Changes

=== [[Banach-Tarski Paradox]] ===

=== [[Banach-Tarski Paradox]] ===

The only miracle described in all four [[Gospels]] is the multiplication of loaves and fish by [[Jesus]], thought for thousands of years to be a mathematical and physical impossibility.  But the 20th century discovery of the so-called [[Banach-Tarski Paradox]] proved a mathematical basis for creating identical copies of an object.

The only miracle described in all four [[Gospels]] is the multiplication of loaves and fish by [[Jesus]], thought for thousands of years to be a mathematical and physical impossibility.  But the 20th century discovery of the so-called [[Banach-Tarski Paradox]] proved a mathematical basis for creating identical copies of an object out of no extra material.

=== Axiomatization of Arithmetic ===

=== Axiomatization of Arithmetic ===