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?
Other Paradoxes based on self-reference
Halbach and Zhang’s Yablo without Gödel
Confusion of levels of language can lead to paradoxes
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:
A The notion of different sizes of infinity leads directly to paradoxes and contradictions
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
Why Gödel’s proof of incompleteness is inherently paradoxical and contradictory
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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