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  1. No Access

    Book

    Introduction to Cardinal Arithmetic

    M. Holz, K. Steffens, E. Weitz in Modern Birkhäuser Classics (1999)

  2. No Access

    Chapter

    The Supremum of pcf μ (a)

    In this chapter we introduce first the essential notion of a control function. Its importance will come out when, in Chapter 8, we determine the cofinalities of partial orderings of the form ([λ]≤k

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  3. No Access

    Chapter

    Applications of pcf-Theory

    As a first application of pcf-theory, we will prove in this section two estimates for cardinal numbers. Initially, if δ is a limit ordinal and μ is a cardinal with μ < ℵ δ then ...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  4. No Access

    Chapter

    Generators of T <λ+(a)

    In this chapter, a denotes an infinite set of regular cardinals. We will prove the following assertion, due to S. Shelah: If |a|+ < min(a) and λ ∈ pcf(a), then there is a set bλ ∈ T<λ+(a) which generates the idea...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  5. No Access

    Chapter

    Local Properties

    In his book [Sh5], Shelah introduces the operator pcf* which satisfies pcf*(pcf*(a)) = pcf*(a). If b is a set of regular cardinals, which is not necessarily progressive, and if every limit point of b is a singula...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  6. No Access

    Chapter

    The Cardinal Function pp(λ)

    In this chapter, we introduce Shelah’s cardinal function pp k (λ) whose properties we now summarize. If λ is a singular cardinal, then the definition of pp k ...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  7. No Access

    Chapter

    Introduction

    If M and N are setsl and if there exists a bijection2 from M onto N, then we say that M and N are equinumerous, and write M ≈ N. To measure the number of members of a set, we will introduce sets of comparison. Wi...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  8. No Access

    Chapter

    Approximation Sequences

    The methods applied in the previous chapters are classical and, in a certain sense, elementary. In this chapter, we will introduce modern model theoretic methods. Notions such as “model of ZFC” and “absolutene...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  9. No Access

    Chapter

    Foundations

    The year of birth of set theory can be regarded as 1872. In this year, Georg Cantor introduced the notion of a transfinite (infinite) ordinal number His investigations, which soon made set theory an independen...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  10. No Access

    Chapter

    Ordinal Functions

    In the Galvin-Hajnal theorem, the question of whether it holds for singular cardinals with countable cofinality was left open. Let ℵδ be singular and v be a cardinal. If δ = ℵδ, then

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  11. No Access

    Chapter

    The Galvin-Hajnal Theorem

    The so-called singular cardinal problem consists of the description of the possible size of the cardinal, % MathType!MTEF!2!1!+- % feaagCart1e...

    M. Holz, K. Steffens, E. Weitz in Introduction to Cardinal Arithmetic (1999)

  12. No Access

    Article

    Eine Unterscheidung zur Unterscheidung und Trennung ein- und mehrbasischer Säuren

    E. Weitz in Zeitschrift für analytische Chemie (1926)