Almost all: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 23:35, 18 April 2024

In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if $X$ is a set, "almost all elements of $X$ " means "all elements of $X$ but those in a negligible subset of $X$ ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.

In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of $X$ " means "a negligible quantity of elements of $X$ ".

Meanings in different areas of mathematics

Prevalent meaning

Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many".^[1]^[2] This use occurs in philosophy as well.^[3] Similarly, "almost all" can mean "all (elements of an uncountable set) except for countably many".^{[sec 1]}

Examples:

Almost all positive integers are greater than 10¹².^[4]^: 293
Almost all prime numbers are odd (2 is the only exception).^[5]
Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and the four Kepler–Poinsot polyhedra).
If P is a nonzero polynomial, then P(x) ≠ 0 for almost all x (if not all x).

Meaning in measure theory

The Cantor function as a function that has zero derivative almost everywhere

When speaking about the reals, sometimes "almost all" can mean "all reals except for a null set".^[6]^[7]^{[sec 2]} Similarly, if S is some set of reals, "almost all numbers in S" can mean "all numbers in S except for those in a null set".^[8] The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an n-dimensional space (where n is a positive integer), these definitions can be generalised to "all points except for those in a null set"^{[sec 3]} or "all points in S except for those in a null set" (this time, S is a set of points in the space).^[9] Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory,^[10]^[11]^{[sec 4]} or in the closely related sense of "almost surely" in probability theory.^[11]^{[sec 5]}

Examples:

In a measure space, such as the real line, countable sets are null. The set of rational numbers is countable, so almost all real numbers are irrational.^[12]
Georg Cantor's first set theory article proved that the set of algebraic numbers is countable as well, so almost all reals are transcendental.^[13]^{[sec 6]}
Almost all reals are normal.^[14]
The Cantor set is also null. Thus, almost all reals are not in it even though it is uncountable.^[6]
The derivative of the Cantor function is 0 for almost all numbers in the unit interval.^[15] It follows from the previous example because the Cantor function is locally constant, and thus has derivative 0 outside the Cantor set.

Meaning in number theory

In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in A below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive integers are in A.^[16]^[17]^{[sec 7]}

More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or the set of primes, if A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A.

Examples:

The natural density of cofinite sets of positive integers is 1, so each of them contains almost all positive integers.
Almost all positive integers are composite.^{[sec 7]}^{[proof 1]}
Almost all even positive numbers can be expressed as the sum of two primes.^[4]^: 489
Almost all primes are isolated. Moreover, for every positive integer $g$ , almost all primes have prime gaps of more than $g$ both to their left and to their right; that is, there is no other prime between $p - g$ and $p + g$ .^[18]

Meaning in graph theory

In graph theory, if A is a set of (finite labelled) Diagramme, it can be said to contain almost all graphs, if the proportion of graphs with n vertices that are in A tends to 1 as n tends to infinity.^[19] However, it is sometimes easier to work with probabilities,^[20] so the definition is reformulated as follows. The proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices (chosen with the uniform distribution) is in A, and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them.^[21] Therefore, equivalently to the preceding definition, the set A contains almost all graphs if the probability that a coin-flip–generated graph with n vertices is in A tends to 1 as n tends to infinity.^[20]^[22] Sometimes, the latter definition is modified so that the graph is chosen randomly in some other way, where not all graphs with n vertices have the same probability,^[21] and those modified definitions are not always equivalent to the main one.

The use of the term "almost all" in graph theory is not standard; the term "asymptotically almost surely" is more commonly used for this concept.^[20]

Example:

Almost all graphs are asymmetric.^[19]
Almost all graphs have diameter 2.^[23]

Meaning in topology

In topology^[24] and especially dynamical systems theory^[25]^[26]^[27] (including applications in economics),^[28] "almost all" of a topological space's points can mean "all of the space's points except for those in a meagre set". Some use a more limited definition, where a subset contains almost all of the space's points only if it contains some open dense set.^[26]^[29]^[30]

Example:

Given an irreducible algebraic variety, the properties that hold for almost all points in the variety are exactly the generic properties.^{[sec 8]} This is due to the fact that in an irreducible algebraic variety equipped with the Zariski topology, all nonempty open sets are dense.

Meaning in algebra

In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X" sometimes means "the elements of some element of U".^[31]^[32]^[33]^[34] For any partition of X into two disjoint sets, one of them will necessarily contain almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X, even if it isn't an ultrafilter.^[34]

Proofs

^ The prime number theorem shows that the number of primes less than or equal to n is asymptotically equal to n/ln(n). Therefore, the proportion of primes is roughly ln(n)/n, which tends to 0 as n tends to infinity, so the proportion of composite numbers less than or equal to n tends to 1 as n tends to infinity.^[17]

References

Primary sources

^ Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996). Integer-Valued Polynomials. Mathematical Surveys and Monographs. Vol. 48. American Mathematical Society. p. xix. ISBN 978-0-8218-0388-2. ISSN 0076-5376.
^ Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". In Hazewinkel, Michiel (ed.). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. Vol. 520. Springer. p. 85. doi:10.1007/978-1-4757-3180-4. ISBN 978-1-4419-4835-9.
^ Gärdenfors, Peter (22 August 2005). The Dynamics of Thought. Synthese Library. Vol. 300. Springer. pp. 190–191. ISBN 978-1-4020-3398-8.
^ ^a ^b Courant, Richard; Robbins, Herbert; Stewart, Ian (18 July 1996). What is Mathematics? An Elementary Approach to Ideas and Methods (2nd ed.). Oxford University Press. ISBN 978-0-19-510519-3.
^ Movshovitz-hadar, Nitsa; Shriki, Atara (2018-10-08). Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook. World Scientific. p. 38. ISBN 978-981-320-864-3. This can also be expressed in the statement: 'Almost all prime numbers are odd.'
^ ^a ^b Korevaar, Jacob (1 January 1968). Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration. Vol. 1. New York: Academic Press. pp. 359–360. ISBN 978-1-4832-2813-6.
^ Natanson, Isidor P. (June 1961). Theory of Functions of a Real Variable. Vol. 1. Translated by Boron, Leo F. (revised ed.). New York: Frederick Ungar Publishing. p. 90. ISBN 978-0-8044-7020-9.
^ Sohrab, Houshang H. (15 November 2014). Basic Real Analysis (2 ed.). Birkhäuser. p. 307. doi:10.1007/978-1-4939-1841-6. ISBN 978-1-4939-1841-6.
^ Helmberg, Gilbert (December 1969). Introduction to Spectral Theory in Hilbert Space. North-Holland Series in Applied Mathematics and Mechanics. Vol. 6 (1st ed.). Amsterdam: North-Holland Publishing Company. p. 320. ISBN 978-0-7204-2356-3.
^ Vestrup, Eric M. (18 September 2003). The Theory of Measures and Integration. Wiley Series in Probability and Statistics. United States: Wiley-Interscience. p. 182. ISBN 978-0-471-24977-1.
^ ^a ^b Billingsley, Patrick (1 May 1995). Probability and Measure (PDF). Wiley Series in Probability and Statistics (3rd ed.). United States: Wiley-Interscience. p. 60. ISBN 978-0-471-00710-4. Archived from the original (PDF) on 23 May 2018.
^ Niven, Ivan (1 June 1956). Irrational Numbers. Carus Mathematical Monographs. Vol. 11. Rahway: Mathematical Association of America. pp. 2–5. ISBN 978-0-88385-011-4.
^ Baker, Alan (1984). A concise introduction to the theory of numbers. Cambridge University Press. p. 53. ISBN 978-0-521-24383-4.
^ Granville, Andrew; Rudnick, Zeev (7 January 2007). Equidistribution in Number Theory, An Introduction. Nato Science Series II. Vol. 237. Springer. p. 11. ISBN 978-1-4020-5404-4.
^ Burk, Frank (3 November 1997). Lebesgue Measure and Integration: An Introduction. A Wiley-Interscience Series of Texts, Monographs, and Tracts. United States: Wiley-Interscience. p. 260. ISBN 978-0-471-17978-8.
^ Hardy, G. H. (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 50.
^ ^a ^b Hardy, G. H.; Wright, E. M. (December 1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 8–9. ISBN 978-0-19-853310-8.
^ Prachar, Karl (1957). Primzahlverteilung. Grundlehren der mathematischen Wissenschaften (in German). Vol. 91. Berlin: Springer. p. 164. Cited in Grosswald, Emil (1 January 1984). Topics from the Theory of Numbers (2nd ed.). Boston: Birkhäuser. p. 30. ISBN 978-0-8176-3044-7.
^ ^a ^b Babai, László (25 December 1995). "Automorphism Groups, Isomorphism, Reconstruction". In Graham, Ronald; Grötschel, Martin; Lovász, László (eds.). Handbook of Combinatorics. Vol. 2. Netherlands: North-Holland Publishing Company. p. 1462. ISBN 978-0-444-82351-9.
^ ^a ^b ^c Spencer, Joel (9 August 2001). The Strange Logic of Random Graphs. Algorithms and Combinatorics. Vol. 22. Springer. pp. 3–4. ISBN 978-3-540-41654-8.
^ ^a ^b Bollobás, Béla (8 October 2001). Random Graphs. Cambridge Studies in Advanced Mathematics. Vol. 73 (2nd ed.). Cambridge University Press. pp. 34–36. ISBN 978-0-521-79722-1.
^ Grädel, Eric; Kolaitis, Phokion G.; Libkin, Leonid; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (11 June 2007). Finite Model Theory and Its Applications. Texts in Theoretical Computer Science (An EATCS Series). Springer. p. 298. ISBN 978-3-540-00428-8.
^ Buckley, Fred; Harary, Frank (21 January 1990). Distance in Graphs. Addison-Wesley. p. 109. ISBN 978-0-201-09591-3.
^ Oxtoby, John C. (1980). Measure and Category. Graduate Texts in Mathematics. Vol. 2 (2nd ed.). United States: Springer. pp. 59, 68. ISBN 978-0-387-90508-2. While Oxtoby does not explicitly define the term there, Babai has borrowed it from Measure and Category in his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham, Grötschel and Lovász's Handbook of Combinatorics (vol. 2), and Broer and Takens note in their book Dynamical Systems and Chaos that Measure and Category compares this meaning of "almost all" to the measure theoretic one in the real line (though Oxtoby's book discusses meagre sets in general topological spaces as well).
^ Baratchart, Laurent (1987). "Recent and New Results in Rational L² Approximation". In Curtain, Ruth F. (ed.). Modelling, Robustness and Sensitivity Reduction in Control Systems. NATO ASI Series F. Vol. 34. Springer. p. 123. doi:10.1007/978-3-642-87516-8. ISBN 978-3-642-87516-8.
^ ^a ^b Broer, Henk; Takens, Floris (28 October 2010). Dynamical Systems and Chaos. Applied Mathematical Sciences. Vol. 172. Springer. p. 245. doi:10.1007/978-1-4419-6870-8. ISBN 978-1-4419-6870-8.
^ Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. (30 April 1997). Dynamics of One-Dimensional Maps. Mathematics and Its Applications. Vol. 407. Springer. p. 33. doi:10.1007/978-94-015-8897-3. ISBN 978-94-015-8897-3.
^ Yuan, George Xian-Zhi (9 February 1999). KKM Theory and Applications in Nonlinear Analysis. Pure and Applied Mathematics; A Series of Monographs and Textbooks. Marcel Dekker. p. 21. ISBN 978-0-8247-0031-7.
^ Albertini, Francesca; Sontag, Eduardo D. (1 September 1991). "Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems". In Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (eds.). Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory. Vol. 8. Birkhäuser. p. 29. doi:10.1007/978-1-4612-3214-8. ISBN 978-1-4612-3214-8.
^ De la Fuente, Angel (28 January 2000). Mathematical Models and Methods for Economists. Cambridge University Press. p. 217. ISBN 978-0-521-58529-3.
^ Komjáth, Péter; Totik, Vilmos (2 May 2006). Problems and Theorems in Classical Set Theory. Problem Books in Mathematics. United States: Springer. p. 75. ISBN 978-0387-30293-5.
^ Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (24 September 2007). The Classical Fields: Structural Features of the Real and Rational Numbers. Encyclopedia of Mathematics and Its Applications. Vol. 112. Cambridge University Press. p. 155. ISBN 978-0-521-86516-6.
^ Schoutens, Hans (2 August 2010). The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics. Vol. 1999. Springer. p. 8. doi:10.1007/978-3-642-13368-8. ISBN 978-3-642-13367-1.
^ ^a ^b Rautenberg, Wolfgang (17 December 2009). A Concise to Mathematical Logic. Universitext (3rd ed.). Springer. pp. 210–212. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1221-3.

Secondary sources

^ Schwartzman, Steven (1 May 1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Spectrum Series. Mathematical Association of America. p. 22. ISBN 978-0-88385-511-9.
^ Clapham, Christopher; Nicholson, James (7 June 2009). The Concise Oxford Dictionary of mathematics. Oxford Paperback References (4th ed.). Oxford University Press. p. 38. ISBN 978-0-19-923594-0.
^ James, Robert C. (31 July 1992). Mathematics Dictionary (5th ed.). Chapman & Hall. p. 269. ISBN 978-0-412-99031-1.
^ Bityutskov, Vadim I. (30 November 1987). "Almost-everywhere". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Vol. 1. Kluwer Academic Publishers. p. 153. doi:10.1007/978-94-015-1239-8. ISBN 978-94-015-1239-8.
^ Itô, Kiyosi, ed. (4 June 1993). Encyclopedic Dictionary of Mathematics. Vol. 2 (2nd ed.). Kingsport: MIT Press. p. 1267. ISBN 978-0-262-09026-1.
^ "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-11.
^ ^a ^b Weisstein, Eric W. "Almost All". MathWorld. See also Weisstein, Eric W. (25 November 1988). CRC Concise Encyclopedia of Mathematics (1st ed.). CRC Press. p. 41. ISBN 978-0-8493-9640-3.
^ Itô, Kiyosi, ed. (4 June 1993). Encyclopedic Dictionary of Mathematics. Vol. 1 (2nd ed.). Kingsport: MIT Press. p. 67. ISBN 978-0-262-09026-1.

[25] The prime number theorem shows that the number of primes less than or equal to n is asymptotically equal to n/ln(n). Therefore, the proportion of primes is roughly ln(n)/n, which tends to 0 as n tends to infinity, so the proportion of composite numbers less than or equal to n tends to 1 as n tends to infinity.^[17]

[Cahen1-1] Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996). Integer-Valued Polynomials. Mathematical Surveys and Monographs. Vol. 48. American Mathematical Society. p. xix. ISBN 978-0-8218-0388-2. ISSN 0076-5376.

[Cahen2-2] Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". In Hazewinkel, Michiel (ed.). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. Vol. 520. Springer. p. 85. doi:10.1007/978-1-4757-3180-4. ISBN 978-1-4419-4835-9.

[Gardenfors-3] Gärdenfors, Peter (22 August 2005). The Dynamics of Thought. Synthese Library. Vol. 300. Springer. pp. 190–191. ISBN 978-1-4020-3398-8.

[Courant-5] Courant, Richard; Robbins, Herbert; Stewart, Ian (18 July 1996). What is Mathematics? An Elementary Approach to Ideas and Methods (2nd ed.). Oxford University Press. ISBN 978-0-19-510519-3.

[6] Movshovitz-hadar, Nitsa; Shriki, Atara (2018-10-08). Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook. World Scientific. p. 38. ISBN 978-981-320-864-3. This can also be expressed in the statement: 'Almost all prime numbers are odd.'

[Korevaar-7] Korevaar, Jacob (1 January 1968). Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration. Vol. 1. New York: Academic Press. pp. 359–360. ISBN 978-1-4832-2813-6.

[Natanson-8] Natanson, Isidor P. (June 1961). Theory of Functions of a Real Variable. Vol. 1. Translated by Boron, Leo F. (revised ed.). New York: Frederick Ungar Publishing. p. 90. ISBN 978-0-8044-7020-9.

[Sohrab-10] Sohrab, Houshang H. (15 November 2014). Basic Real Analysis (2 ed.). Birkhäuser. p. 307. doi:10.1007/978-1-4939-1841-6. ISBN 978-1-4939-1841-6.

[Helmberg-12] Helmberg, Gilbert (December 1969). Introduction to Spectral Theory in Hilbert Space. North-Holland Series in Applied Mathematics and Mechanics. Vol. 6 (1st ed.). Amsterdam: North-Holland Publishing Company. p. 320. ISBN 978-0-7204-2356-3.

[Vestrup-13] Vestrup, Eric M. (18 September 2003). The Theory of Measures and Integration. Wiley Series in Probability and Statistics. United States: Wiley-Interscience. p. 182. ISBN 978-0-471-24977-1.

[Billingsley-14] Billingsley, Patrick (1 May 1995). Probability and Measure (PDF). Wiley Series in Probability and Statistics (3rd ed.). United States: Wiley-Interscience. p. 60. ISBN 978-0-471-00710-4. Archived from the original (PDF) on 23 May 2018.

[Niven-17] Niven, Ivan (1 June 1956). Irrational Numbers. Carus Mathematical Monographs. Vol. 11. Rahway: Mathematical Association of America. pp. 2–5. ISBN 978-0-88385-011-4.

[Baker-18] Baker, Alan (1984). A concise introduction to the theory of numbers. Cambridge University Press. p. 53. ISBN 978-0-521-24383-4.

[Granville-20] Granville, Andrew; Rudnick, Zeev (7 January 2007). Equidistribution in Number Theory, An Introduction. Nato Science Series II. Vol. 237. Springer. p. 11. ISBN 978-1-4020-5404-4.

[Burk-21] Burk, Frank (3 November 1997). Lebesgue Measure and Integration: An Introduction. A Wiley-Interscience Series of Texts, Monographs, and Tracts. United States: Wiley-Interscience. p. 260. ISBN 978-0-471-17978-8.

[Hardy1-22] Hardy, G. H. (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 50.

[Hardy2-23] Hardy, G. H.; Wright, E. M. (December 1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 8–9. ISBN 978-0-19-853310-8.

[Prachar-26] Prachar, Karl (1957). Primzahlverteilung. Grundlehren der mathematischen Wissenschaften (in German). Vol. 91. Berlin: Springer. p. 164. Cited in Grosswald, Emil (1 January 1984). Topics from the Theory of Numbers (2nd ed.). Boston: Birkhäuser. p. 30. ISBN 978-0-8176-3044-7.

[Babai-27] Babai, László (25 December 1995). "Automorphism Groups, Isomorphism, Reconstruction". In Graham, Ronald; Grötschel, Martin; Lovász, László (eds.). Handbook of Combinatorics. Vol. 2. Netherlands: North-Holland Publishing Company. p. 1462. ISBN 978-0-444-82351-9.

[Spencer-28] Spencer, Joel (9 August 2001). The Strange Logic of Random Graphs. Algorithms and Combinatorics. Vol. 22. Springer. pp. 3–4. ISBN 978-3-540-41654-8.

[Bollobas-29] Bollobás, Béla (8 October 2001). Random Graphs. Cambridge Studies in Advanced Mathematics. Vol. 73 (2nd ed.). Cambridge University Press. pp. 34–36. ISBN 978-0-521-79722-1.

[Gradel-30] Grädel, Eric; Kolaitis, Phokion G.; Libkin, Leonid; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (11 June 2007). Finite Model Theory and Its Applications. Texts in Theoretical Computer Science (An EATCS Series). Springer. p. 298. ISBN 978-3-540-00428-8.

[Buckley-31] Buckley, Fred; Harary, Frank (21 January 1990). Distance in Graphs. Addison-Wesley. p. 109. ISBN 978-0-201-09591-3.

[Oxtoby-32] Oxtoby, John C. (1980). Measure and Category. Graduate Texts in Mathematics. Vol. 2 (2nd ed.). United States: Springer. pp. 59, 68. ISBN 978-0-387-90508-2. While Oxtoby does not explicitly define the term there, Babai has borrowed it from Measure and Category in his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham, Grötschel and Lovász's Handbook of Combinatorics (vol. 2), and Broer and Takens note in their book Dynamical Systems and Chaos that Measure and Category compares this meaning of "almost all" to the measure theoretic one in the real line (though Oxtoby's book discusses meagre sets in general topological spaces as well).

[Baratchart-33] Baratchart, Laurent (1987). "Recent and New Results in Rational L² Approximation". In Curtain, Ruth F. (ed.). Modelling, Robustness and Sensitivity Reduction in Control Systems. NATO ASI Series F. Vol. 34. Springer. p. 123. doi:10.1007/978-3-642-87516-8. ISBN 978-3-642-87516-8.

[Broer-34] Broer, Henk; Takens, Floris (28 October 2010). Dynamical Systems and Chaos. Applied Mathematical Sciences. Vol. 172. Springer. p. 245. doi:10.1007/978-1-4419-6870-8. ISBN 978-1-4419-6870-8.

[Sharkovsky-35] Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. (30 April 1997). Dynamics of One-Dimensional Maps. Mathematics and Its Applications. Vol. 407. Springer. p. 33. doi:10.1007/978-94-015-8897-3. ISBN 978-94-015-8897-3.

[Yuan-36] Yuan, George Xian-Zhi (9 February 1999). KKM Theory and Applications in Nonlinear Analysis. Pure and Applied Mathematics; A Series of Monographs and Textbooks. Marcel Dekker. p. 21. ISBN 978-0-8247-0031-7.

[Albertini-37] Albertini, Francesca; Sontag, Eduardo D. (1 September 1991). "Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems". In Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (eds.). Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory. Vol. 8. Birkhäuser. p. 29. doi:10.1007/978-1-4612-3214-8. ISBN 978-1-4612-3214-8.

[Fuente-38] De la Fuente, Angel (28 January 2000). Mathematical Models and Methods for Economists. Cambridge University Press. p. 217. ISBN 978-0-521-58529-3.

[Komjath-40] Komjáth, Péter; Totik, Vilmos (2 May 2006). Problems and Theorems in Classical Set Theory. Problem Books in Mathematics. United States: Springer. p. 75. ISBN 978-0387-30293-5.

[Salzmann-41] Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (24 September 2007). The Classical Fields: Structural Features of the Real and Rational Numbers. Encyclopedia of Mathematics and Its Applications. Vol. 112. Cambridge University Press. p. 155. ISBN 978-0-521-86516-6.

[Schoutens-42] Schoutens, Hans (2 August 2010). The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics. Vol. 1999. Springer. p. 8. doi:10.1007/978-3-642-13368-8. ISBN 978-3-642-13367-1.

[Rautenberg-43] Rautenberg, Wolfgang (17 December 2009). A Concise to Mathematical Logic. Universitext (3rd ed.). Springer. pp. 210–212. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1221-3.

[Schwartzman-4] Schwartzman, Steven (1 May 1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Spectrum Series. Mathematical Association of America. p. 22. ISBN 978-0-88385-511-9.

[Clapham-9] Clapham, Christopher; Nicholson, James (7 June 2009). The Concise Oxford Dictionary of mathematics. Oxford Paperback References (4th ed.). Oxford University Press. p. 38. ISBN 978-0-19-923594-0.

[James-11] James, Robert C. (31 July 1992). Mathematics Dictionary (5th ed.). Chapman & Hall. p. 269. ISBN 978-0-412-99031-1.

[Bityutskov-15] Bityutskov, Vadim I. (30 November 1987). "Almost-everywhere". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Vol. 1. Kluwer Academic Publishers. p. 153. doi:10.1007/978-94-015-1239-8. ISBN 978-94-015-1239-8.

[Ito2-16] Itô, Kiyosi, ed. (4 June 1993). Encyclopedic Dictionary of Mathematics. Vol. 2 (2nd ed.). Kingsport: MIT Press. p. 1267. ISBN 978-0-262-09026-1.

[RealTrans-19] "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-11.

[Weisstein-24] Weisstein, Eric W. "Almost All". MathWorld. See also Weisstein, Eric W. (25 November 1988). CRC Concise Encyclopedia of Mathematics (1st ed.). CRC Press. p. 41. ISBN 978-0-8493-9640-3.

[Ito1-39] Itô, Kiyosi, ed. (4 June 1993). Encyclopedic Dictionary of Mathematics. Vol. 1 (2nd ed.). Kingsport: MIT Press. p. 67. ISBN 978-0-262-09026-1.

[1]

[2]

[3]

[sec 1]

[4]

[5]

[6]

[7]

[sec 2]

[8]

[sec 3]

[9]

[10]

[11]

[sec 4]

[sec 5]

[12]

[13]

[sec 6]

[14]

[15]

[16]

[17]

[sec 7]

[proof 1]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[sec 8]

[31]

[32]

[33]

[34]

@@ Line 1: / Line 1: @@
+{{Short description|In mathematics, with negligible exceptions}}
-In [[mathematics]], the term "'''almost all'''" means "all but a negligible amount". More precisely, if <math>A</math> is a [[set (mathematics)|set]] and <math>B</math> is a [[subset]] of <math>A</math> whose [[complement (set theory)|complement]] in <math>A</math> is [[negligible set|negligible]], then almost all elements of <math>A</math> are in <math>B</math>. The meaning of "negligible" depends on the mathematical context: for instance, it can mean [[finite set|finite]], [[countable set|countable]], or [[null set|null]]. If <math>p</math> is a [[property (mathematics)|property]] and the set of elements of <math>A</math> for which it holds contains almost all elements of <math>A</math>, it is said that <math>p</math> holds for almost all elements of <math>A</math>.
+In [[mathematics]], the term "'''almost all'''" means "all but a negligible quantity". More precisely, if <math>X</math> is a [[set (mathematics)|set]], "almost all elements of <math>X</math>" means "all elements of <math>X</math> but those in a [[negligible set|negligible]] [[subset]] of <math>X</math>". The meaning of "negligible" depends on the mathematical context; for instance, it can mean [[finite set|finite]], [[countable set|countable]], or [[null set|null]].
+In contrast, "'''almost no'''" means "a negligible quantity"; that is, "almost no elements of <math>X</math>" means "a negligible quantity of elements of <math>X</math>".
-==Prevalent meaning==
+==Meanings in different areas of mathematics==
+===Prevalent meaning===
 {{further|Cofinite set}}
-Throughout mathematics, when referring to a subset of an infinite set, "almost all" is sometimes used to mean "all but [[finite set|finite]]ly many".{{r|Cahen1|Cahen2|Halmos}} This use occurs in philosophy as well.{{r|Gardenfors}} Similarly, when referring to a subset of an [[uncountable]] set, "almost all" can mean "all but [[countable set|countably]] many"; that is, a subset <math>B</math> of a set <math>A</math> contains almost all elements of <math>A</math> if it is [[cocountable]].{{r|Schwartzman|group=sec}}
+Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an [[infinite set]]) except for [[finite set|finite]]ly many".{{r|Cahen1|Cahen2}} This use occurs in philosophy as well.{{r|Gardenfors}} Similarly, "almost all" can mean "all (elements of an [[uncountable set]]) except for [[countable set|countably]] many".{{r|Schwartzman|group=sec}}
 Examples:
-* Almost all positive integers are greater than 1,000,000,000,000.{{r|Courant|page=293}}
+* Almost all positive integers are greater than 10<sup>12</sup>.{{r|Courant|page=293}}
+* Almost all [[prime number]]s are odd (2 is the only exception).<ref>{{Cite book|last1=Movshovitz-hadar|first1=Nitsa|url=https://books.google.com/books?id=lp15DwAAQBAJ&q=Almost+all+prime+numbers+are+odd&pg=PA38|title=Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook|last2=Shriki|first2=Atara|date=2018-10-08|publisher=World Scientific|isbn=978-981-320-864-3|pages=38|language=en|quote=This can also be expressed in the statement: 'Almost all prime numbers are odd.'}}</ref>
-* Almost all [[prime number]]s are odd, as 2 is the only exception.
-* Almost all [[polyhedra]] are [[regular polyhedron#The regular polyhedra|irregular]], as there are only nine exceptions: the five [[platonic solid]]s and the four [[Kepler-Poinsot polyhedra]].
+* Almost all [[polyhedra]] are [[regular polyhedron#The regular polyhedra|irregular]] (as there are only nine exceptions: the five [[platonic solid]]s and the four [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]]).
+* If <var>P</var> is a nonzero [[polynomial]], then <var>P(x)</var> &ne; 0 for almost all <var>x</var> (if not all ''x'').
-* If <math>a</math> and <math>b</math> are [[coprime]] positive integers, almost all positive integers can be expressed as <math>ax+by</math> where <math>x</math> and <math>y</math> are positive integers.{{r|coprime|group=proof}}
-* If <math>P</math> is a [[zero polynomial|nonzero]] polynomial, <math>P(x)\neq 0</math> for almost all <math>x</math>.
-==Meaning in measure theory==
+===Meaning in measure theory===
 {{further|Almost everywhere}}
-[[File:CantorEscalier.svg|thumb|right|250px| The Cantor function]]
+[[File:CantorEscalier.svg|thumb|right|250px| The [[Cantor function]] as a function that has zero derivative almost everywhere]]
-When speaking about the [[real number|reals]], sometimes "almost all" means "all reals but a [[null set]]". Similarly, if <math>S</math> is some set of real numbers, "almost all numbers in <math>S</math>" can mean "all numbers in <math>S</math> but those in a null set".{{r|Korevaar|Sohrab|Natanson}} The [[real line]] can be thought of as a one-dimensional [[Euclidean space]]. In the more general case of an <math>n</math>-dimensional space (where <math>n</math> is a positive integer), the definition can be [[generalised]] to "all points in <math>S</math> but those in a null set" (this time, <math>S</math> is a set of points in the space).{{r|Helmberg}}{{r|James|group=sec}} Even more generally, "almost all" is sometimes used in the sense of "[[almost everywhere]]" in [[measure theory]],{{r|Vestrup|Billingsley}}{{r|Bityutskov|group=sec}} or in the closely related sense of "[[almost surely]]" in [[probability theory]].{{r|Billingsley}}{{r|Ito2|group=sec}}
+When speaking about the [[real number|reals]], sometimes "almost all" can mean "all reals except for a [[null set]]".{{r|Korevaar|Natanson}}{{r|Clapham|group=sec}} Similarly, if <var>S</var> is some set of reals, "almost all numbers in <var>S</var>" can mean "all numbers in <var>S</var> except for those in a null set".{{r|Sohrab}} The [[real line]] can be thought of as a one-dimensional [[Euclidean space]]. In the more general case of an <var>n</var>-dimensional space (where <var>n</var> is a positive integer), these definitions can be [[generalised]] to "all points except for those in a null set"{{r|James|group=sec}} or "all points in <var>S</var> except for those in a null set" (this time, <var>S</var> is a set of points in the space).{{r|Helmberg}} Even more generally, "almost all" is sometimes used in the sense of "[[almost everywhere]]" in [[measure theory]],{{r|Vestrup|Billingsley}}{{r|Bityutskov|group=sec}} or in the closely related sense of "[[almost surely]]" in [[probability theory]].{{r|Billingsley}}{{r|Ito2|group=sec}}
 Examples:
-* In a [[measure space]], such as the real line, countable sets are null. The set of [[rational number]]s is countable, and thus almost all real numbers are irrational.{{r|Niven}}
+* In a [[measure space]], such as the real line, countable sets are null. The set of [[rational number]]s is countable, so almost all real numbers are irrational.{{r|Niven}}
-* As [[Georg Cantor]] proved in [[Georg Cantor's first set theory article|his first set theory article]], the set of [[algebraic number]]s is countable as well, so almost all reals are [[transcendental number|transcendental]].{{r|Baker}}
+* Georg [[Cantor's first set theory article]] proved that the set of [[algebraic number]]s is countable as well, so almost all reals are [[transcendental number|transcendental]].{{r|Baker}}{{r|group=sec|RealTrans}}
 * Almost all reals are [[normal number|normal]].{{r|Granville}}
-* The [[Cantor set]] is null as well. Thus, almost all reals are not members of it even though it is uncountable.{{r|Korevaar}}
+* The [[Cantor set]] is also null. Thus, almost all reals are not in it even though it is uncountable.{{r|Korevaar}}
-* The derivative of the [[Cantor function]] is 0 for almost all numbers in the [[unit interval]].{{r|Burk}} It follows from the previous example because the Cantor function is constant—and thus with derivative 0—outside the Cantor set.
+* The derivative of the [[Cantor function]] is 0 for almost all numbers in the [[unit interval]].{{r|Burk}} It follows from the previous example because the Cantor function is [[locally constant function|locally constant]], and thus has derivative 0 outside the Cantor set.
-==Meaning in number theory==
+===Meaning in number theory===
 {{further|Asymptotically almost surely}}
-In [[number theory]], "almost all [[positive integers]]" can mean "a set of positive integers whose [[natural density]] is 1". That is, if <math>A</math> is a set of positive integers, and if the proportion of positive integers below <math>n</math> that are in <math>A</math> [[limit of a sequence|tends to]] 1 as <math>n</math> tends to infinity, then almost all positive integers are in <math>A</math>.{{r|Hardy1|Hardy2}}{{r|Weisstein|group=sec}} More generally, let <math>S</math> be an infinite set of positive integers, such as the set of even positive numbers or of prime numbers. If <math>A</math> is a subset of <math>S</math>, and if the proportion of elements of <math>S</math> below <math>n</math> that are in <math>A</math> tends to 1 as <math>n</math> tends to infinity, then it can be said that almost all elements of <math>S</math> are in <math>A</math>.
+In [[number theory]], "almost all positive integers" can mean "the positive integers in a set whose [[natural density]] is 1". That is, if <var>A</var> is a set of positive integers, and if the proportion of positive integers in ''A'' below <var>n</var> (out of all positive integers below <var>n</var>) [[limit of a sequence|tends to]] 1 as <var>n</var> tends to infinity, then almost all positive integers are in <var>A</var>.{{r|Hardy1|Hardy2}}{{r|Weisstein|group=sec}}
+More generally, let <var>S</var> be an infinite set of positive integers, such as the set of even positive numbers or the set of [[prime number|primes]], if <var>A</var> is a subset of <var>S</var>, and if the proportion of elements of <var>S</var> below <var>n</var> that are in <var>A</var> (out of all elements of <var>S</var> below <var>n</var>) tends to 1 as <var>n</var> tends to infinity, then it can be said that almost all elements of <var>S</var> are in <var>A</var>.
 Examples:
-* The natural density of cofinite sets of positive integers is 1, so each of them contains almost all positive integers.
+* The natural density of [[cofinite set]]s of positive integers is 1, so each of them contains almost all positive integers.
-* Almost all positive integers are [[composite number|composite]].{{r|Weisstein|group=sec}}{{refn |group=proof |According to the [[prime number theorem]], the number of prime numbers less than or equal to <math>n</math> is asymptotically equal to <math>n/\ln n</math>. Therefore, the proportion of primes is roughly <math>1/\ln n</math>, which tends to 0 as <math>n</math> tends to [[infinity]], so the proportion of composite numbers less than or equal to <math>n</math> tends to 1 as <math>n</math> tends to [[infinity]].{{r|Hardy2}}}}
+* Almost all positive integers are [[composite number|composite]].{{r|Weisstein|group=sec}}{{refn |group=proof |The [[prime number theorem]] shows that the number of primes less than or equal to <var>n</var> is asymptotically equal to <var>n</var>/ln(<var>n</var>). Therefore, the proportion of primes is roughly ln(<var>n</var>)/<var>n</var>, which tends to 0 as <var>n</var> tends to [[infinity]], so the proportion of composite numbers less than or equal to <var>n</var> tends to 1 as <var>n</var> tends to infinity.{{r|Hardy2}}}}
 * Almost all even positive numbers can be expressed as the sum of two primes.{{r|Courant|page=489}}
-* Almost all primes are [[twin prime#Isolated prime|isolated]]. Moreover, for every positive integer <math>g</math>, almost all primes have [[prime gap]]s of more than <math>g</math> both to their left and to their right (no other primes between <math>p+g</math> and <math>p-g)</math>.{{r|Prachar}}
+* Almost all primes are [[twin prime#Isolated prime|isolated]]. Moreover, for every positive integer {{mvar|g}}, almost all primes have [[prime gap]]s of more than {{mvar|g}} both to their left and to their right; that is, there is no other prime between {{math|''p'' − ''g''}} and {{math|''p'' + ''g''}}.{{r|Prachar}}
-==Meaning in graph theory==
+===Meaning in graph theory===
-In [[graph theory]], if <math>A</math> is a set of ([[graph labeling|labelled]]) [[graph (discrete mathematics)|graph]]s, it can be said to contain almost all graphs if the proportion of graphs with <math>n</math> vertices that are in <math>A</math> tends to 1 as <math>n</math> tends to infinity.{{r|Graham}}{{efn |group=note |This use of the term "almost all" is not standard; a number of authors use the term "[[asymptotically almost surely]]" for this concept.{{r|Spencer}}}} However, it is sometimes easier to work with probabilities,{{r|Spencer}} so the definition is reformulated as follows. The proportion of graphs with <math>n</math> vertices that are in <math>A</math> equals the probability that a random graph with <math>n</math> vertices (chosen with the [[discrete uniform distribution|uniform distribution]]) is in <math>A</math>, and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them.{{r|Bollobas}} Therefore, equivalently to the preceding definition, <math>A</math> contains almost all graphs if the probability that a coin flip-generated graph with <math>n</math> vertices is in <math>A</math> tends to 1 as <math>n</math> tends to infinity.{{r|Spencer|Gradel}} In the latter definition, sometimes the graph is chosen randomly in some other way, where not all graphs with <math>n</math> vertices have the same probability.{{r|Bollobas}}
+In [[graph theory]], if <var>A</var> is a set of (finite [[graph labeling|labelled]]) [[graph (discrete mathematics)|graph]]s, it can be said to contain almost all graphs, if the proportion of graphs with <var>n</var> vertices that are in <var>A</var> tends to 1 as <var>n</var> tends to infinity.{{r|Babai}} However, it is sometimes easier to work with probabilities,{{r|Spencer}} so the definition is reformulated as follows. The proportion of graphs with <var>n</var> vertices that are in <var>A</var> equals the probability that a random graph with <var>n</var> vertices (chosen with the [[discrete uniform distribution|uniform distribution]]) is in <var>A</var>, and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them.{{r|Bollobas}} Therefore, equivalently to the preceding definition, the set <var>''A''</var> contains almost all graphs if the probability that a coin-flip–generated graph with <var>n</var> vertices is in <var>A</var> tends to 1 as <var>n</var> tends to infinity.{{r|Spencer|Gradel}} Sometimes, the latter definition is modified so that the graph is chosen randomly in some [[random graph#Models|other way]], where not all graphs with <var>n</var> vertices have the same probability,{{r|Bollobas}} and those modified definitions are not always equivalent to the main one.
+The use of the term "almost all" in graph theory is not standard; the term "[[asymptotically almost surely]]" is more commonly used for this concept.{{r|Spencer}}
 Example:
+* Almost all graphs are [[asymmetric graph|asymmetric]].{{r|Babai}}
 * Almost all graphs have [[diameter (graph theory)|diameter]] 2.{{r|Buckley}}
-==Meaning in topology==
+===Meaning in topology===
-In [[topology]]{{r|Oxtoby}} and especially [[dynamical systems theory]]{{r|Baratchart|Broer|Sharkovsky}} (including applications in economy),{{r|Yuan}} subset of a [[topological space]] contains "almost all" of its points if the subset is [[residual set|residual]]. Some authors use a more limited definition, where a subset only contains almost all of the space's points if it contains some [[open set|open]] [[dense set|dense]] set.{{r|Broer|Albertini|Fuente}}
+In [[topology]]{{r|Oxtoby}} and especially [[dynamical systems theory]]{{r|Baratchart|Broer|Sharkovsky}} (including applications in economics),{{r|Yuan}} "almost all" of a [[topological space]]'s points can mean "all of the space's points except for those in a [[meagre set]]". Some use a more limited definition, where a subset contains almost all of the space's points only if it contains some [[Open set|open]] [[dense set]].{{r|Broer|Albertini|Fuente}}
 Example:
-* Given an [[hyperconnected space|irreducible]] [[algebraic variety]], the properties that hold for almost all points in the variety are exactly the [[generic property|generic properties]].{{r|Ito1|group=sec}} It is because in an irreducible algebraic variety equipped with the [[Zariski topology]], all nonempty open sets are dense.
+* Given an [[hyperconnected space|irreducible]] [[algebraic variety]], the [[Property (mathematics)|properties]] that hold for almost all points in the variety are exactly the [[generic property|generic properties]].{{r|Ito1|group=sec}} This is due to the fact that in an irreducible algebraic variety equipped with the [[Zariski topology]], all nonempty open sets are dense.
-==Meaning in algebra==
+===Meaning in algebra===
-In [[abstract algebra]] and [[mathematical logic]], if <math>U</math> is an [[ultrafilter#Special case: ultrafilter on the powerset of a set|ultrafilter]] on a set <math>X</math>, "almost all elements of <math>X</math>" sometimes means "the elements of some ''element'' of <math>U</math>".{{r|Komjath|Salzmann|Schoutens|Rautenberg}} For any [[partition of a set|partition]] of <math>X</math> into two [[disjoint sets|disjoint]] sets, one of them necessarily contains almost all elements of <math>X</math>. It is also possible to think of the elements of a general [[filter (mathematics)#Filter on a set|filter]] on <math>X</math> as containing almost all elements of <math>X</math>.{{r|Rautenberg}}
+In [[abstract algebra]] and [[mathematical logic]], if <var>U</var> is an [[Ultrafilter#Special case: ultrafilter on the powerset of a set|ultrafilter]] on a set <var>X,</var> "almost all elements of <var>X</var>" sometimes means "the elements of some ''element'' of <var>U</var>".{{r|Komjath|Salzmann|Schoutens|Rautenberg}} For any [[Partition of a set|partition]] of <var>X</var> into two [[disjoint sets]], one of them will necessarily contain almost all elements of <var>X.</var> It is possible to think of the elements of a [[Filter (set theory)|filter]] on <var>X</var> as containing almost all elements of <var>X</var>, even if it isn't an ultrafilter.{{r|Rautenberg}}
-==Notes==
-{{reflist |group=note}}
 ==Proofs==
-{{reflist |group=proof |refs=
+{{reflist|group=proof}}
-<ref name=coprime>It follows from [[Bézout's identity]] that all integers&mdash;and thus all positive ones&mdash;can be expressed as <math>ax+by</math> where <math>x</math> and <math>y</math> are integers, though not necessarily positive. However, if one only cares about the case where <math>n=ax+by</math> is positive, at least one of <math>x</math> and <math>y</math> has to be positive: otherwise, <math>n</math> wouldn't be positive either. One can assume [[without loss of generality]] that <math>x</math> is positive. For every nonnegative integer <math>k</math>, <math>n=ax+by=ax-akb+by+bka</math> and so <math>n=a(x-kb)+b(y+ka)</math>. Let <math>k</math> be the least integer such that <math>y+ka</math> is positive. If <math>x-kb</math> is positive as well then <math>a(x-kb)+b(y+ka)</math> is a way of expressing <math>n</math> as a sum of positive integer multiples of <math>a</math> and <math>b</math>. Assume <math>n>ab</math>. Thus, <math>a(x-kb)+b(y+ka)>ab</math>. Because <math>k</math> is the least such integer, <math>y+(k-1)a\le0</math>, so <math>y+ka\le a</math> and thus <math>a(x-kb)>ab-b(y+ka)\ge ab-ba=0</math>. This implies <math>x-kb</math> is positive, so as stated above <math>a(x-kb)+b(y+ka)</math> is a way of expressing <math>n</math> as a sum of positive integer multiples of <math>a</math> and <math>b</math>. Therefore, all positive integers greater than <math>ab</math> can be expressed in this way, so there are at most <math>ab</math> positive integers that cannot. Thus, it is true for almost all positive integers.</ref>
-}}
+==See also==
+* [[Almost]]
+* [[Almost everywhere]]
+* [[Almost surely]]
 ==References==
 ===Primary sources===
 {{reflist |refs=
-<ref name=Cahen1>{{cite book |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |date={{date|1996-12-03}} |title=Integer-Valued Polynomials |series=[[Mathematical Surveys and Monographs]] |volume=48 |publisher=[[American Mathematical Society]] |page=xix |isbn=978-0-8218-0388-2 |issn=0076-5376}}</ref>
+<ref name=Cahen1>{{cite book |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |date=3 December 1996 |title=Integer-Valued Polynomials |series=[[Mathematical Surveys and Monographs]] |volume=48 |publisher=[[American Mathematical Society]] |page=xix |isbn=978-0-8218-0388-2 |issn=0076-5376}}</ref>
-<ref name=Cahen2>{{cite book |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |editor-last=Hazewinkel |editor-first=Michiel |editor-link=Michiel Hazewinkel |date={{date|2010-12-07}} |orig-year=First published 2000 |title=Non-Noetherian Commutative Ring Theory |series=Mathematics and Its Applications |volume=520 |publisher=[[Springer Science+Business Media|Springer]] |page=85 |chapter=Chapter 4: What's New About Integer-Valued Polynomials on a Subset? |doi=10.1007/978-1-4757-3180-4 |isbn=978-1-4419-4835-9}}</ref>
+<ref name=Cahen2>{{cite book |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |editor-last=Hazewinkel |editor-first=Michiel |editor-link=Michiel Hazewinkel |date=7 December 2010 |orig-year=First published 2000 |title=Non-Noetherian Commutative Ring Theory |series=Mathematics and Its Applications |volume=520 |publisher=[[Springer Science+Business Media|Springer]] |page=85 |chapter=Chapter 4: What's New About Integer-Valued Polynomials on a Subset? |doi=10.1007/978-1-4757-3180-4 |isbn=978-1-4419-4835-9}}</ref>
-<ref name=Halmos>{{cite book |last=Halmos |first=Paul R. |author-link=Paul Halmos |date=1962 |title=Algebraic Logic |place=New York |publisher=Chelsea Publishing Company |page=114}}</ref>
+<ref name=Gardenfors>{{cite book |last=Gärdenfors |first=Peter |author-link=Peter Gärdenfors |date=22 August 2005 |title=The Dynamics of Thought |series=Synthese Library |volume=300 |publisher=[[Springer Science+Business Media|Springer]] |pages=190–191 |isbn=978-1-4020-3398-8}}</ref>
-<ref name=Gardenfors>{{cite book |last=Gärdenfors |first=Peter |author-link=Peter Gärdenfors |date={{date|2005-08-22}} |title=The Dynamics of Thought |series=Synthese Library |volume=300 |publisher=[[Springer Science+Business Media|Springer]] |pages=190-191 |isbn=978-1-4020-3398-8}}</ref>
+<ref name=Courant>{{cite book |last1=Courant |first1=Richard |author-link1=Richard Courant |last2=Robbins |first2=Herbert |author-link2=Herbert Robbins |last3=Stewart |first3=Ian |author-link3=Ian Stewart (mathematician) |date=18 July 1996 |title=What is Mathematics? An Elementary Approach to Ideas and Methods |url=https://archive.org/details/WhatIsMathematics |edition=2nd |publisher=[[Oxford University Press]] |isbn=978-0-19-510519-3}}</ref>
+<ref name=Korevaar>{{cite book |last=Korevaar |first=Jacob |author-link=Jacob Korevaar |date=1 January 1968 |title=Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration |volume=1 |place=New York |publisher=[[Academic Press]] |pages=359–360 |isbn=978-1-4832-2813-6}}</ref>
-<ref name=Courant>{{cite book |last1=Courant |first1=Richard |author-link1=Richard Courant |last2=Robbins |first2=Herbert |author-link2=Herbert Robbins |last3=Stewart |first3=Ian |author-link3=Ian Stewart (mathematician) |date={{date|1996-07-18}} |title=What is Mathematics? An Elementary Approach to Ideas and Methods |url=https://ia800401.us.archive.org/16/items/WhatIsMathematics/What%20Is%20Mathematics%20An%20Elementary%20Approach%20to%20Ideas%20and%20Methods.pdf |edition=2nd |publisher=[[Oxford University Press]] |isbn=978-0-19-510519-3}}</ref>
-<ref name=Korevaar>{{cite book |last=Korevaar |first=Jacob |author-link=Jacob Korevaar |date={{date|1968-01-01}} |title=Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration |volume=1 |place=New York |publisher=[[Academic Press]] |pages=359-360 |isbn=978-1-4832-2813-6}}</ref>
+<ref name=Sohrab>{{cite book |last=Sohrab |first=Houshang H. |date=15 November 2014 |title=Basic Real Analysis |edition=2 |publisher=[[Birkhäuser]] |page=307 |doi=10.1007/978-1-4939-1841-6 |isbn=978-1-4939-1841-6}}</ref>
-<ref name=Sohrab>{{cite book |last=Sohrab |first=Houshang H. |date={{date|2014-11-15}} |title=Basic Real Analysis |edition=2 |publisher=[[Birkhäuser]] |page=265 |doi=10.1007/978-1-4939-1841-6 |isbn=978-1-4939-1841-6}}</ref>
 <ref name=Natanson>{{cite book |last=Natanson |first=Isidor P. |author-link=Isidor Natanson |translator-last=Boron |translator-first=Leo F. |date=June 1961 |title=Theory of Functions of a Real Variable |volume=1 |edition=revised |place=New York |publisher=[[Frederick Ungar Publishing Company|Frederick Ungar Publishing]] |page=90 |isbn=978-0-8044-7020-9}}</ref>
 <ref name=Helmberg>{{cite book |last=Helmberg |first=Gilbert |date=December 1969 |title=Introduction to Spectral Theory in Hilbert Space |series=North-Holland Series in Applied Mathematics and Mechanics |volume=6 |edition=1st|place=Amsterdam |publisher=[[North-Holland Publishing Company]] |page=320 |isbn=978-0-7204-2356-3}}</ref>
-<ref name=Vestrup>{{cite book |last=Vestrup |first=Eric M. |date={{date|2003-09-18}} |title=The Theory of Measures and Integration |series=Wiley Series in Probability and Statistics |place=United States |publisher=[[Wiley-Interscience]] |page=182 |isbn=978-0-471-24977-1}}</ref>
+<ref name=Vestrup>{{cite book |last=Vestrup |first=Eric M. |date=18 September 2003 |title=The Theory of Measures and Integration |series=Wiley Series in Probability and Statistics |place=United States |publisher=[[Wiley-Interscience]] |page=182 |isbn=978-0-471-24977-1}}</ref>
-<ref name=Billingsley>{{cite book |last=Billingsley |first=Patrick |author-link=Patrick Billingsley |date={{date|1995-05-01}} |title=Probability and Measure |url=https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf |series=Wiley Series in Probability and Statistics |edition=3rd |place=United States |publisher=[[Wiley-Interscience]] |page=60 |isbn=978-0-471-00710-4 |archive-url=https://web.archive.org/web/20180523011143/https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf |archive-date={{date|2018-05-23}}}}</ref>
+<ref name=Billingsley>{{cite book |last=Billingsley |first=Patrick |author-link=Patrick Billingsley |date=1 May 1995 |title=Probability and Measure |url=https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf |series=Wiley Series in Probability and Statistics |edition=3rd |place=United States |publisher=[[Wiley-Interscience]] |page=60 |isbn=978-0-471-00710-4 |archive-url=https://web.archive.org/web/20180523011143/https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf |archive-date=23 May 2018}}</ref>
-<ref name=Niven>{{cite book |last=Niven |first=Ivan |author-link=Ivan M. Niven |date={{date|1956-06-01}} |title=Irrational Numbers |series=[[Carus Mathematical Monographs]] |volume=11 |place=Rahway |publisher=[[Mathematical Association of America]] |pages=2-5 |isbn=978-0-88385-011-4}}</ref>
+<ref name=Niven>{{cite book |last=Niven |first=Ivan |author-link=Ivan M. Niven |date=1 June 1956 |title=Irrational Numbers |series=[[Carus Mathematical Monographs]] |volume=11 |place=Rahway |publisher=[[Mathematical Association of America]] |pages=2–5 |isbn=978-0-88385-011-4}}</ref>
-<ref name=Baker>{{cite book |last=Baker |first=Alan |author-link=Alan Baker (mathematician) |date=1984 |title=A concise introduction to the theory of numbers |url=https://ia801603.us.archive.org/13/items/AConciseIntroductionToTheTheoryOfNumbers/A%20Concise%20Introduction%20to%20the%20Theory%20of%20Numbers%20.pdf |publisher=[[Cambridge University Press]] |page=53 |isbn=978-0-521-24383-4}}</ref>
+<ref name=Baker>{{cite book |last=Baker |first=Alan |author-link=Alan Baker (mathematician) |date=1984 |title=A concise introduction to the theory of numbers |url=https://archive.org/details/conciseintroduct0000bake/page/53 |publisher=[[Cambridge University Press]] |page=[https://archive.org/details/conciseintroduct0000bake/page/53 53] |isbn=978-0-521-24383-4 }}</ref>
-<ref name=Granville>{{cite book |last1=Granville |first1=Andrew |author-link1=Andrew Granville |last2=Rudnick |first2=Zeev |author-link2=Zeev Rudnick |date={{date|2007-01-07}} |title=Equidistribution in Number Theory, An Introduction |series=Nato Science Series II |volume=237 |publisher=[[Springer Science+Business Media|Springer]] |page=11 |isbn=978-1-4020-5404-4}}</ref>
+<ref name=Granville>{{cite book |last1=Granville |first1=Andrew |author-link1=Andrew Granville |last2=Rudnick |first2=Zeev |author-link2=Zeev Rudnick |date=7 January 2007 |title=Equidistribution in Number Theory, An Introduction |series=Nato Science Series II |volume=237 |publisher=[[Springer Science+Business Media|Springer]] |page=11 |isbn=978-1-4020-5404-4}}</ref>
-<ref name=Burk>{{cite book |last=Burk |first=Frank |date={{date|1997-11-03}} |title=Lebesgue Measure and Integration: An Introduction |series=A Wiley-Interscience Series of Texts, Monographs, and Tracts |place=United States |publisher=[[Wiley-Interscience]] |page=260 |isbn=978-0-471-17978-8}}</ref>
+<ref name=Burk>{{cite book |last=Burk |first=Frank |date=3 November 1997 |title=Lebesgue Measure and Integration: An Introduction |series=A Wiley-Interscience Series of Texts, Monographs, and Tracts |place=United States |publisher=[[Wiley-Interscience]] |page=260 |isbn=978-0-471-17978-8}}</ref>
-<ref name=Hardy1>{{cite book |last=Hardy |first=G. H. |author-link=G. H. Hardy |date=1940 |title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work |publisher=[[Cambridge University Press]] |page=50}}</ref>
+<ref name=Hardy1>{{cite book |last=Hardy |first=G. H. |author-link=G. H. Hardy |date=1940 |title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work |url=https://archive.org/details/pli.kerala.rare.37877 |publisher=[[Cambridge University Press]] |page=50}}</ref>
-<ref name=Hardy2>{{cite book |last1=Hardy |first1=G. H. |author-link1=G. H. Hardy |last2=Wright |first2=E. M. |author-link2=E. M. Wright |date=December 1960 |title=An Introduction to the Theory of Numbers |url=https://ia800201.us.archive.org/23/items/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright.pdf |edition=4th |publisher=[[Oxford University Press]] |pages=8-9 |isbn=978-0-19-853310-8}}</ref>
+<ref name=Hardy2>{{cite book |last1=Hardy |first1=G. H. |author-link1=G. H. Hardy |last2=Wright |first2=E. M. |author-link2=E. M. Wright |date=December 1960 |title=An Introduction to the Theory of Numbers |url=https://archive.org/details/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright |edition=4th |publisher=[[Oxford University Press]] |pages=8–9 |isbn=978-0-19-853310-8}}</ref>
-<ref name=Prachar>{{cite book |last=Prachar |first=Karl |author-link=Karl Prachar |date=1957 |title=Primzahlverteilung |series=Grundlehren der mathematischen Wissenschaften |language=German |volume=91 |place=Berlin |publisher=[[Springer Science+Business Media|Springer]] |page=164}} Cited in {{cite book |last=Grosswald |first=Emil |author-link=Emil Grosswald |date={{date|1984-01-01}} |title=Topics from the Theory of Numbers |edition=2nd |place=Boston |publisher=[[Birkhäuser]] |page=30 |isbn=978-0-8176-3044-7}}</ref>
+<ref name=Prachar>{{cite book |last=Prachar |first=Karl |author-link=Karl Prachar |date=1957 |title=Primzahlverteilung |series=Grundlehren der mathematischen Wissenschaften |language=de |volume=91 |place=Berlin |publisher=[[Springer Science+Business Media|Springer]] |page=164}} Cited in {{cite book |last=Grosswald |first=Emil |author-link=Emil Grosswald |date=1 January 1984 |title=Topics from the Theory of Numbers |edition=2nd |place=Boston |publisher=[[Birkhäuser]] |page=30 |isbn=978-0-8176-3044-7}}</ref>
-<ref name=Bollobas>{{cite book |last=Bollobás |first=Béla |author-link=Béla Bollobás |date={{date|2001-10-08}} |title=Random Graphs |series=Cambridge Studies in Advanced Mathematics |volume=73 |edition=2nd |publisher=[[Cambridge University Press]] |pages=34-36 |isbn=978-0-521-79722-1}}</ref>
+<ref name=Bollobas>{{cite book |last=Bollobás |first=Béla |author-link=Béla Bollobás |date=8 October 2001 |title=Random Graphs |series=Cambridge Studies in Advanced Mathematics |volume=73 |edition=2nd |publisher=[[Cambridge University Press]] |pages=34–36 |isbn=978-0-521-79722-1}}</ref>
-<ref name=Graham>{{cite book |last1=Graham |first1=Ronald |author-link1=Ronald Graham |last2=Grötschel |first2=Martin |author-link2=Martin Grötschel |last3=Lovász |first3=László |author-link3=László Lovász |date={{date|1995-12-25}} |title=Handbook of Combinatorics |volume=2 |publisher=[[North-Holland Publishing Company]] |place=Netherlands |page=1462 |isbn=978-0-444-82351-9}}</ref>
+<ref name=Babai>{{cite book |last=Babai |first=László |author-link=László Babai |editor1-last=Graham |editor1-first=Ronald |editor-link1=Ronald Graham |editor2-last=Grötschel |editor2-first=Martin |editor-link2=Martin Grötschel |editor3-last=Lovász |editor3-first=László |editor-link3=László Lovász |date=25 December 1995 |title=Handbook of Combinatorics |volume=2 |publisher=[[North-Holland Publishing Company]] |place=Netherlands |page=1462 |chapter=Automorphism Groups, Isomorphism, Reconstruction |isbn=978-0-444-82351-9}}</ref>
-<ref name=Spencer>{{cite book |last=Spencer |first=Joel |author-link=Joel Spencer |date={{date|2001-08-09}} |title=The Strange Logic of Random Graphs |series=Algorithms and Combinatorics |volume=22 |publisher=[[Springer Science+Business Media|Springer]] |pages=3-4 |isbn=978-3-540-41654-8}}</ref>
+<ref name=Spencer>{{cite book |last=Spencer |first=Joel |author-link=Joel Spencer |date=9 August 2001 |title=The Strange Logic of Random Graphs |title-link= The Strange Logic of Random Graphs |series=Algorithms and Combinatorics |volume=22 |publisher=[[Springer Science+Business Media|Springer]] |pages=3–4 |isbn=978-3-540-41654-8}}</ref>
-<ref name=Gradel>{{cite book |last1=Grädel |first1=Eric |last2=Kolaitis |first2=Phokion G. |last3=Libkin |first3=Leonid |author-link3=Leonid Libkin |last4=Marx |first4=Maarten |last5=Spencer |first5=Joel |author-link5=Joel Spencer |last6=Vardi |first6=Moshe Y. |author-link6=Moshe Vardi |last7=Venema |first7=Yde |last8=Weinstein |first8=Scott |date={{date|2007-06-11}} |title=Finite Model Theory and Its Applications |series=Texts in Theoretical Computer Science (An [[EATCS]] Series) |publisher=[[Springer Science+Business Media|Springer]] |page=298 |isbn=978-3-540-00428-8}}</ref>
+<ref name=Gradel>{{cite book |last1=Grädel |first1=Eric |last2=Kolaitis |first2=Phokion G. |last3=Libkin |first3=Leonid |author-link3=Leonid Libkin |last4=Marx |first4=Maarten |last5=Spencer |first5=Joel |author-link5=Joel Spencer |last6=Vardi |first6=Moshe Y. |author-link6=Moshe Vardi |last7=Venema |first7=Yde |last8=Weinstein |first8=Scott |date=11 June 2007 |title=Finite Model Theory and Its Applications |series=Texts in Theoretical Computer Science (An [[EATCS]] Series) |publisher=[[Springer Science+Business Media|Springer]] |page=298 |isbn=978-3-540-00428-8}}</ref>
-<ref name=Buckley>{{cite book |last1=Buckley |first1=Fred |last2=Harary |first2=Frank |author-link2=Frank Harary |date={{date|1990-01-21}} |title=Distance in Graphs |publisher=[[Addison-Wesley]] |page=109 |isbn=978-0-201-09591-3}}</ref>
+<ref name=Buckley>{{cite book |last1=Buckley |first1=Fred |last2=Harary |first2=Frank |author-link2=Frank Harary |date=21 January 1990 |title=Distance in Graphs |publisher=[[Addison-Wesley]] |page=109 |isbn=978-0-201-09591-3}}</ref>
-<ref name=Oxtoby>{{cite book |last=Oxtoby |first=John C. |date=1980 |title=Measure and Category |series=[[Graduate Texts in Mathematics]] |volume=2 |edition=2nd |publisher=[[Springer Science+Business Media|Springer]] |place=United States |page=68 |isbn=978-0-387-90508-2}} Cited in {{cite book |last1=Graham |first1=Ronald |author-link1=Ronald Graham |last2=Grötschel |first2=Martin |author-link2=Martin Grötschel |last3=Lovász |first3=László |author-link3=László Lovász |date={{date|1995-12-25}} |title=Handbook of Combinatorics |volume=2 |publisher=[[North-Holland Publishing Company]] |place=Netherlands |page=1474 |isbn=978-0-444-82351-9}}</ref>
+<ref name=Oxtoby>{{cite book |last=Oxtoby |first=John C. |date=1980 |title=Measure and Category |series=[[Graduate Texts in Mathematics]] |volume=2 |edition=2nd |publisher=[[Springer Science+Business Media|Springer]] |place=United States |pages=59,68 |isbn=978-0-387-90508-2}} While Oxtoby does not explicitly define the term there, [[László Babai|Babai]] has borrowed it from ''Measure and Category'' in his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham, [[Martin Grötschel|Grötschel]] and [[László Lovász|Lovász]]'s ''Handbook of Combinatorics'' (vol. 2), and Broer and [[Floris Takens|Takens]] note in their book ''Dynamical Systems and Chaos'' that ''Measure and Category'' compares this meaning of "almost all" to the measure theoretic one in the real line (though Oxtoby's book discusses meagre sets in general topological spaces as well).</ref>
-<ref name=Baratchart>{{cite book |last=Baratchart |first=Laurent |editor-last=Curtain |editor-first=Ruth F. |editor-link=Ruth F. Curtain |date=1987 |title=Modelling, Robustness and Sensitivity Reduction in Control Systems |series=NATO ASI Series F |volume=34 |publisher=[[Springer Science+Business Media|Springer]] |page=123 |doi=10.1007/978-3-642-87516-8 |isbn=978-3-642-87516-8}}</ref>
+<ref name=Baratchart>{{cite book |last=Baratchart |first=Laurent |editor-last=Curtain |editor-first=Ruth F. |editor-link=Ruth F. Curtain |date=1987 |title=Modelling, Robustness and Sensitivity Reduction in Control Systems |series=NATO ASI Series F |volume=34 |publisher=[[Springer Science+Business Media|Springer]] |page=123 |chapter=Recent and New Results in Rational L<sup>2</sup> Approximation |doi=10.1007/978-3-642-87516-8 |isbn=978-3-642-87516-8}}</ref>
-<ref name=Broer>{{cite book |last1=Broer |first1=Henk |last2=Takens |first2=Floris |author-link2=Floris Takens |date={{date|2010-10-28}} |title=Dynamical Systems and Chaos |series=Applied Mathematical Sciences |volume=172 |publisher=[[Springer Science+Business Media|Springer]] |page=245 |doi=10.1007/978-1-4419-6870-8 |isbn=978-1-4419-6870-8}}</ref>
+<ref name=Broer>{{cite book |last1=Broer |first1=Henk |last2=Takens |first2=Floris |author-link2=Floris Takens |date=28 October 2010 |title=Dynamical Systems and Chaos |series=Applied Mathematical Sciences |volume=172 |publisher=[[Springer Science+Business Media|Springer]] |page=245 |doi=10.1007/978-1-4419-6870-8 |isbn=978-1-4419-6870-8}}</ref>
-<ref name=Sharkovsky>{{cite book |last1=Sharkovsky |first1=A. N. |last2=Kolyada |first2=S. F. |last3=Sivak |first3=A. G. |last4=Fedorenko |first4=V. V. |date={{date|1997-04-30}} |title=Dynamics of One-Dimensional Maps |series=Mathematics and Its Applications |volume=407 |publisher=[[Springer Science+Business Media|Springer]] |page=33 |doi=10.1007/978-94-015-8897-3 |isbn=978-94-015-8897-3}}</ref>
+<ref name=Sharkovsky>{{cite book |last1=Sharkovsky |first1=A. N. |last2=Kolyada |first2=S. F. |last3=Sivak |first3=A. G. |last4=Fedorenko |first4=V. V. |date=30 April 1997 |title=Dynamics of One-Dimensional Maps |series=Mathematics and Its Applications |volume=407 |publisher=[[Springer Science+Business Media|Springer]] |page=33 |doi=10.1007/978-94-015-8897-3 |isbn=978-94-015-8897-3}}</ref>
-<ref name=Yuan>{{cite book |last=Yuan |first=George Xian-Zhi |date={{date|1999-02-09}} |title=KMM Theory and Applications in Nonlinear Analysis |series=Pure and Applied Mathematics; A Series of Monographs and Textbooks |publisher=[[Marcel Dekker]] |page=21 |isbn=978-0-8247-0031-7}}</ref>
+<ref name=Yuan>{{cite book |last=Yuan |first=George Xian-Zhi |date=9 February 1999 |title=KKM Theory and Applications in Nonlinear Analysis |series=Pure and Applied Mathematics; A Series of Monographs and Textbooks |publisher=[[Marcel Dekker]] |page=21 |isbn=978-0-8247-0031-7}}</ref>
-<ref name=Albertini>{{cite book |last1=Albertini |first1=Francesca |last2=Sontag |first2=Eduardo D. |author-link2=Eduardo D. Sontag |editor-last1=Bonnard |editor-first1=Bernard |editor-last2=Bride |editor-first2=Bernard |editor-last3=Gauthier |editor-first3=Jean-Paul |editor-last4=Kupka |editor-first4=Ivan |date={{date|1991-09-01}} |title=Analysis of Controlled Dynamical Systems |series=Progress in Systems and Control Theory |volume=8 |publisher=[[Birkhäuser]] |page=29 |chapter=Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems |doi=10.1007/978-1-4612-3214-8 |isbn=978-1-4612-3214-8}}</ref>
+<ref name=Albertini>{{cite book |last1=Albertini |first1=Francesca |last2=Sontag |first2=Eduardo D. |author-link2=Eduardo D. Sontag |editor1-last=Bonnard |editor1-first=Bernard |editor2-last=Bride |editor2-first=Bernard |editor3-last=Gauthier |editor3-first=Jean-Paul |editor4-last=Kupka |editor4-first=Ivan |date=1 September 1991 |title=Analysis of Controlled Dynamical Systems |series=Progress in Systems and Control Theory |volume=8 |publisher=[[Birkhäuser]] |page=29 |chapter=Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems |doi=10.1007/978-1-4612-3214-8 |isbn=978-1-4612-3214-8}}</ref>
-<ref name=Fuente>{{cite book |last=De la Fuente |first=Angel |date={{date|2000-01-28}} |title=Mathematical Models and Methods for Economists |publisher=[[Cambridge University Press]] |page=217 |isbn=978-0-521-58529-3}}</ref>
+<ref name=Fuente>{{cite book |last=De la Fuente |first=Angel |date=28 January 2000 |title=Mathematical Models and Methods for Economists |publisher=[[Cambridge University Press]] |page=217 |isbn=978-0-521-58529-3}}</ref>
-<ref name=Komjath>{{cite book |last1=Komjáth |author-link1=Péter Komjáth |first1=Péter |last2=Totik |first2=Vilmos |author-link2=Vilmos Totik |date={{date|2006-05-02}} |title=Problems and Theorems in Classical Set Theory |series=Problem Books in Mathematics |publisher=[[Springer Science+Business Media|Springer]] |place=United States |page=75 |isbn=978-0387-30293-5}}</ref>
+<ref name=Komjath>{{cite book |last1=Komjáth |author-link1=Péter Komjáth |first1=Péter |last2=Totik |first2=Vilmos |author-link2=Vilmos Totik |date=2 May 2006 |title=Problems and Theorems in Classical Set Theory |series=Problem Books in Mathematics |publisher=[[Springer Science+Business Media|Springer]] |place=United States |page=75 |isbn=978-0387-30293-5}}</ref>
-<ref name=Salzmann>{{cite book |last1=Salzmann |first1=Helmut |last2=Grundhöfer |first2=Theo |last3=Hähl |first3=Hermann |last4=Löwen |first4=Rainer |date={{date|2007-09-24}} |title=The Classical Fields: Structural Features of the Real and Rational Numbers |series=Encyclopedia of Mathematics and Its Applications |volume=112 |publisher=[[Cambridge University Press]] |page=155 |isbn=978-0-521-86516-6}}</ref>
+<ref name=Salzmann>{{cite book |last1=Salzmann |first1=Helmut |last2=Grundhöfer |first2=Theo |last3=Hähl |first3=Hermann |last4=Löwen |first4=Rainer |date=24 September 2007 |title=The Classical Fields: Structural Features of the Real and Rational Numbers |series=Encyclopedia of Mathematics and Its Applications |volume=112 |publisher=[[Cambridge University Press]] |page=[https://archive.org/details/classicalfieldss0000unse/page/155 155] |isbn=978-0-521-86516-6 |url=https://archive.org/details/classicalfieldss0000unse/page/155 }}</ref>
-<ref name=Schoutens>{{cite book |last=Schoutens|first=Hans |date={{date|2010-08-02}} |title=The Use of Ultraproducts in Commutative Algebra |series=[[Lecture Notes in Mathematics]] |volume=1999 |publisher=[[Springer Science+Business Media|Springer]] |page=8 |doi=10.1007/978-3-642-13368-8 |isbn=978-3-642-13367-1}}</ref>
+<ref name=Schoutens>{{cite book |last=Schoutens|first=Hans |date=2 August 2010 |title=The Use of Ultraproducts in Commutative Algebra |series=[[Lecture Notes in Mathematics]] |volume=1999 |publisher=[[Springer Science+Business Media|Springer]] |page=8 |doi=10.1007/978-3-642-13368-8 |isbn=978-3-642-13367-1}}</ref>
-<ref name=Rautenberg>{{cite book |last=Rautenberg |first=Wolfgang |author-link=Wolfgang Rautenberg |date={{date|2009-12-17}} |title=A Concise to Mathematical Logic |series=Universitext |edition=3rd |publisher=[[Springer Science+Business Media|Springer]] |pages=210-212 |doi=10.1007/978-1-4419-1221-3 |isbn=978-1-4419-1221-3}}</ref>
+<ref name=Rautenberg>{{cite book |last=Rautenberg |first=Wolfgang |author-link=Wolfgang Rautenberg |date=17 December 2009 |title=A Concise to Mathematical Logic |series=Universitext |edition=3rd |publisher=[[Springer Science+Business Media|Springer]] |pages=210–212 |doi=10.1007/978-1-4419-1221-3 |isbn=978-1-4419-1221-3}}</ref>
 }}
 ===Secondary sources===
 {{reflist |group=sec |refs=
+<ref name=RealTrans>{{Cite web|url=https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Transcendental|title=Almost All Real Numbers are Transcendental - ProofWiki|website=proofwiki.org|access-date=2019-11-11}}</ref>
-<ref name=Schwartzman>{{cite book |last=Schwartzman |first=Steven |date={{date|1994-05-01}} |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English |series=Spectrum Series |publisher=[[Mathematical Association of America]] |page=22 |isbn=978-0-88385-511-9}}</ref>
-<ref name=James>{{cite book |last=James |first=Robert C. |date={{date|1992-07-31}} |title=Mathematics Dictionary |edition=5th |publisher=[[Chapman & Hall]] |page=269 |isbn=978-0-412-99031-1}}</ref>
+<ref name=Schwartzman>{{cite book |last=Schwartzman |first=Steven |date=1 May 1994 |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English |url=https://archive.org/details/wordsofmathemati0000schw |url-access=registration |series=Spectrum Series |publisher=[[Mathematical Association of America]] |page=[https://archive.org/details/wordsofmathemati0000schw/page/22 22] |isbn=978-0-88385-511-9}}</ref>
+<ref name=Clapham>{{cite book |last1=Clapham |first1=Christopher |last2=Nicholson |first2=James |date=7 June 2009 |title=The Concise Oxford Dictionary of mathematics |series=Oxford Paperback References |edition=4th |page=38 |publisher=[[Oxford University Press]] |isbn=978-0-19-923594-0}}</ref>
-<ref name=Bityutskov>{{cite book |last=Bityutskov |first=Vadim I. |editor-last=Hazewinkel |editor-first=Michiel |editor-link=Michiel Hazewinkel |date={{date|1987-11-30}} |title=[[Encyclopaedia of Mathematics]] |volume=1 |publisher=[[Kluwer Academic Publishers]] |page=153 |chapter=Almost-everywhere |chapter-url=http://www.encyclopediaofmath.org/index.php?title=Almost-everywhere&oldid=31533 |doi=10.1007/978-94-015-1239-8 |isbn=978-94-015-1239-8}}</ref>
-<ref name=Ito2>{{cite book |editor-last=Itô |editor-first=Kiyosi |editor-link=Kiyosi Itô |date={{date|1993-06-04}} |title=[[Encyclopedic Dictionary of Mathematics]] |edition=2nd |volume=2 |publisher=[[MIT Press]] |place=Kingsport |page=1267 |isbn=978-0-262-09026-1}}</ref>
+<ref name=James>{{cite book |last=James |first=Robert C. |author-link=Robert C. James |date=31 July 1992 |title=Mathematics Dictionary |edition=5th |publisher=[[Chapman & Hall]] |page=269 |isbn=978-0-412-99031-1}}</ref>
+<ref name=Bityutskov>{{cite book |last=Bityutskov |first=Vadim I. |editor-last=Hazewinkel |editor-first=Michiel |editor-link=Michiel Hazewinkel |date=30 November 1987 |title=[[Encyclopaedia of Mathematics]] |volume=1 |publisher=[[Kluwer Academic Publishers]] |page=153 |chapter=Almost-everywhere |chapter-url=http://www.encyclopediaofmath.org/index.php?title=Almost-everywhere&oldid=31533 |doi=10.1007/978-94-015-1239-8 |isbn=978-94-015-1239-8}}</ref>
-<ref name=Weisstein>{{MathWorld|title=Almost All|urlname=AlmostAll}} See also {{cite book |last=Weisstein |first=Eric W. |author-link=Eric W. Weisstein |date={{date|1988-11-25}} |title=CRC Concise Encyclopedia of Mathematics |url=https://ia802601.us.archive.org/26/items/CrcEncyclopediaOfMathematics/Weisstein__Eric_W_-_CRC_Concise_Encyclopedia_Mathematics_Vol_1__CRC_1999__4AH.pdf |edition=1st |publisher=[[CRC Press]] |page=41 |isbn=978-0-8493-9640-3}}</ref>
-<ref name=Ito1>{{cite book |editor-last=Itô |editor-first=Kiyosi |editor-link=Kiyosi Itô |date={{date|1993-06-04}} |title=Encyclopedic Dictionary of Mathematics |url=https://archive.org/stream/Ito_Kiyoso_-_Encyclopedic_Dictionary_Of_Math_Volume_1#page/n85/mode/2up |edition=2nd |volume=1 |publisher=[[MIT Press]] |place=Kingsport |page=67 |isbn=978-0-262-09026-1}}</ref>
+<ref name=Ito2>{{cite book |editor-last=Itô |editor-first=Kiyosi |editor-link=Kiyosi Itô |date=4 June 1993 |title=[[Encyclopedic Dictionary of Mathematics]] |edition=2nd |volume=2 |publisher=[[MIT Press]] |place=Kingsport |page=1267 |isbn=978-0-262-09026-1}}</ref>
+<ref name=Weisstein>{{MathWorld|title=Almost All|urlname=AlmostAll}} See also {{cite book |last=Weisstein |first=Eric W. |author-link=Eric W. Weisstein |date=25 November 1988 |title=CRC Concise Encyclopedia of Mathematics |url=https://archive.org/details/CrcEncyclopediaOfMathematics |edition=1st |publisher=[[CRC Press]] |page=41 |isbn=978-0-8493-9640-3}}</ref>
+<ref name=Ito1>{{cite book |editor-last=Itô |editor-first=Kiyosi |editor-link=Kiyosi Itô |date=4 June 1993 |title=Encyclopedic Dictionary of Mathematics |url=https://archive.org/stream/Ito_Kiyoso_-_Encyclopedic_Dictionary_Of_Math_Volume_1#page/n85/mode/2up |edition=2nd |volume=1 |publisher=[[MIT Press]] |place=Kingsport |page=67 |isbn=978-0-262-09026-1}}</ref>
 }}