- January 24, 2011 Recall that Depth(A) is the supremum of cardinalities of subsets of A which are well-
- 27. Main cofinality theorems We will shortly give several proofs involving the important general idea of making elemen-
- On some small cardinals for Boolean algebras Ralph McKenzie and J. Donald Monk
- 3. Cellularity June 8, 2010
- Continuum cardinals June 19, 2004
- Dear Author Here are the proofs of your article.
- Maximal irredundance and maximal ideal independence in Boolean algebras
- ArMaLo ManuskriptNr. (will be inserted by hand later)
- On cellularity in homomorphic images of Boolean algebras
- December, 2010 James Donald Monk
- 0. Introduction 10 January 2011
- 10. Infinite combinatorics In this chapter we survey the most useful theorems of infinite combinatorics; the best
- 20. Isomorphisms and AC In this chapter we prove that if ZF is consistent, then so is AC. First, however, we go
- 21. Embeddings, iterated forcing, and Martin's axiom In this chapter we mainly develop iterated forcing. The idea of iterated forcing is to
- 2. Special classes of Boolean algebras
- The size of maximal almost disjoint families September 20, 2003
- Solutions, exercises in Chapter 5 E5.1 We call B a simple extension of A iff A is a subalgebra of B and there is an x B
- The spectrum of maximal independent subsets of a Boolean algebra
- An Introduction to Cylindric Set J. DONALD MONK, Department of Mathematics, University of
- Depth, -character, and tightness in superatomic Boolean algebras Alan Dow and J. Donald Monk
- 8. Irredundance Clearly Irr(A) |A|. If A is a subalgebra of B, then Irr(A) Irr(B), and Irr can change
- 4. Ordinals In this chapter we introduce the ordinals, prove a general recursion theorem, and develop
- Concerning problems about cardinal invariants on Boolean algebras
- Generalized free products ABSTRACT.A subalgebra B of the direct product
- 3. Elementary set theory Here we will see how the axioms are used to develop very elementary set theory. The axiom
- A brief introduction to Boolean algebras
- 16. Boolean algebras and quasi-orders A Boolean algebra (BA) is a structure A, +, , -, 0, 1 with two binary operations + and
- Solutions, exercises in Chapter 4 E4.1 For any BA A let A : A Clop(Ult(A)) be defined by setting A(a) = S(a). Show
- 2. ZFC axioms Before introducing any set-theoretic axioms at all, we can introduce some important ab-
- ON GENERAL BOUNDEDNESS AND DOMINATING J. DONALD MONK
- Solutions, exercises in Chapter 6 E6.1 Show that Fr() can be isomorphically embedded in any atomless BA.
- 11. Martin's axiom Martin's axiom is not an axiom of ZFC, but it can be added to those axioms. It has many
- 14. Models of set theory In this chapter we introduce the basic notions used for models of set theory: relativization,
- 26. Basic properties of PCF For any set A of regular cardinals define
- 9. Cardinality We denote |A| also by Card(A). The behaviour of this function under algebraic operations
- Problems in the set theory of Boolean algebras J. Donald Monk
- 7. Linearly ordered sets In this chapter we prove some results about linearly ordered sets which form a useful
- 6. Cardinals This chapter is concerned with the basics of cardinal arithmetic.
- Solutions, exercises in Chapter 8 April 12, 2009
- 11. Irredundance May 7, 2009
- Clubs and stationary sets A subset of an ordinal is unbounded iff for every < there is a such that
- 25. Cofinality of posets We begin the study of possible cofinalities of partially ordered sets--the PCF theory. In
- 13. Well-founded relations Here we introduce the usual hierarchy of sets, give a final generalized recursion theorem,
- Index of symbols March 21, 2009
- 3. Homomorphisms May 7, 2009
- 5. Products May 7, 2009
- 7. Complete Boolean algebras May 7, 2009
- Basic facts about ordinals and cardinals This is a brief introduction to elementary facts about ordinals and cardinals. We start
- Index of words March 21, 2009
- Solutions to exercises, Chapter 1 E1.1 Show that for n \2,
- Cardinal invariants on Boolean algebras 8. Cellularity
- 2. Subalgebras May 7, 2009
- Solutions, exercises in Chapter 6 E7.1 Show that any complete BA is isomorphic to RO(X) for some topological space X.
- Minimum-sized in nite partitions of Boolean algebras J. Donald Monk
- In this chapter we study infinite trees. The main things we look at are Konig's tree theorem, Aronszajn trees, and Suslin trees.
- J. Donald Monk Cardinal Invariants on
- 1. Special operations on Boolean algebras
- 5. Topological density We begin with some equivalents of this notion. A set X of non-zero elements of a BA A
- January 25, 2011 If A is a subalgebra of B, then (A) can vary either way from (B); for clearly one can
- 1. First-order logic Here we describe the rigorous logical framework for set theory, indicating some central
- 5. The axiom of choice We give a small number of equivalent forms of the axiom of choice; these forms should be
- 9. Clubs and stationary sets Here we introduce the important notions of clubs and stationary sets. A basic result here
- 17. Generic extensions and forcing In this chapter we give the basic definitions and facts about generic extensions and forcing.
- 18. Powers of regular cardinals In this section we give the first main result using forcing: consistency of the negation of
- 19. Relative constructibility In the next two chapters we give the proof of the consistency of AC. Roughly speaking,
- 23. Proper forcing This section is concerned with the proper forcing axiom (PFA), a generalization of (part
- The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000
- 12. Large cardinals The study, or use, of large cardinals is one of the most active areas of research in set theory
- 15. Constructible sets This chapter is devoted to the exposition of Godel's constructible sets. The development
- Solutions to exercises in Chapter 2 E2.1 Show that for every natural number m there is a BA A generated by a set X with
- Linearly ordered sets We give some results about linear orders which are useful in various constructions of
- May 7, 2009 For any infinite BA A, the depth of A is defined as
- 9. Independence May 7, 2009
- 6. Free algebras and free products May 7, 2009
- 4. Topological duality theory May 7, 2009
- These notes are intended to give a brief introduction to the general theory of Boolean algebras, and then go into the topic of cardinal functions on them. For this topic too these
- Solutions, exercises in Chapter 3 E3.1 Show that I is an ideal in a BA A iff I is an ideal in the associated ring Ar
- 22. Various forcing partial orders In this section we briefly survey various partial orders which have been used in forcing
- Notes on pseudo-tree algebras May 18, 2003
- Towers and maximal chains in Boolean algebras J. Donald Monk
- An interval Boolean algebra A such that a(A) ! t(A)
- 24. More examples of iterated forcing We give some more examples of iterated forcing. These are concerned with a certain partial
- Solutions for first practice exam 1 Define even integer, prime integer, n!.
- Review for first test The test will cover sections 19.
- Assignment #2 solutions Page 25, 4.1 Prove that the sum of two odd integers is even.
- section 003 MWF2, STAD135
- Assignment #1 solutions Page 2, 1.1 Simplify the following algebraic expression
- Assignment #4 solutions Page 63, 10.1 Write the following sentences using the quantifier notation (i.e., use the
- Index of names and words September 25, 2011
- Assignment #5 solutions Page 96, 14.4 Complete the proof of Theorem 14.5; that is, prove that congruence modulo
- Assignment #8 solutions Page 215, 25.1 We list several pairs of functions f and g. For each pair, please do the
- Index of symbols September 25, 2011
- Assignment #7 solutions Page 169, 21.8 The following sequences of numbers are recursively defined. Answer the
- Solutions for second practice exam 1 Write out the following in quantifier notation, using and , assuming that , , x, y
- Solutions for test 2 1. Write out the following using , , , and then write the negation of the sentence,
- Assignment #6 solutions Page 141, 19.1 Please state the contrapositive of each of the following statements
- Solutions for test 1 1. Define the following: (a) prime integer. (b) a|b. (c) (n)k.
- Assignment #3 solutions Page 43, 7.2 Airports have names, but they also have three-letter codes. For example,
- Assignment #9 solutions Page 264, 31.2 A pair of dice are rolled. What is the probability that neither die shows a
- Test 2a, take-home make-up Math 2001, section 003
- 4. First-order logic In this chapter we finish introducing the notion of first-order logic, and connect this notion
- 6. Basic model theory We survey important notions and results in model theory.
- 2. Terms and varieties We now make one important step towards full model theory: terms and equations. Let be
- 3. Sentential logic Here we discuss some more components of our final first-order logic: the logic surrounding
- 7. Morley's theorem This chapter is devoted to the proof of Morley's theorem, which says that in a countable
- 5. The compactness theorem Here we prove the compactness theorem: If a set of sentences is such that every finite
- Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of
- Model theory (Math 6000)