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Technique invented by Paul Cohen for proving consistency and independence results
In set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand
Forcing_(mathematics)
In mathematics, forcing is a method of constructing new models M[G] of set theory by adding a generic subset G of a poset P to a model M. The poset P used
List_of_forcing_notions
In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable
Proper_forcing_axiom
Collection of sets in which every two sets have the same intersection
Unsolved problem in mathematics For any sunflower size, does every set of uniformly sized sets which is of cardinality greater than some exponential in
Sunflower_(mathematics)
Topics referred to by the same term
Look up forcing in Wiktionary, the free dictionary. Forcing may refer to: Forcing (mathematics), a technique for obtaining independence proofs for set
Forcing
In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by Cohen (1963) to prove the independence of
Ramified_forcing
Method for constructing models of set theory
In the mathematical discipline of set theory, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a
Iterated_forcing
Condition in order theory and topology
the statement of Martin's axiom. In the theory of forcing, ccc partial orders are used because forcing with any generic set over such an order preserves
Countable_chain_condition
Mathematical theory
unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra was studied
Random_algebra
Proposition in mathematical logic
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Continuum_hypothesis
Subfield of mathematics
of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. The mathematical field
Mathematical_logic
Mathematical lemma
one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a poset (P, ≤) is called dense in P if for any
Rasiowa–Sikorski_lemma
Branch of mathematics that studies sets
of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins
Set_theory
American philosopher
among the "ten best" of their year by the Philosopher's Annual. Forcing (mathematics) Kenny Easwaran at Texas A&M University "The Philosopher's Annual"
Kenny_Easwaran
Set theory concept
syntactic forcing A forcing relation p ⊩ ϕ {\displaystyle p\Vdash \phi } is defined between elements p of the poset and formulas φ of the forcing language
Boolean-valued_model
A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a symbol,
Mathematical_object
a branch of T {\displaystyle T} . A forcing notion is said to have the Laver property if and only if the forcing extension has the Laver property over
Laver_property
In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete. The countable
Cantor_algebra
Difference in electric potential between two points in space
the electric field is not conservative. For more, see Conservative force § Mathematical description. For example, in the Lorenz gauge, the electric potential
Voltage
of a notion of forcing that preserves stationary subsets of ω1, then there is a D-generic filter. Forcing with a ccc notion of forcing preserves stationary
Martin's_maximum
In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially
Generic_filter
Mathematical theorem in set theory
{\displaystyle G} . The proof of Easton's theorem uses forcing with a proper class of forcing conditions over a model satisfying the generalized continuum
Easton's_theorem
Boolean algebra with all operators and laws forming a complete logical system
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are
Complete_Boolean_algebra
name is used in forcing to impose an upper bound on the number of subsets in the generic model. It is used in the context of forcing to prove independence
Nice_name
In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used
Collapsing_algebra
Type of Boolean algebra
In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in the theory of forcing. A Cohen algebra is a Boolean
Cohen_algebra
Basic framework of mathematics
Foundations of mathematics are the logical and mathematical frameworks that allow the development of mathematics without generating self-contradictory
Foundations_of_mathematics
Standard system of axiomatic set theory
Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create
Zermelo–Fraenkel_set_theory
Criticism of non-standard analysis Standard part function Set theory Forcing (mathematics) Boolean-valued model Kripke semantics General frame Predicate logic
List of mathematical logic topics
List_of_mathematical_logic_topics
Infinite cardinal number
beginner's guide to forcing". arXiv:0712.1320 [math.LO]. Harris, Kenneth A. (April 6, 2009). "Lecture 31" (PDF). Department of Mathematics. kaharris.org. Intro
Aleph_number
Collection of mathematical objects
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects:
Set_(mathematics)
Algorithm for training neural networks
into the RNN after each step, thus forcing the RNN to stay close to the ground-truth sequence. The term "teacher forcing" can be motivated by comparing the
Teacher_forcing
In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition
Suslin_algebra
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern
History_of_mathematics
invariants in forcing arguments. It is named for Gerald Enoch Sacks. A forcing notion is said to have the Sacks property if and only if the forcing extension
Sacks_property
In mathematics, the amoeba order is the partial order of open subsets of 2ω of measure less than 1/2, ordered by reverse inclusion. Amoeba forcing is
Amoeba_order
In mathematics, a closed preordered set is one whose anti-well-ordered subsets have lower bounds. Let κ {\displaystyle \kappa } be a cardinal. A preordered
Closed_preordered_set
Form of mathematical proof
Mathematical induction is a method for proving that a statement P ( n ) {\displaystyle P(n)} is true for every natural number n {\displaystyle n} , that
Mathematical_induction
Relationships between music and mathematics
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and
Music_and_mathematics
Reasoning for mathematical statements
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The
Mathematical_proof
American mathematician (1942–2016)
California–Berkeley Jack Silver at the Mathematics Genealogy Project Cummings, James (2009). "Iterated Forcing and Elementary Embeddings". In Handbook
Jack_Silver
American mathematician
lottery preparation as a general method of forcing indestructibility. Hamkins introduced the modal logic of forcing and proved with Benedikt Löwe that if ZFC
Joel_David_Hamkins
Symbol representing a mathematical object
In mathematics, a variable (from Latin variabilis 'changeable') is a symbol, typically a letter, that refers to an unspecified mathematical object. One
Variable_(mathematics)
Theorem for proving more complex theorems
In mathematics and other fields, a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement.
Lemma_(mathematics)
Branch of mathematical logic
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining
Reverse_mathematics
Catalan mathematician
contributions concerning forcing, large cardinals, infinite combinatorics and their applications to other areas of mathematics. Bagaria was born in 1958
Joan_Bagaria
Method using forcing to construct sets with desired properties in computability theory
Forcing in computability theory is a modification of Paul Cohen's original set-theoretic technique of forcing to deal with computability concerns. Conceptually
Forcing_(computability)
Index of articles associated with the same name
Stratification has several usages in mathematics. In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing
Stratification_(mathematics)
Czech mathematician
theory, with particular emphasis on the method of forcing, Springer-Verlag Lecture Notes in Mathematics 217 (1971) (ISBN 978-3540055648) The axiom of choice
Thomas_Jech
Overview of and topical guide to logic
Effective enumeration Element (mathematics) Empty function Empty set Enumeration Extensionality Finite set Forcing (mathematics) Function (set theory) Function
Outline_of_logic
3-volume treatise on mathematics, 1910–1913
(often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and
Principia_Mathematica
Branch of applied mathematics
development of mathematical ideas inspired by physics, known as physical mathematics. There are several distinct branches of mathematical physics, and these
Mathematical_physics
Influence that can change motion of an object
mechanics, force makes ideas like pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector
Force
Statement that is taken to be true
modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical
Axiom
Mathematician (1845–1918)
the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between
Georg_Cantor
Coincidence in mathematics
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation
Mathematical_coincidence
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and
Matrix_(mathematics)
Development of mathematics in South Asia
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400
Indian_mathematics
Property of two varying quantities with a constant ratio
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant
Proportionality_(mathematics)
Concept in model theory
v t e Mathematical logic General Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma
Strength_(mathematical_logic)
Basic notion of sameness in mathematics
In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical
Equality_(mathematics)
1960 article by Eugene Wigner
Unreasonable Effectiveness of Mathematics in the Natural Sciences" was the title of the 1959 Richard Courant Lecture in Mathematical Sciences, delivered at New
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
Symbolic description of a mathematical object
In mathematics, an expression is an arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can
Expression_(mathematics)
Additional mathematical object
In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation
Mathematical_structure
Class of graphs
1989. Forcing graphs play an important role in the study of pseudorandomness in graph sequences. The forcing conjecture states that the forcing graphs
Forcing_graph
Mathematical set containing no elements
In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic
Empty_set
Polish mathematician and statistician (1932–2008)
convention, Trybula transfers, Wesolowski texas, Gawrys fourth suit forcing). "Mathematics Genealogy Project". Retrieved 9 February 2021. "Grave record for
Stanisław_Trybuła
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly
Philosophy_of_mathematics
Axiom of set theory
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection
Axiom_of_choice
American mathematician
countable. This important independence result was the first when a forcing (see Laver forcing), adding a real, was iterated with countable support iteration
Richard_Laver
Limitative results in mathematical logic
published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics. The theorems are interpreted as showing that Hilbert's
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
2.71828...; base of natural logarithms
The number e is a mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm and exponential function. It is sometimes
E_(mathematical_constant)
Laws in physics about force and motion
by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), originally published in 1687. Newton
Newton's_laws_of_motion
Broad concept generalizing scalars in mathematics and physics
In mathematics and physics, a vector is a generalization of a single number. It may denote a vector quantity, i.e., physical quantity that cannot be expressed
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
2015 film by Matthew Brown
labour, his employers notice that he seems to have exceptional skills in mathematics and they begin to make use of him for rudimentary accounting tasks. It
The_Man_Who_Knew_Infinity
A list of articles with mathematical proofs: Bertrand's postulate and a proof Estimation of covariance matrices Fermat's little theorem and some proofs
List_of_mathematical_proofs
Logical principle
of the finiteness of the basis of the invariant system was simply not mathematics. Hilbert, on the other hand, throughout his life was to insist that if
Law_of_excluded_middle
Algebraic structure modeling logical operations
others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models. A
Boolean_algebra_(structure)
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned
Mathematics_and_art
A timeline of mathematical logic; see also history of logic. 1847 – George Boole proposes symbolic logic in The Mathematical Analysis of Logic, defining
Timeline of mathematical logic
Timeline_of_mathematical_logic
Israeli mathematician
of mathematics at the University of California, Los Angeles. He has made major contributions to the theory of inner models, determinacy and forcing. Neeman
Itay_Neeman
Classical statement of gravity as force
Newton's work Philosophiæ Naturalis Principia Mathematica (Latin for 'Mathematical Principles of Natural Philosophy' (the Principia)), first published on
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
Umbrella term for technical disciplines
mathematics (STEM) is an umbrella term used to group together the related technical disciplines of science, technology, engineering, and mathematics.
Science, technology, engineering, and mathematics
Science,_technology,_engineering,_and_mathematics
Turning force around an axis
is typically represented mathematically using the lowercase Greek letter tau (𝜏). When being referred to as moment of force, it is commonly denoted by
Torque
American logician (1933–2019)
contributions were in recursion theory. Named after him is Sacks forcing, a forcing notion based on perfect sets and the Sacks Density Theorem, which
Gerald_Sacks
Contraction of an astronomical object due to the influence of its gravity
balance with the potential energy of the internal gravitational force. Mathematically this is expressed using the virial theorem, which states that to
Gravitational_collapse
Technique to make a model more generalizable and transferable
In mathematics, statistics, finance, and computer science, particularly in machine learning and inverse problems, regularization is a process that converts
Regularization_(mathematics)
American mathematician
dissertation on forcing supervised by Abraham Robinson. At Wesleyan, she served three times as department chair. She was an American Mathematical Society (AMS)
Carol_Wood
Diagram that shows all possible logical relations between a collection of sets
(April 2023). "The Venn Behind the Diagram". Mathematics Today. Vol. 59, no. 2. Institute of Mathematics and its Applications. pp. 53–55. Lewis, Clarence
Venn_diagram
Topics referred to by the same term
band Europe Force (A Certain Ratio album), 1986 Force (Superfly album), 2012 "Force" (Superfly song) "Force" (Alan Walker song), 2015 Forcing (magic), a
Force_(disambiguation)
In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean
Equivalent definitions of mathematical structures
Equivalent_definitions_of_mathematical_structures
Paradox in set theory
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician
Russell's_paradox
Branch of mathematics
Calculus is the branch of mathematics that studies continuous change, and is the principal precursor of modern mathematical analysis. Originally called
Calculus
American mathematician (1934–2007)
Uppsala University, Sweden. Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis
Paul_Cohen
Binding of quarks in subatomic particles
particle physics. Mathematically, QCD is a non-abelian gauge theory based on a local (gauge) symmetry group called SU(3). The force carrier particle of
Strong_interaction
Axiom of projective determinacy Axiom of real determinacy Empty set Forcing (mathematics) Fuzzy set Hereditary set Internal set theory Intersection (set theory)
List_of_set_theory_topics
American mathematician
mathematician whose research concerns mathematical logic, set theory, forcing, and recursion theory. She is a professor of mathematics at Dartmouth College. As a
Marcia_Groszek
1994 mathematics book
two chapters on "Forcing and sets of reals" and "Aspects of measurability". The main topic of the first of these chapters is forcing, a technique introduced
The_Higher_Infinite
Fundamental interaction between charged particles
By determining a force law for the interaction between elements of electric current, Ampère placed the subject on a solid mathematical foundation. A theory
Electromagnetism
9th-century Arabic work on algebra
abbreviated Al-Jabr or Algebra (Arabic: الجبر), is an Arabic-language mathematical treatise on algebra written in Baghdad around 820 by the Persian polymath
Al-Jabr
FORCING MATHEMATICS
FORCING MATHEMATICS
Girl/Female
Hindu, Indian
Forming the Crest
Girl/Female
Tamil
Morning
Surname or Lastname
English (mainly Lancashire and Cheshire)
English (mainly Lancashire and Cheshire) : unexplained.Probably an altered form of German Dornig, which is probably a nickname for someone with a sharp tongue, from an adjectival derivative of Middle High German, Middle Low German dorn ‘thorn’. The suffixes -ig and -ing were often interchanged in Pennsylvania German and elsewhere. The name may also refer to a sloe bush.
Surname or Lastname
English
English : habitational name from places in Oxfordshire and West Sussex named Goring, from Old English GÄringas ‘people of GÄra’, a short form of the various compound names with the first element gÄr ‘spear’.German (Göring) : see Goering.
Boy/Male
Tamil
Morning
Surname or Lastname
English (Somerset)
English (Somerset) : unexplained.
Girl/Female
Tamil
Pratushya | பà¯à®°à®¤à¯à®‚à®·à¯à®¯à®¾Â
Morning
Pratushya | பà¯à®°à®¤à¯à®‚à®·à¯à®¯à®¾Â
Surname or Lastname
English
English : ethnic name from Old French Lohereng ‘man from Lorraine’ (see Lorraine).
Girl/Female
Arabic, Muslim
Golden Morning; Beautiful Morning
Girl/Female
American, British, English, Latin
Farmer; Variant of Georgina
Boy/Male
German French
Famous in battle.
Girl/Female
Tamil
Prattysha | பà¯à®°à®¤à¯à®¤à¯à®¯à®·à®¾
Morning
Prattysha | பà¯à®°à®¤à¯à®¤à¯à®¯à®·à®¾
Boy/Male
Latin
Strong; fortunate.
Boy/Male
French, German
Renowned Warrior's Son; From Lorraine; Son of the Famous Warrior
Surname or Lastname
English
English : patronymic from Dear 1.German (Döring) : see Doering.
Boy/Male
German
Renowned Warrior's Son
Surname or Lastname
English
English : habitational name from a place in West Sussex, so named from the Old English personal name Fēra + -ingas ‘people of’, ‘family of’, or ‘followers of’.
Girl/Female
English Latin
Girl/Female
Tamil
Prajisha | பà¯à®°à®œà¯€à®·à®¾Â
Morning
Prajisha | பà¯à®°à®œà¯€à®·à®¾Â
Surname or Lastname
English
English : from a diminutive of Moore 2, 3.North German (Möring) : patronymic from the nickname Mohr (see Mohr 2).North German (Möring) : habitational name from Möringen or Möhringen near Stendal and Stettin.Dutch : variant of Morin.
FORCING MATHEMATICS
FORCING MATHEMATICS
Boy/Male
Arabic
Servant; Attendant
Boy/Male
Buddhist, Indian
Wisdom Gate
Boy/Male
Hindu
Silken, Talent
Girl/Female
Hindu, Indian, Marathi, Telugu
Rarest of the Rare; Extremely Beautiful
Boy/Male
American, British, English
Lives at the Castle
Girl/Female
Muslim/Islamic
Abiding Forever
Boy/Male
American, Australian, Finnish, French, German, Latin, Portuguese, Spanish
Moorish; Dark-skinned; A Moor; Dark Skinned
Girl/Female
Hindu
Lustrous
Girl/Female
Hindu, Indian, Marathi
Gift of God; God of Wealth
Girl/Female
Muslim
FORCING MATHEMATICS
FORCING MATHEMATICS
FORCING MATHEMATICS
FORCING MATHEMATICS
FORCING MATHEMATICS
a.
Pertaining to, or characteristic of, working days, or workdays; everyday; hence, plodding; hard-working.
n.
The act of putting a foot to anything; also, that which is added as a foot; as, the footing of a stocking.
n.
The morning of yesterday.
n.
The act of counterfeiting.
v. i.
The aggregate of the fences put up for inclosure or protection; as, the fencing of a farm.
n.
The act or process of giving form or shape to anything; as, in shipbuilding, the exact shaping of partially shaped timbers.
p. pr. & vb. n.
of Form
a.
Pertaining to the first part or early part of the day; being in the early part of the day; as, morning dew; morning light; morning service.
a.
Not native or belonging to a certain country; born in or belonging to another country, nation, sovereignty, or locality; as, a foreign language; foreign fruits.
a.
Outside; extraneous; separated; alien; as, a foreign country; a foreign government.
a.
Pertaining to agriculture; devoted to, adapted to, or engaged in, farming; as, farming tools; farming land; a farming community.
p. pr. & vb. n.
of Force
a.
Remote; distant; strange; not belonging; not connected; not pertaining or pertient; not appropriate; not harmonious; not agreeable; not congenial; -- with to or from; as, foreign to the purpose; foreign to one's nature.
n.
The art of raising plants, flowers, and fruits at an earlier season than the natural one, as in a hitbed or by the use of artificial heat.
p. pr. & vb. n.
of Fork
n.
See Furring.
n.
A piece of forged work in metal; -- a general name for a piece of hammered iron or steel.
p. pr. & vb. n.
of Farce
n.
Stuffing; forcemeat.
n.
The accomplishing of any purpose violently, precipitately, prematurely, or with unusual expedition.