with social or professional functioning are excessive fantasizing, peculiar thinking, diffuse identity, unstable goals and nonconformity with the demands of the society.

High openness is characteristic to schizotypal personality disorder (odd and fragmented thinking), narcissistic personality disorder (excessive self-valuation) and paranoid personality disorder (sensitivity to external hostility). Lack of insight (shows low openness) is characteristic to all personality disorders and could explain the persistence of maladaptive behavioral patterns.

The problems associated with low openness are difficulties adapting to change, low tolerance for different worldview or lifestyles, emotional flattening, alexithymia and a narrow range of interests. Rigidity is the most obvious aspect of (low) openness among personality disorders and that shows lack of knowledge of one's emotional experiences. It is most characteristic of obsessive-compulsive personality disorder. Its opposite, known as impulsivity (here: an aspect of openness that shows a tendency to behave unusually or autistically), is characteristic of schizotypal and borderline personality disorders.

Religiosity and spirituality

Openness to experience has mixed relationships with different types of religiosity and spirituality. General religiosity has a weak association with low openness. Religious fundamentalism has a somewhat more substantial relationship with low openness. Mystical experiences occasioned by the use of psilocybin were found to increase openness significantly (see 'Drug Use,' below).

Gender

A study examining gender differences in big five personality traits in 55 nations found that across nations there were negligible average differences between men and women in openness to experience. By contrast, across nations women were found to be significantly higher than men in average neuroticism, extraversion, agreeableness, and conscientiousness. In 8 cultures, men were significantly higher than women in openness, but in 4 cultures women were significantly higher than men. Previous research has found that women tend to be higher on the feelings facet of openness, whereas men tend to be higher on the ideas facet, although the 55 nation study did not assess individual facets.

Dream recall

A study on individual differences in the frequency of dream recall found that openness to experience was the only big five personality trait related to dream recall. Dream recall frequency has also been related to similar personality traits, such as absorption and dissociation. The relationship between dream recall and these traits has been considered as evidence of the continuity theory of consciousness. Specifically, people who have vivid and unusual experiences during the day, such as those who are high in these traits, tend to have more memorable dream content and hence better dream recall.

Sexuality

Openness is related to many aspects of sexuality. Men and women high in openness are more well-informed about sex, have wider sexual experience, stronger sex drives, and more liberal sexual attitudes. In married couples, wives' but not husbands' level of openness is related to sexual satisfaction. This might be because open wives are more willing to explore a variety of new sexual experiences, leading to greater satisfaction for both spouses. Compared to heterosexuals, people who are homosexual, asexual, or bisexual—particularly bisexuals—average higher in openness.

Genes and physiology

Openness to experience, like the other traits in the five factor model, is believed to have a genetic component. Identical twins (who have the same DNA) show similar scores on openness to experience, even when they have been adopted into different families and raised in very different environments. One genetic study with 86 subjects found Openness to experience related to the 5-HTTLPR polymorphism associated with the serotonin transporter gene. A meta-analysis by Bouchard and McGue of four twin studies found openness to be the most heritable (mean = 57%) of the Big Five traits.

Higher levels of openness have been linked to activity in the ascending dopaminergic system and the dorsolateral prefrontal cortex. Openness is the only personality trait that correlates with neuropsychological tests of dorsolateral prefrontal cortical function, supporting theoretical links among openness, cognitive functioning, and IQ.

Geography

An Italian study found that people who lived on Tyrrhenian islands tended to be less open to experience than those living on the nearby mainland, and that people whose ancestors had inhabited the islands for twenty generations tended to be less open to experience than more recent arrivals. Additionally, people who emigrated from the islands to the mainland tended to be more open to experience than people who stayed on the islands, and than those who immigrated to the islands.

People living in the eastern and western parts of the United States tend to score higher on openness to experience than those living in the Midwestern United States and the Southern United States. The highest average scores on openness are found in the states of New York, Oregon, Massachusetts, Washington, and California. Lowest average scores come from North Dakota, Wyoming, Alaska, Alabama, and Wisconsin.

Drug use

Psychologists in the early 1970s used the concept of openness to experience to describe people who are more likely to use marijuana. Openness was defined in these studies as high creativity, adventuresomeness, internal sensation novelty seeking, and low authoritarianism. Several correlational studies confirmed that young people who score high on this cluster of traits are more likely to use marijuana. More recent research has replicated this finding using contemporary measures of openness.

Cross-cultural studies have found that cultures high in Openness to experience have higher rates of use of the drug ecstasy, although a study at the individual level in the Netherlands found no differences in openness levels between users and non-users. Ecstasy users tended to be higher in extraversion and lower in conscientiousness than non-users.

A 2011 study found Openness (and not other traits) increased with the use of psilocybin, an effect that held even after 14 months. The study found that individual differences in levels of mystical experience while taking psilocybin were correlated with increases in Openness. Participants who met criteria for a 'complete mystical experience' experienced a significant mean increase in Openness, whereas those participants who did not meet the criteria experienced no mean change in Openness. Five of the six facets of Openness (all except Actions) showed this pattern of increase associated with having a mystical experience. Increases in Openness (including facets as well as total score) among those whose had a complete mystical experience were maintained more than a year after taking the drug. Participants who had a complete mystical experience changed more than 4 T-score points between baseline and follow up. By comparison, Openness has been found to normally decrease with ageing by 1 T-score point per decade.

Infinity

From Wikipedia, the free encyclopedia

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

Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them.

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol ${\displaystyle \mathrm{\infty }}$.

Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the Universe is spatially infinite is an open question.

History

Further information: Infinity (philosophy)

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".

Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving the infinitude of the prime numbers, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated:

If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.

Zeno: Achilles and the tortoise

Main article: Zeno's paradoxes § Achilles and the tortoise

Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".

Achilles races a tortoise, giving the latter a head start.

• Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
• Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
• Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
• Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.

Etc.

Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.

Zeno was not attempting to make a point about infinity. As a member of the Eleatics school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.

Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,

$${\displaystyle a+ax+a{x}^2+a{x}^3+a{x}^4+a{x}^5+\cdots =\frac{a}{1-x}.}$$

Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him

$${\displaystyle 10+0.1+0.001+0.00001+\cdots =\frac{10}{1-0.01}=\frac{10}{0.99}=10.10101\dots \text{ seconds}.}$$

Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

• Enumerable: lowest, intermediate, and highest
• Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
• Infinite: nearly infinite, truly infinite, infinitely infinite

17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation ${\displaystyle \mathrm{\infty }}$ for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ${\displaystyle \frac{1}{\mathrm{\infty }}.}$ But in Arithmetica infinitorum (1656), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."

In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.

Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:

Mathematics is the science of the infinite.

Symbol

Main article: Infinity symbol

The infinity symbol ${\displaystyle \mathrm{\infty }}$ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (&infin;) and in LaTeX as \infty.

It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.

Calculus

Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of continuity.

Real analysis

In real analysis, the symbol ${\displaystyle \mathrm{\infty }}$, called "infinity", is used to denote an unbounded limit. The notation ${\displaystyle x\to \mathrm{\infty }}$ means that ${\displaystyle x}$ increases without bound, and ${\displaystyle x\to -\mathrm{\infty }}$ means that ${\displaystyle x}$ decreases without bound. For example, if ${\displaystyle f(t)\ge 0}$ for every ${\displaystyle t}$, then

• ${\displaystyle {\int }_a^{b}f(t)dt=\mathrm{\infty }}$ means that ${\displaystyle f(t)}$ does not bound a finite area from ${\displaystyle a}$ to ${\displaystyle b.}$
• ${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=\mathrm{\infty }}$ means that the area under ${\displaystyle f(t)}$ is infinite.
• ${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=a}$ means that the total area under ${\displaystyle f(t)}$ is finite, and is equal to ${\displaystyle a.}$

Infinity can also be used to describe infinite series, as follows:

• ${\displaystyle \sum _{i=0}^{\mathrm{\infty }}f(i)=a}$ means that the sum of the infinite series converges to some real value ${\displaystyle a.}$
• ${\displaystyle \sum _{i=0}^{\mathrm{\infty }}f(i)=\mathrm{\infty }}$ means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.

In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled ${\displaystyle +\mathrm{\infty }}$ and ${\displaystyle -\mathrm{\infty }}$ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat ${\displaystyle +\mathrm{\infty }}$ and ${\displaystyle -\mathrm{\infty }}$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.

Complex analysis

By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.

In complex analysis the symbol ${\displaystyle \mathrm{\infty }}$, called "infinity", denotes an unsigned infinite limit. The expression ${\displaystyle x\to \mathrm{\infty }}$ means that the magnitude ${\displaystyle |x|}$ of ${\displaystyle x}$ grows beyond any assigned value. A point labeled ${\displaystyle \mathrm{\infty }}$ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely ${\displaystyle z/0=\mathrm{\infty }}$ for any nonzero complex number ${\displaystyle z}$. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of ${\displaystyle \mathrm{\infty }}$ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).

Nonstandard analysis

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).

Set theory

Main articles: Cardinality and Ordinal number

One-to-one correspondence between an infinite set and its proper subset

A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ₀), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.

Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Cardinality of the continuum

Main article: Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum ${\displaystyle \mathbf{c}}$ is greater than that of the natural numbers ${\displaystyle {\mathrm{\aleph }}_0}$; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that ${\displaystyle \mathbf{c}=2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0}$.

Further information: Cantor's diagonal argument and Cantor's first set theory article

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, ${\displaystyle \mathbf{c}={\mathrm{\aleph }}_1={\beth }_1}$.

Further information: Beth number § Beth one

This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R.

See also: Hilbert's paradox of the Grand Hotel

The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.

Geometry

Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).

One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.

Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).

In particular, in modern mathematics, lines are infinite sets.

Infinite dimension

The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.

In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.

Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.

Mathematics without infinity

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.

Physics

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.^[46] Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."

Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is still an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.

The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.

The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes. Also, cyclic models posit an infinite amount of Big Bangs, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.

Logic

In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."

Computing

The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.

Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.

In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.

In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely.

Arts, games, and cognitive sciences

Perspective artwork uses the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

Variations of chess played on an unbounded board are called infinite chess.

Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.