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In the
philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
, the
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstr ...
of actual
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
involves the acceptance (if the
axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...
is included) of infinite entities as given, actual and completed objects. These might include the set of
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
,
extended real numbers In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
,
transfinite number In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...
s, or even an infinite sequence of
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. As a result, potential infinity is often formalized using the concept of
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
.


Anaximander

The ancient Greek term for the potential or improper infinite was ''
apeiron ''Apeiron'' (; ) is a Greek word meaning "(that which is) unlimited," "boundless", "infinite", or "indefinite" from ''a-'', "without" and ''peirar'', "end, limit", "boundary", the Ionic Greek form of ''peras'', "end, limit, boundary". Origin ...
'' (unlimited or indefinite), in contrast to the actual or proper infinite ''aphorismenon''. ''Apeiron'' stands opposed to that which has a ''peras'' (limit). These notions are today denoted by ''potentially infinite'' and ''actually infinite'', respectively.
Anaximander Anaximander (; grc-gre, Ἀναξίμανδρος ''Anaximandros''; ) was a pre-Socratic Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newnes, 1961, Vol. 1, p. 403. a city of Ionia (in moder ...
(610–546 BC) held that the'' apeiron'' was the principle or main element composing all things. Clearly, the 'apeiron' was some sort of basic substance.
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's notion of the ''apeiron'' is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues ''Parmenides'' and the ''Philebus''.


Aristotle

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
sums up the views of his predecessors on infinity as follows:
"Only the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also." (Aristotle)
The theme was brought forward by Aristotle's consideration of the apeiron—in the context of mathematics and physics (the study of nature):
"Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'." (Aristotle)
Belief in the existence of the infinite comes mainly from five considerations: # From the nature of time – for it is infinite. # From the division of magnitudes – for the mathematicians also use the notion of the infinite. # If coming to be and passing away do not give out, it is only because that from which things come to be is infinite. # Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself. # Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody – not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought. (Aristotle) Aristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than the heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude.


Aristotle's potential–actual distinction

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
handled the topic of infinity in ''Physics'' and in ''Metaphysics''. He distinguished between ''actual'' and ''potential'' infinity. ''Actual infinity'' is completed and definite, and consists of infinitely many elements. ''Potential infinity'' is never complete: elements can be always added, but never infinitely many. Aristotle distinguished between infinity with respect to addition and division.
"As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed."
With respect to division, a potentially infinite sequence of divisions might start, for example, 1, 1/2, 1/4, 1/8, 1/16, but the process of division cannot be exhausted or completed. Aristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the ctualinfinite and do not use it" (''Phys.'' III 2079 29).


Scholastic, Renaissance and Enlightenment thinkers

The overwhelming majority of
scholastic philosophers Scholastic may refer to: * a philosopher or theologian in the tradition of scholasticism * ''Scholastic'' (Notre Dame publication) * Scholastic Corporation, an American publishing company of educational materials * Scholastic Building, in New Yor ...
adhered to the motto ''Infinitum actu non datur''. This means there is only a (developing, improper, "syncategorematic") ''potential infinity'' but not a (fixed, proper, "categorematic") ''actual infinity''. There were exceptions, however, for example in England.
It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle. ( G. Cantor)
Actual infinity exists in number, time and quantity. (J. Baconthorpe , p. 96
During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.
The continuum actually consists of infinitely many indivisibles ( G. Galilei , p. 97
I am so in favour of actual infinity. ( G.W. Leibniz , p. 97
However, the majority of pre-modern thinkers agreed with the well-known quote of Gauss:
I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. ( C.F. Gauss n a letter to Schumacher, 12 July 1831


Modern era

Actual infinity is now commonly accepted. The drastic change was initialized by Bolzano and Cantor in the 19th century.
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liber ...
, who introduced the notion of ''set'' (in German: ''Menge''), and Georg Cantor, who introduced
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, opposed the general attitude. Cantor distinguished three realms of infinity: (1) the infinity of God (which he called the "absolutum"), (2) the infinity of reality (which he called "nature") and (3) the transfinite numbers and sets of mathematics.
A multitude which is larger than any finite multitude, i.e., a multitude with the property that every finite set f members of the kind in questionis only a part of it, I will call an infinite multitude. (B. Bolzano , p. 6
Accordingly I distinguish an eternal uncreated infinity or absolutum, which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. (Georg Cantor) (G. Cantor , p. 252
The numbers are a free creation of human mind. ( R. Dedekind a, p. III
One proof is based on the notion of God. First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened. (G. Cantor , p. 400
Cantor distinguished two types of actual infinity; the transfinite and the absolute, about which he affirmed:
These concepts are to be strictly differentiated, insofar the former is, to be sure, ''infinite'', yet capable of ''increase'', whereas the latter is ''incapable of increase'' and is therefore ''indeterminable'' as a mathematical concept. This mistake we find, for example, in ''
Pantheism Pantheism is the belief that reality, the universe and the cosmos are identical with divinity and a supreme supernatural being or entity, pointing to the universe as being an immanent creator deity still expanding and creating, which has ex ...
''. (G. Cantor, ''Über verschiedene Standpunkte in bezug auf das aktuelle Unendliche'', in ''Gesammelte Abhandlungen mathematischen und philosophischen Inhalts'', pp. 375, 378)


Current mathematical practice

Actual infinity is now commonly accepted, because mathematicians have learned how to construct algebraic statements using it. For example, one may write down a symbol, \omega, with the verbal description that "\omega stands for completed (
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
) infinity". This symbol may be added as an
ur-element In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory There ...
to any set. One may also provide
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s that define addition, multiplication and inequality; specifically,
ordinal arithmetic In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
, such that expressions like n<\omega can be interpreted as "any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
is less than completed infinity". Even "common sense" statements such as \omega < \omega+1 are possible and consistent. The theory is sufficiently well developed, that rather complex algebraic expressions, such as \omega^2, \omega^\omega and even 2^\omega can be interpreted as valid algebraic expressions, can be given a verbal description, and can be used in a wide variety of theorems and claims in a consistent and meaningful fashion. The ability to define
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s in a consistent, meaningful way, renders much of the debate moot; whatever personal opinion one may hold about infinity or constructability, the existence of a rich theory for working with infinities using the tools of algebra and logic is clearly in hand.


Opposition from the Intuitionist school

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential,Kleene 1952/1971:48. but not to be mistaken for ''physically existing''. The question of whether
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
or
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
form definite sets is therefore independent of the question of whether infinite things exist physically in
nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...
. Proponents of
intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand,
constructive analysis In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more comm ...
does accept the existence of the completed infinity of the integers. For intuitionists, infinity is described as ''potential''; terms synonymous with this notion are ''becoming'' or ''constructive''. For example,
Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
describes the notion of a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
tape as "a linear 'tape', (potentially) infinite in both directions." To access memory on the tape, a Turing machine moves a ''read head'' along it in finitely many steps: the tape is therefore only "potentially" infinite, since while there is always the ability to take another step, infinity itself is never actually reached. Mathematicians generally accept actual infinities.Actual infinity follows from, for example, the acceptance of the notion of the integers as a set, see J J O'Connor and E F Robertson
"Infinity"
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
is the most significant mathematician who defended actual infinities, equating the
Absolute Infinite The Absolute Infinite (''symbol'': Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfin ...
with God. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. The present-day conventional finitist interpretation of ordinal and
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s is that they consist of a collection of special symbols, and an associated
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symb ...
, within which statements may be made. All such statements are necessarily finite in length. The soundness of the manipulations is founded only on the basic principles of a formal language:
term algebra In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set ''X'' of variables is exa ...
s,
term rewriting In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...
, and so on. More abstractly, both (finite)
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
and
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...
offer the needed tools to work with infinities. One does not have to "believe" in infinity in order write down algebraically valid expressions employing symbols for infinity.


Classical set theory

The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound. Classical set theory accepts the notion of actual, completed infinities. However, some
finitist Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are ac ...
philosophers of mathematics and constructivists object to the notion.
If the positive number ''n'' becomes infinitely great, the expression 1/''n'' goes to naught (or gets infinitely small). In this sense one speaks of the improper or potential infinite. In sharp and clear contrast the set just considered is a readily finished, locked infinite set, fixed in itself, containing infinitely many exactly defined elements (the natural numbers) none more and none less. ( A. Fraenkel , p. 6
Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics. (A. Fraenkel , p. 245
To look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i.e., we are able to "produce" bigger and bigger sets. (A. Fraenkel et al. , p. 118
(
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...
) maintains that a veritable continuum which is not denumerable can be obtained as a medium of free development; that is to say, besides the points which exist (are ready) on account of their definition by laws, such as e, pi, etc. other points of the continuum are not ready but develop as so-called choice sequences. (A. Fraenkel et al. , p. 255
Intuitionists reject the very notion of an arbitrary sequence of integers, as denoting something finished and definite as illegitimate. Such a sequence is considered to be a growing object only and not a finished one. (A. Fraenkel et al. , p. 236
Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for "actual infinity." The arguments using infinity, including the Differential
Calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
of
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
, do not require the use of infinite sets. (T. Jec

Owing to the gigantic simultaneous efforts of Gottlob Frege, Frege,
Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
and Cantor, the infinite was set on a throne and revelled in its total triumph. In its daring flight the infinite reached dizzying heights of success. ( D. Hilbert , p. 169
One of the most vigorous and fruitful branches of mathematics ..a paradise created by Cantor from which nobody shall ever expel us ..the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity. (D. Hilbert on set theory
Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking. (D. Hilbert , 190
Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. ( A. Robinson 0, p. 507
Indeed, I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world. (A. Robinson)
Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory. ( Y. Maninbr>
Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities. (Y. Mani

There is no actual infinity, that the Cantorians have forgotten and have been trapped by contradictions. ( Henri Poincaré, H. Poincaré es mathématiques et la logique III, Rev. métaphys. morale 14 (1906) p. 316
When the objects of discussion are linguistic entities ..then that collection of entities may vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday. (D. Isle

There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity... regarding the numbers as an incomplete infinity offers a viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. (E. Nelso

During the renaissance, particularly with Giordano Bruno, Bruno, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. ( P. Lorenzen


See also

*
Limit (mathematics) In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integral ...
*
Cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...


References


Sources


"Infinity" at The MacTutor History of Mathematics archive
treating the history of the notion of infinity, including the problem of actual infinity. *
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
, ''Physics'

*
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liber ...
, 1851, ''Paradoxien des Unendlichen'', Reclam, Leipzig. * Bernard Bolzano 1837, ''Wissenschaftslehre'', Sulzbach. *
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
in E. Zermelo (ed.) 1966, ''Gesammelte Abhandlungen mathematischen und philosophischen Inhalts'', Olms, Hildesheim. *
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
in 1960 ''Was sind und was sollen die Zahlen?'', Vieweg, Braunschweig. * Adolf Abraham Fraenkel 1923, ''Einleitung in die Mengenlehre'', Springer, Berlin. * Adolf Abraham Fraenkel, Y. Bar-Hillel, A. Levy 1984, ''Foundations of Set Theory'', 2nd edn., North Holland, Amsterdam New York. *
Stephen C. Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
1952 (1971 edition, 10th printing), ''Introduction to Metamathematics'', North-Holland Publishing Company, Amsterdam New York. . * H. Meschkowski 1981, ''Georg Cantor: Leben, Werk und Wirkung'' (2. Aufl.), BI, Mannheim. * H. Meschkowski, W. Nilson (Hrsg.) 1991, ''Georg Cantor – Briefe'', Springer, Berlin. *
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
1979, ''Selected Papers'', Vol. 2, W.A.J. Luxemburg, S. Koerner (Hrsg.), North Holland, Amsterdam. {{Infinity Infinity Metaphysics Philosophy of mathematics