axiom of power set

The axiom of choice differs from other axioms of ZF by stating existence of a set (i.e., a choice function) without defining it (unlike, for instance, the axiom of pairing or the axiom of power set).

1978, Thomas Jech, Set Theory, Academic Press, page 38:

Verifying that the axiom of power set is in #92;#123;#92;phi#92;in#92;mathcal#123;L#125;#42;#58;S#92;vdash#92;mathcal#123;I#125;(#92;phi)#92;#125; relies on some rudimentary comprehension axioms.

2003, Thomas Forster, Reasoning About Theoretical Entities, World Scientific, page 51:

But the ZF axioms of which the hierarchy is an intuitive model involve impredicative quantifications. Most striking is the axiom of power set in tandem with the axiom of separation.

2011, Adam Rieger, “9: Paradox, ZF, and the Axiom of Foundation”, in David DeVidi, Michael Hallett, Peter Clark, editors, Logic, Mathematics, Philosophy: Vintage Enthusiasms: Essays in Honour of John L. Bell, Springer, page 183:

2012, A. H. Lightstone, H. B. Enderton (editor), Mathematical Logic: An Introduction to Model Theory, Plenum Press, Softcover, page 292,
The Axiom of Power Set asserts that the collection of all subsets of a set is a set. […] Adding the Axiom of Power Set compels the collection empty to be a set.

(set theory) The axiom that the power set of any set exists and is a valid set, which appears in the standard axiomatisation of set theory, ZFC.


order type

order types (plural)
order-type (alternative)
ordertype (alternative)

1965 [John Wiley], Raymond L. Wilder, Introduction to the Foundations of Mathematics: 2nd Edition, 2012, Dover, page 116,
Another way of putting this is to state that the order type is that aspect of the arrangement of the elements of a simply ordered set, which remains unchanged when any two elements are exchanged. […] As in the case of cardinal numbers, order types may be denoted by suitable symbols called ordinal numerals.

In [13] Chajoth studied how the order type of a chain can alter if we change the position of elements in a linearly ordered set, resp. if we introduce a new element in a linearly ordered set.

2005, Egbert Harzheim, Ordered Sets, Springer, page 332:

2011, Douglas Cenzer, Valentina Harizanov, Jeffrey B. Remmel, Effective Categoricity of Injection Structures, Benedikt Löwe, Dag Normann, Ivan Soskov, Alexandra Soskova (editors, Models of Computation in Context: 7th Conference on Computability in Europe, CiE 2011, Proceedings, Springer, LNCS 6735, page 51,
We let ω denote the order type of N under the usual ordering and Z denote the order type of Z under the usual ordering.

equivalence relation
equivalence relation

(set theory) In the context of sets equipped with an order (especially, the context of totally ordered sets), the characteristic of being a member of some equivalence class of such sets under the equivalence relation "existence of an order-preserving bijection".

järjestystyyppi

characteristic of being a member of an equivalence class of ordered sets


axiom of countable choice

The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable. One example is the axiom of countable choice, which states that if A#95;0,A#95;1,#92;dotsA#95;n#92;dots form a denumerable set of nonempty sets, their product is nonempty.[…]The axiom of countable choice is constantly used in analysis; it is often hidden so as not to sow confusion in the minds of the students (who are inclined to accept anything desired) or of the professors (who do not like to shake the foundations of the discipline).

2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, page 169:

we have no way to infer
∃R∀n[P0n→∀x(R_nx→Fx)∧n=Nx:R_nx)]
without an axiom of countable choice.

2012, Richard G. Heck, Jr., Reading Frege's Grundgesetze, Oxford University Press, page 271, But, once again, while we can easily prove ∀n[P0n→∃G(∀x(Gx→Fx)∧n=Nx:Gx)]

2013, Valentin Blot, Colin Riba, On Bar Recursion and Choice in a Classical Setting, Chung-chien Shan (editor), Programming Languages and Systems: 11th International Symposium, APLAS 2013, Proceedings, Springer, LNCS 8301, page 349,
We show how Modified Bar-Recursion, a variant of Spector's Bar-Recursion due to Berger and Oliva can be used to realize the Axiom of Countable Choice in Parigot's Lambda-Mu-calculus, a direct-style language for the representation and evaluation of classical proofs.

(set theory) A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function; equivalently, the statement that the direct product of a countable collection of nonempty sets is nonempty.


Zermelo set theory

After German mathematician Ernst Zermelo, who introduced the theory in a paper in 1908.

1998, Geoffrey Hellman, 8: Beyond Definitionism—But Not Too Far Beyond, Matthias Schirn (editor), 2003, Paperback, The Philosophy of Mathematics Today, Oxford University Press (Clarendon Press), page 215,
The system can be claimed to transcend certain critical limitations of predicative analysis and certain awkwardness of known nominalistic approaches to mathematics at once, and would seem to provide a well-motivated alternative to Zermelo set theory as a framework for scientifically applicable mathematics.

According to the great logician Alonzo Church, in spite of superficial differences, these two "widely accepted symbolic systems," Zermelo set theory and simple type theory, are "in their currently accepted forms essentially similar" [1939, 69-70].

2001, José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Springer (Birkhäuser), page 338:

Z is a formal specification language founded on Zermelo set theory.

2006, Gerard O'Regan, Mathematical Approaches to Software Quality, Springer, page 20:

(set theory) An early and influential axiomatisation of set theory.


axiom of regularity

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that every non-empty set contains a member that is disjoint from that set.


axiom of pairing

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that if two sets exist, there exists a set with those two sets as its sole elements.


axiom of union

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that a union exists for any set, containing exactly all the elements contained within the sets within that set.


axiom of infinity

(set theory) One of the axioms in axiomatic set theory that guarantees the existence of an infinite set.


Sperner's theorem

Named after Emanuel Sperner, who published it in 1928.

(set theory) A theorem that describes the largest possible families of finite sets none of which contain any other sets in the family.


transfinite induction

en-noun

transfinite induction (plural transfinite inductions)

The validity of the principle of transfinite induction for well-ordered sets enables us to carry out proofs by transfinite induction and definitions by transfinite induction. A proof by transfinite induction is a direct application of the principle when it is required to show that each element of a well-ordered set A has a property P.[…]To understand the method of definition by transfinite induction some preparation is necessary.

1967, Kam-Tim Leung, Doris Lai-chue Chen, Elementary Set Theory, Parts I and II, Hong Kong University Press, page 108:

1970 [Addison-Wesley], Howard DeLong, A Profile of Mathematical Logic, Dover, 2004, page 218,
Just what kinds of transfinite inductions are to be considered finitary is debatable. Transfinite induction up to an arbitrary ordinal is certainly not finitary. However, it can be shown that certain transfinite inductions are reducible to ordinary mathematical inductions. For example, induction up to ω^ω is reducible to ordinary induction. Gentzen in his proof used transfinite induction up to ε₀.

The published version of 1936 had a different proof based on the famous principle of transfinite induction up to the ordinal #92;varepsilon#95;0 by which consistency^([of Peano arithmetic]) followed. A third proof of 1938 used transfinite induction but the logical system was a sequent calculus.

2009, Jan von Plato, “Gentzen's Logic”, in Dov M. Gabbay, John Woods, editors, Handbook of the History of Logic, Volume 5: Logic from Russell to Church, Elsevier (North-Holland), page 667:

mathematical induction
mathematical induction

(mathematics, set theory) An extension of mathematical induction to well-ordered sets of transfinite cardinality, such as sets of ordinal numbers or cardinal numbers.


epsilon number

From the Greek letter ε (“epsilon”), used to denote the numbers.

More generally, the epsilon numbers are the ordinals #92;varepsilon for which #92;varepsilon#61;#92;omega#92;varepsilon. The smallest epsilon number is #92;varepsilon#95;0. It is a countable ordinal, being the countable union of countable sets. By the Veblen fixed-point theorem, the class of epsilon numbers is unbounded.

1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240:

We show that the associated ordinals are the #92;varepsilon#95;0th epsilon number and the first #92;varepsilon#95;0-critical number, respectively.

1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17:

2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, [Revision of 1971 Addison-Wesley edition], page 203,
Thus there is at least one epsilon number, namely ε₀; we can easily show, in fact, that ε₀ is the least epsilon number.

(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.


L-fuzzy

(set theory) Relating to a generalized variant of the notion of fuzzy sets, with membership functions taking values in a (fixed or variable) algebra or structure L of a given kind, usually at least a poset or lattice.


Goguen category

Goguen category (plural Goguen categories)

(set theory) A fuzzy kind of category in which membership is determined not in binary (true-or-false) fashion but by a more general interval.


Zermelo's theorem

Ernst Zermelo
Zermelo's theorem
Zermelo's theorem (game theory)


supertransitive

(set theory) Describing a transitive set that contains all subsets of all its elements


π-system

π-systems (plural)
pi-system (alternative)

To see this, first check that #92;sigma(X#95;0,X#95;1,#92;dots)#61;#92;sigma(#92;mathcalF#95;0), where #92;textstyle#92;mathcalF#95;0#58;#61;#92;bigcup#92;infty#95;#123;k#61;0#125;#92;sigma(X#95;0,#92;dots,X#95;k) is a field and, in particular, a #92;boldsymbol#92;pi-system.

2007, Rabi Bhattacharya, Edward C. Waymire, A Basic Course in Probability Theory, Springer, page 49:

We start with a basis of simple roots #92;Delta of #92;Phi. Then we apply all possible elementary transformations and add the resulting #92;boldsymbol#92;pi-systems to the list. Of course, if #92;Gamma is a #92;boldsymbol#92;pi-system, and #92;Gamma' is a #92;boldsymbol#92;pi-system obtained from it by an elementary transformation and the diagrams of #92;Gamma and #92;Gamma' are the same, the root subsystems they span are the same, and therefore we do not add #92;Gamma'.

2017, Willem Adriaan de Graaf, Computation with Linear Algebraic Groups, Taylor & Francis (CRC Press), page 221:

Clearly the definitions for a #92;boldsymbol#92;pi-system and a #92;lambda-system are both satisfied by a #92;sigma-algebra.[…]
Proposition 4.1.8 Let #92;Omega be a set and B be a collection of subsets of #92;Omega. The collection B is a #92;sigma-algebra if and only if B is a #92;lambda-system and a #92;boldsymbol#92;pi-system.

2021, Jeremy J. Becnel, Tools for Infinite Dimensional Analysis, Taylor & Francis (CRC Press):

(set theory, measure theory, probability theory) A non-empty collection of subsets of a given set Ώ that is closed under non-empty finite intersections.

π-система

collection of subsets closed under non-empty finite intersections


Cantor's theorem

Named after German mathematician George Cantor (1845-1918) who stated and proved this theorem.

(set theory) A theorem in set theory which states for any set A, the set of all subsets of A (the power set of A, denoted by 𝒫(A)) has a strictly greater cardinality than A itself.


limit cardinal

(set theory) A cardinal number that cannot be reached from another by repeated successor operations (a weak limit cardinal) or by repeated power set operations (a strong limit cardinal).


left total

Coordinate term: right total

(set theory, of a binary relation R on A×B) Such that every element of the domain is related to at least one element of the codomain: such that ∀a∈A;;∃b∈B:(a,b)∈R


right total

Coordinate term: left total

(set theory) Of a binary relation, to have every element of the right set occur at least once: ∀b∈B∃a∈A:(a,b)∈R⊆A×B


ramified forcing

axiom of constructibility
axiom of constructibility

Zermelo-Fraenkel set theory
Zermelo-Fraenkel set theory

partially ordered set
partially ordered set

(set theory) The original form of forcing, starting with a model M of set theory in which the axiom of constructibility, V = L, holds, and then building up a larger model M[G] of Zermelo-Fraenkel set theory by adding a generic subset G of a partially ordered set to M, imitating Kurt Gödel's constructible hierarchy.


Cartesian square

Cartesian square (plural Cartesian squares)

(set theory) For a set X, the Cartesian product X² = X × X.


Hausdorff gap

Named after German mathematician Felix Hausdorff (1868–1942), who published proof of the first example in 1909.

The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.^([Wikipedia])

Of course, Hausdorff did not talk of models of set theory or prove absoluteness results. He gave a direct construction in ZFC of what is now called a Hausdorff gap and proved properties 1 through 4 for his construction. The notion of a Hausdorff gap and the proof that Hausdorff gaps are indestructible under cardinal preserving extensions (property 5 above) are due to Kunen (unpublished).

1995, Robert M. Solovay, “*1970 [Godel's ontological proof]”, in Kurt Gödel, edited by Solomon Feferman, John W. Dawson Jr., Warren Goldfarb, Charles Parsons, and Robert M. Solovay, Kurt Gödel: Collected Works: Volume III, Oxford University Press, page 419:

Let #92;langle#92;langlea#95;#92;epsilon#58;#92;epsilonlt;#92;omega#95;1#92;rangle#92;langleb#95;#92;epsilon#58;#92;epsilonlt;#92;omega#95;1#92;rangle#92;rangle be a Hausdorff gap.

1997, Winfried Just, Martin Weese, Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 118:

2013, Ilijas Farah, Eric Wofsey, 3: Set theory and operator algebras, James Cummings, Ernest Schimmerling (editors), Appalachian Set Theory: 2006-2012, London Mathematical Society, Cambridge University Press, page 101,
This family is one of the instances of incompactness of ω₁ that are provable in ZFC, along with Hausdorff gaps, special Aronszajn trees, or nontrivial coherent families of partial functions.

(set theory, order theory) A pair of collections of integer sequences such that there is no integer sequence lying between the two.


omegath

(set theory, rare) In the position known as omega (a transfinite ordinal number referring to the next position after ordering a countably infinite set).


infinite hotel paradox

(set theory) A thought experiment illustrating a counterintuitive property of infinite sets, showing that a fully occupied hotel with infinitely many rooms can still accommodate any number of additional guests.


Lévy hierarchy

(set theory, logic) A hierarchy of formulas in the formal language of the Zermelo-Fraenkel set theory. Its first level contains only formulas with no unbounded quantifiers and is denoted by Δ₀=Σ₀=Π₀. Subsequent levels are given by finding a formula in prenex normal form which is provably equivalent over ZFC, and counting the number of changes of quantifiers.


semiset

(set theory) In set theory, a semiset is a proper class that is a subclass of a set.


finite set

(set theory) A set with only finitely many members. This includes the empty set.


clubsuit

(mathematics, set theory) A family of combinatorial principles that is a weaker version of the corresponding diamond principle.


abstract polytope

abstract polytope (plural abstract polytopes)

(set theory, order theory) A partially ordered set that captures the dyadic properties of a normal polytope without specifying its geometric properties.


complete partial order

complete partial order (plural complete partial orders)

(set theory, order theory) Three similar yet distinct classes of partially ordered sets, characterized by their completeness.