Sunflower (Mathematics)

Sunower (mathematics)From Wikipedia, the free encyclopedia

Contents

1 Combinatorics 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Approaches and subelds of combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Enumerative combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Analytic combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Partition theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.5 Design theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Finite geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.7 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.8 Matroid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.9 Extremal combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.10 Probabilistic combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.11 Algebraic combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.12 Combinatorics on words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.13 Geometric combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.14 Topological combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.15 Arithmetic combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.16 Innitary combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Related elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Combinatorial optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Coding theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Discrete and computational geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.4 Combinatorics and dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.5 Combinatorics and physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Continuum hypothesis 172.1 Cardinality of innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Independence from ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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2.3 Arguments for and against CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 The generalized continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Implications of GCH for cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . 202.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Forcing (mathematics) 223.1 Intuitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Forcing posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 P-names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Countable transitive models and generic lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 Cohen forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 The countable chain condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 Easton forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.9 Random reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.10 Boolean-valued models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.11 Meta-mathematical explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.12 Logical explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.15 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Partially ordered set 304.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 324.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.10 Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.11 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.12 Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.13 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.14 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

CONTENTS iii

4.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.17 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Sunower (mathematics) 375.1 lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 lemma for !2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Sunower lemma and conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Upper and lower bounds 396.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3 Bounds of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.4 Tight bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 ZermeloFraenkel set theory 417.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 427.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 1

Combinatorics

Not to be confused with combinatoriality.

Combinatorics is a branch of mathematics concerning the study of nite or countable discrete structures. Aspects ofcombinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding whencertain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designsand matroid theory), nding largest, smallest, or optimal objects (extremal combinatorics and combinatorialoptimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniquesto combinatorial problems (algebraic combinatorics).Combinatorial problems arise inmany areas of puremathematics, notably in algebra, probability theory, topology, andgeometry,[1] and combinatorics also has many applications in mathematical optimization, computer science, ergodictheory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving anad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerfuland general theoretical methods were developed, making combinatorics into an independent branch of mathematics inits own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerousnatural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas andestimates in the analysis of algorithms.A mathematician who studies combinatorics is called a combinatorialist or a combinatorist.

1.1 HistoryMain article: History of combinatorics

Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In 6th century BCE,ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 dierenttastes, taken one at a time, two at a time, etc., thus computing all 26 1 possibilities. Greek historian Plutarchdiscusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicateenumerative problem, which was later shown to be related to Schrder numbers.[2][3] In theOstomachion, Archimedes(3rd century BCE) considers a tiling puzzle.In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indianmathematician Mahvra (c. 850) provided formulae for the number of permutations and combinations,[4][5] andthese formulas may have been familiar to Indian mathematicians as early as the 6th century CE.[6] The philosopherand astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coecients, while a closedformula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in1321.[7] The arithmetical triangle a graphical diagram showing relationships among the binomial coecientswas presented by mathematicians in treatises dating as far back as the 10th century, and would eventually becomeknown as Pascals triangle. Later, in Medieval England, campanology provided examples of what is now known asHamiltonian cycles in certain Cayley graphs on permutations.[8]

During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth.

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2 CHAPTER 1. COMBINATORICS

Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging eld. In modern times,the works of J. J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) laid the foundation forenumerative and algebraic combinatorics. Graph theory also enjoyed an explosion of interest at the same time,especially in connection with the four color problem.In the second half of 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens ofnew journals and conferences in the subject.[9] In part, the growth was spurred by new connections and applicationsto other elds, ranging from algebra to probability, from functional analysis to number theory, etc. These connectionsshed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at thesame time led to a partial fragmentation of the eld.

1.2 Approaches and subelds of combinatorics

1.2.1 Enumerative combinatorics

Main article: Enumerative combinatorics

Enumerative combinatorics is the most classical area of combinatorics, and concentrates on counting the numberof certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematicalproblem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonaccinumbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a uniedframework for counting permutations, combinations and partitions.

1.2.2 Analytic combinatorics

Main article: Analytic combinatorics

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis andprobability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae andgenerating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.

1.2.3 Partition theory

Main article: Partition theory

Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closelyrelated to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, itis now considered a part of combinatorics or an independent eld. It incorporates the bijective approach and varioustools in analysis, analytic number theory, and has connections with statistical mechanics.

1.2.4 Graph theory

Main article: Graph theory

Graphs are basic objects in combinatorics. The questions range from counting (e.g., the number of graphs on nvertices with k edges) to structural (e.g., which graphs contain Hamiltonian cycles) to algebraic questions (e.g., givena graph G and two numbers x and y, does the Tutte polynomial TG(x,y) have a combinatorial interpretation?). Itshould be noted that while there are very strong connections between graph theory and combinatorics, these two aresometimes thought of as separate subjects.[10]

1.2. APPROACHES AND SUBFIELDS OF COMBINATORICS 3

1.2.5 Design theoryMain article: Combinatorial design

Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties.Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such asin Kirkmans schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system,which systems play an important role in the classication of nite simple groups. The area has further connections tocoding theory and geometric combinatorics.

1.2.6 Finite geometryMain article: Finite geometry

Finite geometry is the study of geometric systems having only a nite number of points. Structures analogous tothose found in continuous geometries (Euclidean plane, real projective space, etc.) but dened combinatorially arethe main items studied. This area provides a rich source of examples for design theory. It should not be confusedwith discrete geometry (combinatorial geometry).

1.2.7 Order theoryMain article: Order theory

Order theory is the study of partially ordered sets, both nite and innite. Various examples of partial orders appearin algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examplesof partial orders include lattices and Boolean algebras.

1.2.8 Matroid theoryMain article: Matroid theory

Matroid theory abstracts part of geometry. It studies the properties of sets (usually, nite sets) of vectors in a vectorspace that do not depend on the particular coecients in a linear dependence relation. Not only the structure but alsoenumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied asa part of the order theory. It is now an independent eld of study with a number of connections with other parts ofcombinatorics.

1.2.9 Extremal combinatoricsMain article: Extremal combinatorics

Extremal combinatorics studies extremal questions on set systems. The types of questions addressed in this case areabout the largest possible graph which satises certain properties. For example, the largest triangle-free graph on 2nvertices is a complete bipartite graph Kn,n. Often it is too hard even to nd the extremal answer f(n) exactly and onecan only give an asymptotic estimate.Ramsey theory is another part of extremal combinatorics. It states that any suciently large conguration will containsome sort of order. It is an advanced generalization of the pigeonhole principle.

1.2.10 Probabilistic combinatoricsMain article: Probabilistic method


In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain propertyfor a random discrete object, such as a random graph? For instance, what is the average number of triangles in arandom graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certainprescribed properties (for which explicit examples might be dicult to nd), simply by observing that the probabilityof randomly selecting an object with those properties is greater than 0. This approach (often referred to as theprobabilistic method) proved highly eective in applications to extremal combinatorics and graph theory. A closelyrelated area is the study of nite Markov chains, especially on combinatorial objects. Here again probabilistic toolsare used to estimate the mixing time.Often associated with Paul Erds, who did the pioneer work on the subject, probabilistic combinatorics was tradi-tionally viewed as a set of tools to study problems in other parts of combinatorics. However, with the growth ofapplications to analysis of algorithms in computer science, as well as classical probability, additive and probabilisticnumber theory, the area recently grew to become an independent eld of combinatorics.

1.2.11 Algebraic combinatorics

Main article: Algebraic combinatorics

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theoryand representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques toproblems in algebra. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, andcan be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularlystrong and signicant.

1.2.12 Combinatorics on words

Main article: Combinatorics on words

Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics,including number theory, group theory and probability. It has applications to enumerative combinatorics, fractalanalysis, theoretical computer science, automata theory and linguistics. While many applications are new, the classicalChomskySchtzenberger hierarchy of classes of formal grammars is perhaps the best known result in the eld.

1.2.13 Geometric combinatorics

Main article: Geometric combinatorics

Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks,for example, how many faces of each dimension can a convex polytope have. Metric properties of polytopes play animportant role as well, e.g. the Cauchy theorem on rigidity of convex polytopes. Special polytopes are also considered,such as permutohedra, associahedra and Birkho polytopes. We should note that combinatorial geometry is an oldfashioned name for discrete geometry.

1.2.14 Topological combinatorics

Main article: Topological combinatorics

Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions,partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused withcombinatorial topology which is an older name for algebraic topology.

1.3. RELATED FIELDS 5

1.2.15 Arithmetic combinatoricsMain article: Arithmetic combinatorics

Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory andharmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction,multiplication, and division). Additive combinatorics refers to the special case when only the operations of additionand subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamicalsystems.

1.2.16 Innitary combinatoricsMain article: Innitary combinatorics

Innitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to innite sets. Itis a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremalcombinatorics.Gian-Carlo Rota used the name continuous combinatorics[11] to describe probability and measure theory, since thereare many analogies between counting and measure.

1.3 Related elds

1.3.1 Combinatorial optimizationCombinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part ofcombinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, relatedto operations research, algorithm theory and computational complexity theory.

1.3.2 Coding theoryCoding theory started as a part of design theory with early combinatorial constructions of error-correcting codes.The main idea of the subject is to design ecient and reliable methods of data transmission. It is now a large eld ofstudy, part of information theory.

1.3.3 Discrete and computational geometryDiscrete geometry (also called combinatorial geometry) also began a part of combinatorics, with early results onconvex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computationalgeometry, these two elds partially merged and became a separate eld of study. There remain many connectionswith geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discretegeometry.

1.3.4 Combinatorics and dynamical systemsCombinatorial aspects of dynamical systems is another emerging eld. Here dynamical systems can be dened oncombinatorial objects. See for example graph dynamical system.

1.3.5 Combinatorics and physicsThere are increasing interactions between combinatorics and physics, particularly statistical physics. Examples includean exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic andTutte polynomials on the other hand.


1.4 See also Combinatorial biology Combinatorial chemistry Combinatorial data analysis Combinatorial game theory Combinatorial group theory List of combinatorics topics Phylogenetics

1.5 Notes[1] Bjrner and Stanley, p. 2[2] Stanley, Richard P.; Hipparchus, Plutarch, Schrder, and Hough, American Mathematical Monthly 104 (1997), no. 4,

344350.[3] Habsieger, Laurent; Kazarian, Maxim; and Lando, Sergei; On the Second Number of Plutarch, American Mathematical

Monthly 105 (1998), no. 5, 446.[4] O'Connor, John J.; Robertson, Edmund F., Combinatorics, MacTutor History of Mathematics archive, University of St

Andrews.[5] Puttaswamy, Tumkur K. (2000), The Mathematical Accomplishments of Ancient Indian Mathematicians, in Selin,

Helaine, Mathematics Across Cultures: The History of Non-Western Mathematics, Netherlands: Kluwer Academic Pub-lishers, p. 417, ISBN 978-1-4020-0260-1

[6] Biggs, Norman L. (1979). The Roots of Combinatorics. Historia Mathematica 6: 109136.[7] Maistrov, L. E. (1974), Probability Theory: A Historical Sketch, Academic Press, p. 35, ISBN 9781483218632. (Transla-

tion from 1967 Russian ed.)[8] White, Arthur T.; Ringing the Cosets, American Mathematical Monthly, 94 (1987), no. 8, 721746; White, Arthur T.;

Fabian Stedman: The First Group Theorist?", American Mathematical Monthly, 103 (1996), no. 9, 771778.[9] See Journals in Combinatorics and Graph Theory[10] Sanders, Daniel P.; 2-Digit MSC Comparison[11] Continuous and pronite combinatorics

1.6 References Bjrner, Anders; and Stanley, Richard P.; (2010); A Combinatorial Miscellany Bna, Mikls; (2011); A Walk Through Combinatorics (3rd Edition). ISBN 978-981-4335-23-2, ISBN 978-981-4460-00-2(pbk)

Graham, Ronald L.; Groetschel, Martin; and Lovsz, Lszl; eds. (1996); Handbook of Combinatorics, Vol-umes 1 and 2. Amsterdam, NL, and Cambridge, MA: Elsevier (North-Holland) and MIT Press. ISBN 0-262-07169-X

Lindner, Charles C.; and Rodger, Christopher A.; eds. (1997); Design Theory, CRC-Press; 1st. edition (Oc-tober 31, 1997). ISBN 0-8493-3986-3.

Riordan, John (1958); An Introduction to Combinatorial Analysis, New York, NY: Wiley & Sons (republished) Stanley, Richard P. (1997, 1999); Enumerative Combinatorics, Volumes 1 and 2, Cambridge University Press.ISBN 0-521-55309-1, ISBN 0-521-56069-1

van Lint, Jacobus H.; and Wilson, Richard M.; (2001); A Course in Combinatorics, 2nd Edition, CambridgeUniversity Press. ISBN 0-521-80340-3

1.7. EXTERNAL LINKS 7

1.7 External links Hazewinkel, Michiel, ed. (2001), Combinatorial analysis, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Combinatorial Analysis an article in Encyclopdia Britannica Eleventh Edition Combinatorics, a MathWorld article with many references. Combinatorics, from a MathPages.com portal. The Hyperbook of Combinatorics, a collection of math articles links. The Two Cultures of Mathematics by W. T. Gowers, article on problem solving vs theory building.


An example of change ringing (with six bells), one of the earliest nontrivial results in Graph Theory.


Five binary trees on three vertices, an example of Catalan numbers.

A plane partition.


Petersen graph.


{x,y,z}

{y,z}{x,z}{x,y}

{y} {z}{x}

Hasse diagram of the powerset of {x,y,z} ordered by inclusion.


Self-avoiding walk in a square grid graph.


Young diagram of a partition (5,4,1).

Construction of a ThueMorse innite word.


An icosahedron.


Splitting a necklace with two cuts.


Kissing spheres are connected to both coding theory and discrete geometry.

Chapter 2

Continuum hypothesis

This article is about the hypothesis in set theory. For the assumption in uid mechanics, see Fluid mechanics.

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of innite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the rstof Hilberts 23 problems presented in the year 1900. he answer to this problem is independent of ZFC set theory,so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resultingtheory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen,complementing earlier work by Kurt Gdel in 1940.The name of the hypothesis comes from the term the continuum for the real numbers. It is abbreviated CH.

2.1 Cardinality of innite setsMain article: Cardinal number

Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspon-dence) between them. Intuitively, for two sets S and T to have the same cardinality means that it is possible to pairo elements of S with elements of T in such a fashion that every element of S is paired o with exactly one elementof T and vice versa. Hence, the set fbanana; apple; pearg has the same cardinality as fyellow; red; greeng .With innite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate.The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a propersubset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rationalnumbers than integers, and more real numbers than rational numbers. However, this intuitive analysis does not takeaccount of the fact that all three sets are innite. It turns out the rational numbers can actually be placed in one-to-onecorrespondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set ofintegers: they are both countable sets.Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers(see Cantors rst uncountability proof and Cantors diagonal argument). His proofs, however, give no indication ofthe extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuumhypothesis as a possible solution to this question.The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinalityof the set of integers. Equivalently, as the cardinality of the integers is @0 ("aleph-naught") and the cardinality of thereal numbers is 2@0 (i.e. it equals the cardinality of the power set of the integers), the continuum hypothesis says thatthere is no set S for which

@0 < jSj < 2@0 :

17

18 CHAPTER 2. CONTINUUM HYPOTHESIS

Assuming the axiom of choice, there is a smallest cardinal number @1 greater than @0 , and the continuum hypothesisis in turn equivalent to the equality

2@0 = @1:A consequence of the continuum hypothesis is that every innite subset of the real numbers either has the samecardinality as the integers or the same cardinality as the entire set of the reals.There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH)which says that for all ordinals

2@ = @+1:That is, GCH asserts that the cardinality of the power set of any innite set is the smallest cardinality greater thanthat of the set.

2.2 Independence from ZFCCantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain (Dauben 1990). Itbecame the rst on David Hilberts list of important open questions that was presented at the International Congressof Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated.Kurt Gdel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standardZermeloFraenkel set theory (ZF), even if the axiom of choice is adopted (ZFC) (Gdel (1940)). Paul Cohen showedin 1963 that CH cannot be proven from those same axioms either (Cohen (1963) & Cohen (1964)). Hence, CH isindependent of ZFC. Both of these results assume that the ZermeloFraenkel axioms are consistent; this assumptionis widely believed to be true. Cohen was awarded the Fields Medal in 1966 for his proof.The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory.As a result of its independence, many substantial conjectures in those elds have subsequently been shown to beindependent as well.So far, CH appears to be independent of all known large cardinal axioms in the context of ZFC.The independence from ZFC means that proving or disproving the CH within ZFC is impossible. Gdel and Cohensnegative results are not universally accepted as disposing of the hypothesis. Hilberts problem remains an active topicof research; see Woodin (2001) and Koellner (2011a) for an overview of the current research status.The continuum hypothesis was not the rst statement shown to be independent of ZFC. An immediate consequenceof Gdels incompleteness theorem, which was published in 1931, is that there is a formal statement (one for eachappropriate Gdel numbering scheme) expressing the consistency of ZFC that is independent of ZFC, assuming thatZFC is consistent. The continuum hypothesis and the axiom of choice were among the rst mathematical statementsshown to be independent of ZF set theory. These proofs of independence were not completed until Paul Cohendeveloped forcing in the 1960s. They all rely on the assumption that ZF is consistent. These proofs are called proofsof relative consistency (see Forcing (mathematics)).

2.3 Arguments for and against CHGdel believed that CH is false and that his proof that CH is consistent with ZFC only shows that the ZermeloFraenkel axioms do not adequately characterize the universe of sets. Gdel was a platonist and therefore had noproblems with asserting the truth and falsehood of statements independent of their provability. Cohen, though aformalist (Goodman 1979), also tended towards rejecting CH.Historically, mathematicians who favored a rich and large universe of sets were against CH, while those favoringa neat and controllable universe favored CH. Parallel arguments were made for and against the axiom of con-structibility, which implies CH. More recently, Matthew Foreman has pointed out that ontological maximalism canactually be used to argue in favor of CH, because among models that have the same reals, models with more setsof reals have a better chance of satisfying CH (Maddy 1988, p. 500).

2.4. THE GENERALIZED CONTINUUM HYPOTHESIS 19

Another viewpoint is that the conception of set is not specic enough to determine whether CH is true or false. Thisviewpoint was advanced as early as 1923 by Skolem, even beforeGdels rst incompleteness theorem. Skolem arguedon the basis of what is now known as Skolems paradox, and it was later supported by the independence of CH fromthe axioms of ZFC, since these axioms are enough to establish the elementary properties of sets and cardinalities. Inorder to argue against this viewpoint, it would be sucient to demonstrate new axioms that are supported by intuitionand resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generallyconsidered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171).At least two other axioms have been proposed that have implications for the continuum hypothesis, although theseaxioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presentedan argument against CH by showing that the negation of CH is equivalent to Freilings axiom of symmetry, a statementabout probabilities. Freiling believes this axiom is intuitively true but others have disagreed. A dicult argumentagainst CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000 (Woodin 2001a,2001b). Foreman (2003) does not reject Woodins argument outright but urges caution.Solomon Feferman (2011) has made a complex philosophical argument that CH is not a denite mathematical prob-lem. He proposes a theory of deniteness using a semi-intuitionistic subsystem of ZF that accepts classical logic forbounded quantiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition is mathemati-cally denite if the semi-intuitionistic theory can prove (_:) . He conjectures that CH is not denite accordingto this notion, and proposes that CH should therefore be considered not to have a truth value. Peter Koellner (2011b)wrote a critical commentary on Fefermans article.Joel David Hamkins proposes a multiverse approach to set theory and argues that the continuum hypothesis is settledon the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can nolonger be settled in the manner formerly hoped for. (Hamkins 2012). In a related vein, Saharon Shelah wrote thathe does not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we justhave to discover the additional axiom. My mental picture is that we have many possible set theories, all conformingto ZFC. (Shelah 2003).

2.4 The generalized continuum hypothesisThe generalized continuum hypothesis (GCH) states that if an innite sets cardinality lies between that of an inniteset S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as thepower set of S. That is, for any innite cardinal there is no cardinal such that < < 2: GCH is equivalentto:

@+1 = 2@ for every ordinal : (occasionally called Cantors aleph hypothesis)

The beth numbers provide an alternate notation for this condition: @ = i for every ordinal :This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power setof the integers. It was rst suggested by Jourdain (1905).Like CH, GCH is also independent of ZFC, but Sierpiski proved that ZF + GCH implies the axiom of choice (AC),so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. Toprove this, Sierpiski showed GCH implies that every cardinality n is smaller than some Aleph number, and thus canbe ordered. This is done by showing that n is smaller than 2@0+n which is smaller than its own Hartogs number this uses the equality 2@0+n = 2 2@0+n ; for the full proof, see Gillman (2002).Kurt Gdel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to theordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohens model in which CH fails is a modelin which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developedby Cohen to prove Eastons theorem, which shows it is consistent with ZFC for arbitrarily large cardinals @ tofail to satisfy 2@ = @+1: Much later, Foreman and Woodin proved that (assuming the consistency of very largecardinals) it is consistent that 2 > + holds for every innite cardinal : Later Woodin extended this by showingthe consistency of 2 = ++ for every . A recent result of Carmi Merimovich shows that, for each n1, it isconsistent with ZFC that for each , 2 is the nth successor of . On the other hand, Lszl Patai (1930) proved, thatif is an ordinal and for each innite cardinal , 2 is the th successor of , then is nite.For any innite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsetsof B. Thus for any innite cardinals A and B,

20 CHAPTER 2. CONTINUUM HYPOTHESIS

A < B ! 2A 2B:

If A and B are nite, the stronger inequality

A < B ! 2A < 2B

holds. GCH implies that this strict, stronger inequality holds for innite cardinals as well as nite cardinals.

2.4.1 Implications of GCH for cardinal exponentiationAlthough the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, onecan deduce from it the values of cardinal exponentiation in all cases. It implies that @@ is (see: Hayden & Kennison(1968), page 147, exercise 76):

@+1 when +1;@ when +1 < and @ < cf(@) where cf is the conality operation; and@+1 when +1 < and @ cf(@) .

2.5 See also Aleph number Beth number Cardinality -logic Wetzels problem

2.6 References Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: DoverPublications. ISBN 978-0-486-46921-8.

Cohen, Paul J. (December 15, 1963). The Independence of the Continuum Hypothesis. Proceedings of theNational Academy of Sciences of the United States of America 50 (6): 11431148. doi:10.1073/pnas.50.6.1143.JSTOR 71858. PMC 221287. PMID 16578557.

Cohen, Paul J. (January 15, 1964). The Independence of the Continuum Hypothesis, II. Proceedings of theNational Academy of Sciences of the United States of America 51 (1): 105110. doi:10.1073/pnas.51.1.105.JSTOR 72252. PMC 300611. PMID 16591132.

Dales, H. G.; Woodin, W. H. (1987). An Introduction to Independence for Analysts. Cambridge. Dauben, Joseph Warren (1990). Georg Cantor: His Mathematics and Philosophy of the Innite. PrincetonUniversity Press. pp. 134137. ISBN 9780691024479.

Enderton, Herbert (1977). Elements of Set Theory. Academic Press. Feferman, Solomon (2011). Is the ContinuumHypothesis a denitemathematical problem?" (PDF).Exploring

the Frontiers of Independence (Harvard lecture series).

Foreman, Matt (2003). Has the Continuum Hypothesis been Settled?" (PDF). Retrieved February 25, 2006.


Freiling, Chris (1986). Axioms of Symmetry: Throwing Darts at the Real Number Line. Journal of SymbolicLogic (Association for Symbolic Logic) 51 (1): 190200. doi:10.2307/2273955. JSTOR 2273955.

Gdel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press. Gillman, Leonard (2002). Two Classical Surprises Concerning the Axiom of Choice and the ContinuumHypothesis (PDF). American Mathematical Monthly 109.

Gdel, K.: What is Cantors Continuum Problem?, reprinted in Benacerraf and Putnams collection Philosophyof Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gdels arguments against CH.

Goodman, Nicolas D. (1979). Mathematics as an objective science. The American Mathematical Monthly86 (7): 540551. doi:10.2307/2320581. MR 542765. This view is often called formalism. Positions more orless like this may be found in Haskell Curry [5], Abraham Robinson [17], and Paul Cohen [4].

Joel David Hamkins. The set-theoretic multiverse. Rev. Symb. Log. 5 (2012), no. 3, 416449. Seymour Hayden and John F. Kennison: ZermeloFraenkel Set Theory (1968), Charles E. Merrill PublishingCompany, Columbus, Ohio.

Jourdain, Philip E. B. (1905). On transnite cardinal numbers of the exponential form. Philosophical Mag-azine, Series 6 9: 4256. doi:10.1080/14786440509463254.

Koellner, Peter (2011a). The Continuum Hypothesis (PDF). Exploring the Frontiers of Independence (Har-vard lecture series).

Koellner, Peter (2011b). Feferman On the Indeniteness of CH (PDF). Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North-Holland.ISBN 978-0-444-85401-8.

Maddy, Penelope (June 1988). Believing the Axioms, I. Journal of Symbolic Logic (Association for SymbolicLogic) 53 (2): 481511. doi:10.2307/2274520. JSTOR 2274520.

Martin, D. (1976). Hilberts rst problem: the continuum hypothesis, inMathematical Developments ArisingfromHilberts Problems, Proceedings of Symposia in PureMathematics XXVIII, F. Browder, editor. AmericanMathematical Society, 1976, pp. 8192. ISBN 0-8218-1428-1

McGough, Nancy. The Continuum Hypothesis. Merimovich, Carmi (2007). A power function with a xed nite gap everywhere. Journal of Symbolic Logic72 (2): 361417. doi:10.2178/jsl/1185803615.

Moore, Gregory H. (2011). Early history of the generalized continuum hypothesis: 18781938. Bull. Sym-bolic Logic 17 (4): 489532. doi:10.2178/bsl/1318855631. MR 2896574.

Shelah, Saharon (2003). Logical dreams. Bull. Amer. Math. Soc. (N.S.) 40 (2): 203228. doi:10.1090/s0273-0979-03-00981-9.

Woodin, W. Hugh (2001a). The Continuum Hypothesis, Part I (PDF). Notices of the AMS 48 (6): 567576. Woodin, W. Hugh (2001b). The ContinuumHypothesis, Part II (PDF).Notices of the AMS 48 (7): 681690.

German literature

Cantor, Georg (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal fr die Reine und Angewandte Math-ematik 84: 242258. doi:10.1515/crll.1878.84.242.

Patai, L. (1930). Untersuchungen ber die -reihe. Mathematische und naturwissenschaftliche Berichte ausUngarn 37: 127142.

2.7 External links Szudzik, Matthew and Weisstein, Eric W., Continuum Hypothesis, MathWorld.

This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 3

Forcing (mathematics)

For the use of forcing in recursion theory, see Forcing (recursion theory).

In the mathematical discipline of set theory, forcing is a technique discovered by Paul Cohen for proving consistencyand independence results. It was rst used, in 1963, to prove the independence of the axiom of choice and thecontinuum hypothesis from ZermeloFraenkel set theory. Forcing was considerably reworked and simplied in thefollowing years, and has since served as a powerful technique both in set theory and in areas of mathematical logicsuch as recursion theory.Descriptive set theory uses the notion of forcing from both recursion theory and set theory. Forcing has also beenused in model theory but it is common in model theory to dene genericity directly without mention of forcing.

3.1 IntuitionsForcing is equivalent to the method of Boolean-valued models, which some feel is conceptually more natural andintuitive, but usually much more dicult to apply.Intuitively, forcing consists of expanding the set theoretical universeV to a larger universeV*. In this bigger universe,for example, one might have lots of new subsets of = {0,1,2,} that were not there in the old universe, and therebyviolate the continuum hypothesis. While impossible on the face of it, this is just another version of Cantors paradoxabout innity. In principle, one could consider

V = V f0; 1g;identify x 2 V with (x; 0) , and then introduce an expanded membership relation involving the new sets of theform (x; 1) . Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set,and allowing for ne control over the properties of the expanded universe.Cohens original technique, now called ramied forcing, is slightly dierent from the unramied forcing expoundedhere.

3.2 Forcing posetsA forcing poset is an ordered triple, (P, , 1), where is a preorder on P that satises following splitting condition:

For all p P, there are q, r P such that q, r p with no s P such that s q, r

The largest element of P is 1, that is, p 1 for all p P.Members of P are called forcing conditions or just conditions.One reads p q as p is stronger than q. Intuitively, the smaller condition provides more information, just as thesmaller interval [3.1415926,3.1415927] provides more information about the number than the interval [3.1,3.2]does.

22

3.2. FORCING POSETS 23

There are various conventions in use. Some authors require to also be antisymmetric, so that the relation is a partialorder. Some use the term partial order anyway, conicting with standard terminology, while some use the termpreorder. The largest element can be dispensed with. The reverse ordering is also used, most notably by SaharonShelah and his co-authors.

3.2.1 P-namesAssociated with a forcing poset P is the class V(P) of P-names. P-names are sets of the form

{(u, p) : u is a P-name and p P and (some criterion involving u and p)}

Using transnite recursion, one denes

Name(0) = {} , Name( + 1) = the power set of (Name() P), Name() = {Name() : < for a limit ordinal} ,

and then the class of P-names is dened by

V(P) = {Name() : is an ordinal} .

The P-names are, in fact, an expansion of the universe. Given x V, one denes x to be the P-name

x = {(y, 1) : y x} .

Again, this is really a denition by transnite recursion.

3.2.2 InterpretationGiven any subset G of P, one next denes the interpretation or valuation map from P-names by

val(u, G) = {val(v, G) : p G , (v, p) u} .

(Again a denition by transnite recursion.) Note that if 1 is in G, then

val(x, G) = x.

One denes

G = {(p, p) : p G} ,

so that

val(G,G) = G.

3.2.3 ExampleA good example of a forcing poset is (Bor(I) , , I ) where I = [0,1] and Bor(I) are the Borel subsets of I havingnon-zero Lebesgue measure. In this case, one can talk about the conditions as being probabilities, and a Bor(I)-nameassigns membership in a probabilistic sense. Because of the ready intuition this example can provide, probabilisticlanguage is sometimes used with other forcing posets.

24 CHAPTER 3. FORCING (MATHEMATICS)

3.3 Countable transitive models and generic ltersThe key step in forcing is, given a ZFC universe V, to nd appropriate G not in V. The resulting class of all interpre-tations of P-names will turn out to be a model of ZFC, properly extending the original V (since GV).Instead of working with V, one considers a countable transitive model M with (P,,1) M. By model, we mean amodel of set theory, either of all of ZFC, or a model of a large but nite subset of the ZFC axioms, or some variantthereof. Transitivity means that if x y M, then x M. TheMostowski collapsing theorem says this can be assumedif the membership relation is well-founded. The eect of transitivity is that membership and other elementary notionscan be handled intuitively. Countability of the model relies on the LwenheimSkolem theorem.Since M is a set, there are sets not in M this follows from Russells paradox. The appropriate set G to pick, andadjoin toM, is a generic lter on P. The lter condition means that GP and

1 G ; if p q G, then p G ; if p,q G, then r G, r p and r q ;

For G to be generic means

if D M is a dense subset of P (that is, p P implies q D, q p) then GD 0 .

The existence of a generic lter G follows from the RasiowaSikorski lemma. In fact, slightly more is true: given acondition p P, one can nd a generic lter G such that p G. Due to the splitting condition, if G is lter, then P\Gis dense. If G is inM then P\G is inM becauseM is model of set theory. By this reason, a generic lter is never inM.

3.4 ForcingGiven a generic lterGP, one proceeds as follows. The subclass ofP-names inM is denotedM(P). LetM[G]={val(u,G):uM(P)}.To reduce the study of the set theory ofM[G] to that ofM, one works with the forcing language, which is built uplike ordinary rst-order logic, with membership as binary relation and all the names as constants.Dene p M;P (u1,,un) (read "p forces in model M with poset P) where p is a condition, is a formulain the forcing language, and the ui are names, to mean that if G is a generic lter containing p, then M[G] (val(u1,G),,val(un,G)). The special case 1 M;P is often written P M;P or M;P . Such statementsare true inM[G] no matter what G is.What is important is that this external denition of the forcing relation p M;P is equivalent to an internaldenition, dened by transnite induction over the names on instances of u v and u = v, and then by ordinaryinduction over the complexity of formulas. This has the eect that all the properties ofM[G] are really properties ofM, and the verication of ZFC inM[G] becomes straightforward. This is usually summarized as three key properties:

Truth: M[G] (val(u1,G),,val(un,G)) if and only if it is forced by G, that is, for some condition p G, p

M;P (u1,,un).

Denability: The statement "p M;P (u1,,un)" is denable inM. Coherence: If p M;P (u1,,un) and q p, then q M;P (u1,,un).

We dene the forcing relation in V by induction on complexity, in which we simultaneously dene forcing of atomicformulas by -induction and then we dene it by induction on formula complexity.1. p P a 2 b if (8q p)(9r q)(9s; c)((s; c) 2 b ^ r s ^ r P a = c) .2. p P a = b if (8q p)(8c 2 V P )(q P c 2 a , q P c 2 b) .3. p P :f if :(9q p)q P f .4. p P (f ^ g) if (p P f ^ p P g) .

3.5. CONSISTENCY 25

5. p P (8x)f if (8x 2 V P )p P f .In 15 p is an arbitrary condition. In 1 and 2 a and a are arbitrary names and in 35 f and g are arbitrary formulaswhere all free occurrences of variables referring names. This denition is syntax transform of formulas. This meansthat for any given formula f(x1; : : : ; xn) the formula p P f(x1; : : : ; xn)with free variables p; P; x1; : : : ; xn is welldened. In fact this syntax transform has following properties: any equivalence given by 1-5 is theorem (single theo-rem per formula) and for any formula f(x1; : : : ; xn) following formula (8p; P; x1; : : : ; xn)(p P f(x1; : : : ; xn))(po(P ) ^ p 2 dom(P ) ^ x1; : : : ; xn 2 V P ) is theorem where po(P ) means that P is partial order with splittingcondition. The bit of denition is existence of syntax transform with these properties.This denition provides the possibility of working in V without any countable transitive model M . The followingstatement gives announced denability:(8M;P; x1; : : : ; xn)(ctm(M)^po(P )^P 2M^p 2 dom(P )^x1; : : : ; xn 2MP ) (p M;P f(x1; : : : ; xn) ,M j= p P f(x1; : : : ; xn)))where ctm(M) means thatM is countable transitive model satisfying some nite part of ZF axioms depending onformula f .(Where no confusion is possible we simply write .)

3.5 ConsistencyThe above can be summarized by saying the fundamental consistency result is that given a forcing poset P, we mayassume that there exists a generic lter G, not in the universe V, such that V[G] is again a set theoretic universe,modelling ZFC. Furthermore, all truths in V[G] can be reduced to truths in V regarding the forcing relation.Both styles, adjoining G to a countable transitive model M or to the whole universe V, are commonly used. Lesscommonly seen is the approach using the internal denition of forcing, and no mention of set or class models ismade. This was Cohens original method, and in one elaboration, it becomes the method of Boolean-valued analysis.

3.6 Cohen forcingThe simplest nontrivial forcing poset is ( Fin(,2), , 0 ), the nite partial functions from to 2={0,1} under reverseinclusion. That is, a condition p is essentially two disjoint nite subsets p1[1] and p1[0] of , to be thought of asthe yes and no parts of p, with no information provided on values outside the domain of p. q is stronger than pmeans that q p, in other words, the yes and no parts of q are supersets of the yes and no parts of p, and inthat sense, provide more information.Let G be a generic lter for this poset. If p and q are both in G, then pq is a condition, because G is a lter. Thismeans that g=G is a well-dened partial function from to 2, because any two conditions in G agree on theircommon domain.g is in fact a total function. Given n , let Dn={ p : p(n) is dened }, then Dn is dense. (Given any p, if n is not inps domain, adjoin a value for n, the result is in Dn.) A condition p GDn has n in its domain, and since p g, g(n)is dened.Let X=g1[1], the set of all yes members of the generic conditions. It is possible to give a name for X directly.Let X = { ( n, p ) : p(n)=1 }, then val( X, G ) = X. Now suppose A in V. We claim that XA. Let DA = { p :n, ndom(p) and p(n)=1 if and only if nA }. DA is dense. (Given any p, if n is not in ps domain, adjoin a valuefor n contrary to the status of "nA".) Then any pGDA witnesses XA. To summarize, X is a new subset of ,necessarily innite.Replacing with 2, that is, consider instead nite partial functions whose inputs are of the form (n,), with n


The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does notcollapse cardinals. For this, a sucient combinatorial property is that all of the antichains of this poset are countable.

3.7 The countable chain conditionMain article: Countable chain condition

An antichain A of P is a subset such that if p and q are in A, then p and q are incompatible (written p q), meaningthere is no r in P such that r p and r q. In the Borel sets example, incompatibility means pq has measure zero.In the nite partial functions example, incompatibility means that pq is not a function, in other words, p and q assigndierent values to some domain input.P satises the countable chain condition (c.c.c.) if every antichain in P is countable. (The name, which is obviouslyinappropriate, is a holdover from older terminology. Some mathematicians write c.a.c. for countable antichaincondition.)It is easy to see that Bor(I) satises the c.c.c., because the measures add up to at most 1. Fin(E,2) is also c.c.c., butthe proof is more dicult.Given an uncountable subfamilyW Fin(E,2), shrinkW to an uncountable subfamilyW0 of sets of size n, for somen

3.9. RANDOM REALS 27

At one time, it was thought that more sophisticated forcing would also allow arbitrary variation in the powers ofsingular cardinals. But this has turned out to be a dicult, subtle and even surprising problem, with several morerestrictions provable in ZFC, and with the forcing models depending on the consistency of various large cardinalproperties. Many open problems remain.

3.9 Random realsMain article: random algebra

In the Borel sets ( Bor(I), , I ) example, the generic lter converges to a real number r, called a random real. Aname for the decimal expansion of r (in the sense of the canonical set of decimal intervals that converge to r) can begiven by letting r = { ( E, E ) : E = [ k10n, (k + 1)10n ], 0 k < 10n }. This is, in some sense, just a subname ofG.To recover G from r, one takes those Borel subsets of I that contain r. Since the forcing poset is in V, but r is not inV, this containment is actually impossible. But there is a natural sense in which the interval [0.5, 0.6] in V containsa random real whose decimal expansion begins 0.5. This is formalized by the notion of Borel code.Every Borel set can, nonuniquely, be built up, starting from intervals with rational endpoints and applying the opera-tions of complement and countable unions, a countable number of times. The record of such a construction is calleda Borel code. Given a Borel set B in V, one recovers a Borel code, and then applies the same construction sequencein V[G], getting a Borel set B*. One can prove that one gets the same set independent of the construction of B, andthat basic properties are preserved. For example, if BC, then B*C*. If B has measure zero, then B* has measurezero.So given r, a random real, one can show thatG = { B (inV) : rB* (inV[G]) }. Because of the mutual interdenabilitybetween r and G, one generally writes V[r] for V[G].A dierent interpretation of reals in V[G] was provided by Dana Scott. Rational numbers in V[G] have names thatcorrespond to countably many distinct rational values assigned to a maximal antichain of Borel sets, in other words,a certain rational-valued function on I = [0,1]. Real numbers in V[G] then correspond to Dedekind cuts of suchfunctions, that is, measurable functions.

3.10 Boolean-valued modelsMain article: Boolean-valued model

Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In these, any statement isassigned a truth value from some complete atomless Boolean algebra, rather than just a true/false value. Then anultralter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point isthat the resulting theory has a model which contains this ultralter, which can be understood as a new model obtainedby extending the old one with this ultralter. By picking a Boolean-valued model in an appropriate way, we can get amodel that has the desired property. In it, only statements which must be true (are forced to be true) will be true,in a sense (since it has this extension/minimality property).

3.11 Meta-mathematical explanationIn forcing we usually seek to show some sentence is consistent with ZFC (or optionally some extension of ZFC). Oneway to interpret the argument is that we assume ZFC is consistent and use it to prove ZFC combined with our newsentence is also consistent.Each condition is a nite piece of information the idea is that only nite pieces are relevant for consistency, sinceby the compactness theorem a theory is satisable if and only if every nite subset of its axioms is satisable. Then,we can pick an innite set of consistent conditions to extend our model. Thus, assuming consistency of set theory,we prove consistency of the theory extended with this innite set.


3.12 Logical explanationBy Gdels incompleteness theorem one cannot prove the consistency of any suciently strong formal theory, such asZFC, using only the axioms of the theory itself, unless the theory itself is inconsistent. Consequently mathematiciansdo not attempt to prove the consistency of ZFC using only the axioms of ZFC, or to prove ZFC+H is consistent forany hypothesis H using only ZFC+H. For this reason the aim of a consistency proof is to prove the consistency ofZFC + H relative to consistency of ZFC. Such problems are known as problems of relative consistency. In fact oneproves(*) ZFC ` Con(ZFC)! Con(ZFC +H):We will give the general schema of relative consistency proofs. Because any proof is nite it uses nite number ofaxioms.

ZFC + :Con(ZFC +H) ` 9T (Fin(T ) ^ T ZFC ^ (T ` :H)):

For any given proof ZFC can verify validity of this proof. This is provable by induction by the length of the proof.

ZFC ` 8T ((T ` :H)! (ZFC ` (T ` :H))):

Now we obtain

ZFC + :Con(ZFC +H) ` 9T (Fin(T ) ^ T ZFC ^ (ZFC ` (T ` :H))):

If we prove the following(**) ZFC ` 8T (Fin(T ) ^ T ZFC ! (ZFC ` Con(T +H)))we can conclude that

ZFC + :Con(ZFC +H) ` 9T (Fin(T ) ^ T ZFC ^ (ZFC ` (T ` :H)) ^ (ZFC ` Con(T +H)))

which is equivalent to

ZFC + :Con(ZFC +H) ` :Con(ZFC)

which gives (*). The core of the relative consistency proof is proving (**). One has to construct a ZFC proof ofCon(T + H) for any given nite set T of ZFC axioms (by ZFC instruments of course). (No universal proof of Con(T+ H) of course.)In ZFC it is provable that for any condition p the set of formulas (evaluated by names) forced by p is deductivelyclosed. Also, for any ZFC axiom, ZFC proves that this axiom is forced by 1. Then it suces to prove that there is atleast one condition which forces H.In the case of Boolean valued forcing, the procedure is similar one has to prove that the Boolean value of H is not0.Another approach uses the reection theorem. For any given nite set of ZFC axioms there is ZFC proof that thisset of axioms has a countable transitive model. For any given nite set T of ZFC axioms there is nite set T' of ZFCaxioms such that ZFC proves that if a countable transitive model M satises T' then M[G] satises T. One has toprove that there is nite set T of ZFC axioms such that if a countable transitive model M satises T then M[G]satises the hypothesis H. Then, for any given nite set T of ZFC axioms, ZFC proves Con(T + H).Sometimes in (**) some stronger theory S than ZFC is used for proving Con(T + H). Then we have proof of consis-tency of ZFC + H relative to the consistency of S. Note that ZFC ` Con(ZFC)$ Con(ZFL) , where ZFL is ZF+ V = L (axiom of constructibility).

3.13. SEE ALSO 29

3.13 See also List of forcing notions Nice name

3.14 References Bell, J. L. (1985) Boolean-ValuedModels and Independence Proofs in Set Theory, Oxford. ISBN 0-19-853241-5

Cohen, P. J. (1966). Set theory and the continuum hypothesis. AddisonWesley. ISBN 0-8053-2327-9. Grishin, V.N. (2001), Forcing method, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.

3.15 External links Nik Weavers book Forcing for Mathematicians was written for mathematicians who want to learn the basicmachinery of forcing. No background in logic is assumed, beyond the facility with formal syntax which shouldbe second nature to any well-trained mathematician.

Tim Chows article A Beginners Guide to Forcing is a good introduction to the concepts of forcing that avoidsa lot of technical detail. This paper grew out of Chows newsgroup article Forcing for dummies. In addition toimproved exposition, the Beginners Guide includes a section on Boolean Valued Models.

See also Kenny Easwarans article A Cheerful Introduction to Forcing and the Continuum Hypothesis, whichis also aimed at the beginner but includes more technical details than Chows article.

The Independence of the Continuum Hypothesis Paul J. Cohen, Proceedings of the National Academy ofSciences of the United States of America, Vol. 50, No. 6. (Dec. 15, 1963), pp. 11431148.

The Independence of the Continuum Hypothesis, II Paul J. Cohen Proceedings of the National Academy ofSciences of the United States of America, Vol. 51, No. 1. (Jan. 15, 1964), pp. 105110.

Paul Cohen gave a historical lecture The Discovery of Forcing (Rocky Mountain J. Math. Volume 32, Number4 (2002), 10711100) about how he developed his independence proof. The linked page has a download linkfor an open access PDF but your browser must send a referer header from the linked page to retrieve it.

Weisstein, Eric W., Forcing, MathWorld.

Chapter 4

Partially ordered set

{x,y,z}

{y,z}{x,z}{x,y}

{y} {z}{x}

The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal leveldon't share a precedence relationship. Other pairs, such as {x} and {y,z}, do not either.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitiveconcept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together witha binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.Such a relation is called a partial order to reect the fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiartotal orders, in which every pair is related. A nite poset can be visualized through its Hasse diagram, which depictsthe ordering relation.[1]

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy.Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

30

4.1. FORMAL DEFINITION 31

4.1 Formal denitionA (non-strict) partial order[2] is a binary relation "" over a set P which is reexive, antisymmetric, and transitive,i.e., which satises for all a, b, and c in P:

a a (reexivity); if a b and b a then a = b (antisymmetry); if a b and b c then a c (transitivity).

In other words, a partial order is an antisymmetric preorder.A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimesalso used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered setscan also be referred to as ordered sets, especially in areas where these structures are more common than posets.For a, b, elements of a partially ordered set P, if a b or b a, then a and b are comparable. Otherwise they areincomparable. In the gure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partialorder under which every pair of elements is comparable is called a total order or linear order; a totally orderedset is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no twodistinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-rightgure). An element a is said to be covered by another element b, written a c < d ...

4.3 ExtremaThere are several notions of greatest and least element in a poset P, notably:

32 CHAPTER 4. PARTIALLY ORDERED SET

Greatest element and least element: An element g in P is a greatest element if for every element a in P, a g.An element m in P is a least element if for every element a in P, a m. A poset can only have one greatest orleast element.

Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a inP such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a

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