Paradoxes

Paradoxes in general:

A brief overview of paradoxes

The Liar Paradox:

The original self-referential paradox: “This sentence is false”

The Surprise Examination paradox:

A seemingly challenging paradox that is easily resolved by logical analysis.

The Balls in the Urn Paradox (The Ross-Littlewood paradox):

How woolly thinking about infinity can lead to contradictions

The Platonist Rod Paradox:

A paradox inspired by the Balls in the Urn Paradox and which demonstrates how a belief in the “existence” of an actual infinity leads to contradictions

The Drinker Paradox:

A paradox based on the pretense that one can apply classical formal logic to a constantly changing real world situation

Gödel on Church’s Paradox:

A paradox that is reliant on the confusion of different levels of language

Curry’s Paradox (Lob’s paradox):

A paradox that relies on an inconsistent system for its creation

Goodman’s Paradox (The New Riddle of Induction):

Despite claims that the paradox arises from scientific considerations, in fact it is a result of ambiguity of definition

The Bertrand Paradox:

How a question that isn’t sufficiently well-defined can result in two or more different answers that can all be correct

Wang’s Paradox (The Sorites paradox):

If you keep removing grains of sand from a heap of sand, when does the heap stop being a heap?

Language, Logic and Intuition:

A brief summary of the need for logic rather than intuition in the use of language to arrive at conclusions

Logic and Language:

Why logic is not independent of language

Natural Language and Reality:

How the failure to understand the limitations of natural language lead to absurdities

Infinity and the Infinite:

Paradoxes and contradictions arising from ideas on infinity in conventional mathematics

Cardinal numbers and different sizes of infinity:

How the notion of different sizes of infinity leads directly to paradoxes and contradictions

Indefinable numbers:

Why the notion of “indefinable numbers” is inherently paradoxical and contradictory

The origins of transfinite numbers:

How Cantor started the notion of a number bigger than any other number

Proof of more real numbers than natural numbers?

The notion that there are more real numbers than natural numbers, as well as having no proof, is inherently paradoxical

The Courant & Robbins Paradox:

A contradiction in a proof that demonstrates the inherent contradictions that result from illogical assumptions about the infinite

Gödel’s paradox:

Why Gödel’s proof of incompleteness is inherently paradoxical and contradictory