# Logic & Mathematical Paradoxes by Sindy Dunbar

By Sindy Dunbar

Desk of Contents
Chapter 2 - All Horses are an analogous colour & endless Regress
Chapter four - Paradoxes of fabric Implication
Chapter 6 - unforeseen placing Paradox
Chapter nine - Gabriel's Horn & lacking sq. Puzzle
Chapter 14 - Godel's Incompleteness Theorems
Chapter 15 - Gambler's Fallacy

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Extra info for Logic & Mathematical Paradoxes

Example text

Overlaying the hypotenuses from both figures results in a very thin parallelogram with the area of exactly one grid square, the same area "missing" from the second figure. According to Martin Gardner, the puzzle was invented by a New York City amateur magician Paul Curry in 1953. The principle of a dissection paradox has however been known since the 1860s. The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers. Many other geometric dissection puzzles are based on a few simple properties of the famous Fibonacci sequence.

Existential presuppositions A number of authors have argued that propositions of the form "All A are B" presuppose that there are objects which are A. This analysis has been applied to the raven paradox: ... H1: "All ravens are black" and H2: "All nonblack things are nonravens" are not strictly equivalent ... due to their different existential presuppositions. Moreover, although H1 and H2 describe the same regularity - the nonexistence of nonblack ravens - they have different logical forms. The two hypotheses have different senses and incorporate different procedures for testing the regularity they describe.

Using this terminology, the Banach–Tarski paradox can be reformulated as follows: A three-dimensional Euclidean ball is equidecomposable with two copies of itself. In fact, there is a sharp result in this case, due to Robinson: doubling the ball can be accomplished with five pieces, and fewer than five pieces will not suffice. The strong version of the paradox claims: Any two bounded subsets of 3-dimensional Euclidean space with non-empty interiors are equidecomposable. While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of Bernstein–Schroeder theorem due to Banach that implies that if A is equidecomposable with a subset of B and B is equidecomposable with a subset of A, then A and B are equidecomposable.