Classical Mathematical Logic

The Semantic Foundations of Logic

Richard L. Epstein

• Publisher: Princeton University Press
• Year: 2011
• ISBN:
  9781400841554
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations.
The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference. Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

XIX	Two-Dimensional Euclidean Geometry
	in collaboration with Leslaw Szczerba

A. The Axiom System E2

• Exercises for Section A

B. Deriving Geometric Notions

1. Basic properties of the primitive notions

2. Lines

3. One-dimensional geometry and point symmetries

4. Line symmetry

5. Perpendicular lines

6. Parallel lines

• Exercises for Sections B.1–B.6

7. Parallel projection

8. The Pappus-Pascal theorem

9. Multiplication of points

C. Betweenness and Congruence Expressed Algebraically

D. Ordered Fields and Cartesian Planes

E. The Real Numbers

• Exercises for Sections C–E

Historical Remarks

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A. The Axiom System E2

In this chapter we'll continue our geometric analysis of the real numbers by formalizing the geometry of flat surfaces. Our goal is to give a theory that is equivalent to the theory of real numbers presented in Chapter XVII.

Our axiomatization of two-dimensional geometry will use the same primitives as for one-dimension: points and the relations of betweenness and congruence. Lines and other geometric figures and relations, which others often take as primitive, will be definable. Roughly, since two points determine a line, we can define a line as all those points lying in the betweenness relation with respect to two given points. Then we can quantify over lines as “pseudo-variables” by quantifying over pairs of points.

So, as in Chapter XVIII, our formal language will be L( = ; P₀³, P₀⁴), which again we can write as L(=; B, = ) with the sam...