Chapter 6 on Set Theory Part I

This is a first posting on Chapter 6; there may be others.

Mathematics is not about sets; it is about measurement of the world. But this book needs to assess set theory because:

  • Set theory is common currency in mathematics. It is presupposed and taken for granted in all advanced mathematical writings since the early 20th century
  • It is often alleged to be the foundation of mathematics. Most attempts to find a realist approach to mathematics ultimately connect to set theoretic axioms
  • The validity of set theory in general or, at least, the formal axiomatic approach to set theory, is highly questionable in a reality-based approach.
  • If set theory is valid, or can properly be rehabilitated, a reality-based answer to the question: “What is a set?” is essential.
  • One needs to address the key question: What, properly, would be the measurement function performed by sets?
  • It is essential to assess the status of the proclaimed axioms of set theory
  • If there is a valid concept of mathematical set, it remains to assess its value. The clearest way to offer an affirmative answer is to provide a non-trivial application to something of demonstrable value

These are a lot of bases to cover and even readers sympathetic to my approach are likely to find Chapter 6 the most difficult one in the book. The chapter begins with a positive treatment of mathematical sets and only offers its critical review of formal axiomatic set theory at the end. Chapter 6 covers such topics as:

  1. What is a mathematical set? How does a set differ from a concept? In what sense is a set open-ended, like a concept? In what sense is it not open-ended? What is its proper sphere? What measurement-related function does it serve?
  2. Proper domain of set theory. Set theory as specifically applicable to mathematics. Mathematics conceptualizes the world with respect to measurement of differences. Set theory as presupposing mathematical concepts. An indication of proper hierarchy in mathematics.
  3. Sets as a) primitive measurements b) performing a function of isolation c) a perspective on distinguished objects of measurement. A reality perspective on set theoretic “constructions.” Constructions as recognitions of relationships among quantities.
  4. Point set topology as a non-trivial and important application of sets. What is the purpose of point set topology and why is it important?
  5. Axioms of set theory, context and nature of the ZF axioms and their application to mathematics. Critical comments.
  6. Why has mathematics survived? (I claim that it has.)
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2 thoughts on “Chapter 6 on Set Theory Part I

  1. On page 295 you state “The concept of man is open-ended. But the referents of the concept man do not constitute a set.”

    I am trying to figure out why not. Are you saying we cannot determine the referents in some cases? e.g. when a fetus becomes a man? Or when prior species evolved to be called human(i.e. man)? Or is it something else entirely.

    On the surface it seems that you want to discount infinite sets for concretes. But if I say the “set of all men,” that may only be finite–most likely finite. It seems we can delimit them based on DNA.

    I am not challenging the statement, just trying to follow where you are going with this.

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    • Trent,

      The challenge of Chapter 6 is to offer a reality-oriented alternative to standard axiomatic set theory and one first step is to distinguish mathematical sets from concepts.

      Regarding the concept ‘man’, I don’t consider the existence of borderline cases to be the fundamental issue, nor do I have a problem with sets of specifically demarcated concretes. But a concept, in general, specifies a kind of thing, not a set of things. One can, indeed, specify a set of existing men or even of men who have lived in the past, but to specify a set of men does not end with forming the concept of man.

      The units or referents of a concept, such as ‘man’ or ‘balloon’ or ‘galaxy’, include all men or balloons galaxies that have ever existed, exist today, or will ever, in the fullness of time, ever come into existence. The concepts apply to these referents, not by virtue of specifically identifying them, as such, but by virtue of the kind of concretes that these referents are. One can refer to future potential existents, but one cannot specifically identify them.

      A mathematical set, by contrast, is a specific demarcation of possible values within a specific constellation of dimensions, pertaining, generally and in some fashion, to measurement. A set is, in this sense, “well-defined” and is distinguished specifically by its membership: Two different characterizations specify the same set if they specify the same elements.

      The virtue of a concept is that it is open-ended, that it applies to the not-yet known or encountered and, also, that it embraces characteristics, currently unknown, that may one day be discovered. The virtue of a set is that it is well-defined, as specific values among a specific range of possibilities, possibilities that, by some conceptual means, are identified.

      In the course of Chapter 6, I discuss how the concept of set arises in mathematics, what it refers to, why it is needed, how it relates to measurement, and what it accomplishes.

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