Titlebook: An Outline of Set Theory; James M. Henle Book 1986 Springer-Verlag New York Inc. 1986 Finite.calculus.cardinals.mathematics.ordinal.set th

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The Natural NumbersThe object of this chapter is to define a set to represent the numbers 0, 1, 2, .... To be complete, we must also show how to add and multiply these numbers and prove all the usual laws: commutative, associative, etc. The most important idea contained in our construction is that of mathematical induction.

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The IntegersIn this chapter we will construct a set to represent the positive and negative integers. As before, we will define addition and multiplication. In addition to the properties proved for ℕ , we will now have additive inverses. The key idea in our construction is the use of equivalence classes.

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The RationalsOur next goal is to construct the rational numbers. The method is very much like that of the previous chapter.

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The Real NumbersWe complete our construction of the standard number systems with Dedekind’s approach to the real numbers. For various reasons, there is a lot more work involved in this task, so we will limit ourselves to the definition of ℝ, +. and 0., and some examination of the difficulties of proceeding further.

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The OrdinalsWe wish to extend ℕ, our set of counting numbers, to a larger class of numbers we can use to count infinite sets. These will be our first type of infinite number, and they will be used to measure the “lengths” of large sets.

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The UniverseWe now explore some pure set theory, examining the structure of the universe of sets. A crucial concept will be that of a set which in itself is a universe of sets, that is, all the axioms of ZF are true about the members of this set.

Mets552

Choice and InfinitesimalsWe prove here Theorem 7.10 which offers three equivalent forms of the Axiom of Choice. We then use AC to construct a system of numbers called the Hyperreal numbers (ℍℝ). This system extends ℝ as ℝ extended ℚ and ℚ extended ℤ. ℍℝ contains both infinite numbers and infinitesimals.

GORGE

The Integers # 13. 3.1. As you try to prove transitivity you will realize that you are missing an important fact about ℕ, a cancellation law:

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The Ordinals # 24. We meet here yet another of the many faces of induction. Under ordinary circumstances the following principles (on any linearly ordered set) are the same: