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Readers here have a chance to test their own mathematical knowledge, wisdom,
integrity...
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Is standard set theory — e.g. ZF, ZFC, von Neumann... Cantor... — standardly
consistent?
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How sure are you?
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Would you consider mathematics a failure, or yourself a
failure as a mathematician, if there existed standard
contradictions/inconsistencies (the theoretically fatal kind, as opposed to “mere paradoxes”)
— i.e.
serious... oversights — in set theory? Would
you if they were trivial... oversights?
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What should be done about people who question or try
to find fault with fundamental mathematical beliefs such as the “intuitively
obvious” consistency of set theory?
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Rather look at
pictures?!
The Good
Shepherd’s Paradox
A New Paradox of Infinity
In Set Theory
by Michael Hugh Knowles
Intro: A
Game of Golf
An old Scottish Pro Golfer, a Banker, and a Man of the Cloth are playing a round
of golf with a Mathematician. The Mathematician fills their ears with his
glorious vision of the wonders of transfinite set theory. (This is intro
material that can be skipped if you are impatient to start gnawing on...
The Golfer’s Paradox
At the 19th hole, Willie, the old Scottish Pro, shows
that trying to add a lone golf ball to an infinity of golf balls paired
1-to-1 with
“wee small glasses” is... too paradoxical.
The Banker’s (or
is it The Bean Counter’s) Paradox
Doing Willie one better, the Banker shows that if you
assume that you start
with “infinity + 1” golf balls “paired”
with only “infinity” “wee small glasses”... again,
too paradoxical.
The Man of
the Cloth’s Paradox
Our Man of the Cloth inspires us all with his vision of how even just a single sheep
cannot be truly lost, even in an absolute infinity of...
The Good
Shepherd’s Paradox
The Last Word on the subject...
Analysis — The Golfer’s Paradox
examined more closely and more formally.
Analysis — The Banker’s Paradox
examined more closely and more formally.
Final Analysis — The Good Shepherd’s Paradox.
Figures 1-3 from
The Golfer’s
Paradox.
(See
Figure 1. One Homeless Golf Ball, One Golf Ball in One Wee Small Glass)
(See Figure 2. One Homeless Golf Ball, Two Golf Balls in Two Wee Small Glasses)
(See Figure 3. Homeless Golf Ball, Aleph‑null Golf Balls in Wee Small Glasses)
Figure 4-7 from
The
Banker’s Paradox
(See Figure 4. Homeless Golf Ball “0”, Many Golf Balls in Wee Small Glasses)
(See
Figure 5. Many Golf Balls in Wee Small Glasses, All Mixed Up...)
(See
Figure 6. Many Golf Balls in Wee Small Glasses, All Mixed Up Except
1-1)
(See Figure 7. Many Golf Balls in Wee Small Glasses,
All Matched Up... Except 0?)
From the sub-section on
The Good Shepherd’s Paradox
(See The Good Shepherd...
?)
Copyright © 1995 - 2002,
Michael Hugh Knowles. All rights reserved.
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