The Platonic Dream
Essays in Fundamental Mathematics
On the Set of All Numbers
These first essays may seem redundant, because I have covered this subject at the beginning of The Map of Physics. But it well behooves us to begin a series of essays on number theory with a discussion of the subject of the theory and of the object of the theorizing.
When we talk about a set we must begin by defining the set; that is, by establishing the criterion by which we include an element in the set or exclude it from the set. Having done that for the set of all numbers, we will then develop number theory by examining the logical relationships among the members of the set and of various subsets.
GO TO: Logical Proof
GO TO: Mathematical Induction
GO TO: The Axioms of Set Theory
GO TO: The Natural Numbers and The Integers
GO TO: The Integers
GO TO: Elementary Infinity
GO TO: The Real Numbers
GO TO: Advanced Infinity
GO TO: An Addendum to Advanced Infinity
GO TO: The Semantics of Infinity
GO TO: Hilbert's First Problem
GO TO: Infinitesimal
GO TO: The Complex Numbers
GO TO: The Fundamental Theorem of Algebra
Algebraic Playbook
GO TO: The Algebra Playbook
GO TO: Factoring X2+1
GO TO: Everting the Doppler Shift
GO TO: Cardano's Solution of the Depressed Cubic Equation
Number Theory
GO TO: Pythagorean Triples
GO TO: The Bernoulli Numbers
GO TO: Gaussian Sums
GO TO: More on Gaussian Sums
GO TO: Gaussian Sums of Powers
GO TO: Enmeshments
GO TO: The Theorem of Finite Differences
GO TO: Katy's Theorem
Infinite Series
GO TO: Euler's Theorem
GO TO: A Minor Point on Infinite Series
GO TO: Powers and Roots of The Simple Geometric Series
GO TO: Powers and Roots of The Alternating Geometric Series
Analysis
GO TO: Leibniz's Theorem
GO TO: The Dirac Delta
GO TO: Differentials, Exact and Inexact
GO TO: Vector Multiplication
GO TO: Vector Differentiation
GO TO: Liouville's Theorem
Geometry
GO TO: Area of an Elliptic Sector
GO TO: Coordinate Transformations
GO TO: The Levi-Civita Tensor
Differential Equations
GO TO: The Classical Harmonic Oscillator
GO TO: Solving The Wave Equation
GO TO: Solving Hermite's Equation
GO TO: The Hermite Polynomials
GO TO: Solving Bessel's Equation
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