The Infinite

Cantor’s 1891 Diagonal proof:

A comprehensive logical analysis that demonstrates how various untenable illogical assumptions have been added onto the proof itself

The Power Set Proof:

A proof based on the idea behind Cantor’s 1891 Diagonal Proof

Non-Diagonal Proofs and Enumerations:

Why an enumeration can be possible outside of a mathematical system even though it is not possible within the system

Infinity and the Infinite:

How conventional mathematics makes illogical assumptions that lead to contradictions

Why do people believe weird things?

Why do intelligent people continue to defend beliefs that have no logical basis?

Are there different sizes of infinity?

A look at the conventional definition of Cardinal Numbers and Cardinality

Indefinable numbers:

A demonstration of contradictions that arise from the notion of “indefinable numbers”

The origins of transfinite numbers:

How Cantor started the notion of limitlessly many numbers that are bigger than a limitlessly large number

The Axiom of Choice and Well-Ordering:

What the terms “axiom of choice” and “well-ordering” really mean

The Continuum Hypothesis:

How the notion of different levels of infinity created an unsolvable problem

Proof of more real numbers than natural numbers?

Has there ever actually been any proof that there are more real numbers than natural numbers?

The Courant & Robbins Contradiction:

A contradiction in a proof that provides a demonstration of the inherent contradictions that typically result from illogical assumptions about the infinite

Lebesgue Measure Theory:

A demonstration by a logical analysis that reveals the inherent contradictions in the Lebesgue theory of measure

Lebesgue Measure Cranks:

A look at cranky attempts to justify Lebesgue measure theory

Understanding Limits and Infinity:

How the correct application of limit states to sets composed of infinitely many intervals can eliminate contradictions.

Understanding sets of decreasing intervals:

The crucial points that enable full understanding of sets of decreasing intervals

Real numbers and infinity:

An analysis of what real numbers are in terms of a logical consideration of language

Cantor’s invented numbers:

How Cantor’s invention of transfinite numbers ignored obvious contradictions and which continue to be ignored

Cantor’s religious beliefs:

How Cantor’s religious beliefs influenced his invention of transfinite numbers

A list of real numbers with no diagonal number:

How to define a list of real numbers for which no Diagonal number can be defined

Sums of infinitely many fractions, Part 1:

The contradiction inherent in the concept that infinitely many numbers can be summated

Sums of infinitely many fractions, Part 2:

Another demonstration of an inherent contradiction in the concept that infinitely many numbers can be summated

On the Infinite:

A critical look at David Hilbert’s 1925 lecture On the Infinite (Über das Unendliche)

Actual and Potential Infinity:

Are there two types of infinity, actual completed infinity and potential infinity?

Proving Goodstein’s theorem:

Are transfinite numbers really necessary to prove that every Goodstein sequence terminates?

Formal papers:

A selection of formal papers related to the infinite

Cantor: On infinite linear point-sets, Part 1:

(Über unendliche lineare Punktmannig-faltigkeiten)

Cantor: On infinite linear point-sets, Part 2:

(Über unendliche lineare Punktmannig-faltigkeiten)

Cantor: On infinite linear point-sets, Part 3:

(Über unendliche lineare Punktmannig-faltigkeiten)

Cantor: Grundlagen - Foundations of a general theory of sets:

(Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Part 5 of Über unendliche lineare Punktmannig-faltigkeiten)

Cantor’s 1874 “On a Property of the Set of all Real Algebraic Numbers ”:

(Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen)

Cantor’s 1895 “Contributions to the Foundations of the Theory Of Transfinite Numbers”

(Beiträge zur Begründung der transfiniten Mengenlehre)

Cantor: “A Contribution to the Theory of Sets”:

(Ein Beitrag zur Mannigfaltigkeitslehre)

Cantor’s 1891 Diagonal paper “On an elementary question of set theory”:

(Über eine elemtare Frage de Mannigfaltigkeitslehre)

Julius König’s On the Foundations of Set Theory:

An online English translation of the paper that initiated the notion that there must be undefinable real numbers

Giuseppe Vitali’s On the problem of measuring sets of points:

An online English translation of Giuseppe Vitali’s 1905 paper that raised the problem of immeasurable sets of points

Platonism:

How Platonism has been the source of irrational assumptions about the infinite

Logic and Language:

How the failure to realize that logic is not independent of the language it is stated in has led to contradictions

Mathematical Proofs:

Despite claims to the contrary, the reality is that much of mathematics is not supported by logical proof

The Platonist Rod Paradox:

A paradox demonstrating the pitfalls that can result from the failure to apply limits to cases involving infinity

Misunderstanding the Infinite:

A review of how Shaughan Lavine’s book Understanding the Infinite demonstrates the author’s failure to understand his subject

A failure to understand:

A demonstration of a fairly typical failure to understand Cantor’s Diagonal proof