Banach-Tarski and other similar "paradoxes" are NOT about the Dogma of Choice. What these paradoxes actually demonstrate is the ABSURDITY of viewing geometrical objects as completed infinites of points, i.e. sets of infinitely many points. Examples of equally as ridiculous demonstrations that don't require choice are:
- Sierpinski–Mazurkiewicz Paradox, which states that there are infinite subsets of the plane which can be decomposed into several parts which are then translated by rigid motions to form two identical copies of the original subset. No choice required!
- Exactly the same but the subset is a subset of the RATIONAL sphere (i.e. points on the 2-sphere in R^3 with rational coordinates) and the rigid motions are motions of the sphere.
and many other similar "paradoxes". When the modern mathematician is presented with them, his retort is to say "oh well that just means we don't understand infinity that well". Yeah, sure, you don't find infinity intuitive yet you are comfortable with pretending to have completed the infinite and with manipulating it like any other finite object! How ridiculous!
These so-called paradoxes clearly demonstrate that their infinitist schizo mathematics have NOTHING to do with modelling nature and do NOT represent intuitive ideas of how the world works. Everything about it is pure COPE.
- Sierpinski–Mazurkiewicz Paradox, which states that there are infinite subsets of the plane which can be decomposed into several parts which are then translated by rigid motions to form two identical copies of the original subset. No choice required!
- Exactly the same but the subset is a subset of the RATIONAL sphere (i.e. points on the 2-sphere in R^3 with rational coordinates) and the rigid motions are motions of the sphere.
and many other similar "paradoxes". When the modern mathematician is presented with them, his retort is to say "oh well that just means we don't understand infinity that well". Yeah, sure, you don't find infinity intuitive yet you are comfortable with pretending to have completed the infinite and with manipulating it like any other finite object! How ridiculous!
These so-called paradoxes clearly demonstrate that their infinitist schizo mathematics have NOTHING to do with modelling nature and do NOT represent intuitive ideas of how the world works. Everything about it is pure COPE.
