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The Good Shepherd’s Paradox
A New Paradox of Infinity
In Set Theory
(cont.)
The Banker’s (... make that)
The Bean Counter’s Paradox
The Banker finally broke the
silence. He had been as stunned as the others by Willie’s demonstration. But he,
too, now had a gleam in his eye.
“Willie! I’ll go you one better.
Since I’m a Banker,...”
“Bean Counter!” they
simultaneously agreed, or disagreed, as the case may be.
“... I’ll call your paradox the
head of the ‘coin’ of this paradox. I’ll supply the tail. The golf balls will
represent coins and...”
“Beans!” the other two both said
at once. The Mathematician seemed wrapped in silence.
“... the glasses will represent
slots that can each hold one and only one coin.”
“Bean.”
“Willie, you didn’t
give the glasses — slots — numbers; you didn’t need to, but I will, here. Let’s
assume that before the Professor starts switching the coins in their slots, like
they do in set theory to get aleph‑null + 1 to... biject, isn’t it Professor?
biject 1-to-1 with aleph‑null — I know, I know, it’s probably the sets that biject, not the aleph‑nulls
or cardinalities per se — let’s assume that each coin has
the same unique serial number that its slot has. Nothing fancy, we’ll just go
with 1, 2, 3... like the Professor did.”
Figure 4. Homeless Golf Ball “0”, Many Golf Balls in Wee Small Glasses
The Banker arranged a few “wee
small glasses” in a row, each with its one-and-only-one golf ball. He took out a
marking pen and numbered several of the glasses and balls with numbers in pairs:
‘1’, ‘1’; ‘2’, ‘2’; ‘3’, ‘3’;... Then he took another ball and wrote a big ‘0’ on
it.
Then he mixed up the balls in
the glasses... with one ball — bean — left over.
“OK! Let’s assume that somehow
all the aleph‑null + 1 balls — coins — have been somehow bijected
1-to-1
with the aleph‑null slots. Let’s look at slot 1. If slot 1 does not have coin 1
in it... by the way, this is a Bank, so I call this... auditing! So as
we start our... auditing procedure,” he seemed to derive pleasure from saying
the word... “auditing”, “if slot 1 doesn’t have coin 1 in it, then we find
the slot that has coin 1 in it. Slot 1 will have some other coin in it, it
doesn’t matter which. We have only 2 pairs of coins and slots selected for this
stage of the auditing procedure: coin j in slot 1, and coin 1 in slot
k. No others are involved, or need to be involved at this stage.
Figure 5. Many Golf Balls in Wee Small Glasses, All Mixed Up...
“Without adversely affecting the
bijection, the global 1-to-1, we can switch those two coins so that coin 1
is in slot 1 and coin j is in slot
k; and the others — that we
don’t care about yet — are still paired just as they were. Any problem so far?!”
Figure 6. Many Golf Balls in Wee Small Glasses, All Mixed Up Except 1-1
The others shook their heads and
said “no.” The Banker proceeded.
“This procedure is general. For
any number except 0, there is both a coin with that number and a
slot with that number, involving at most 2 pairs of coins and slots. So, for ANY
number EXCEPT 0, we can switch the pairings of those 2 pairs, so that the
common number — it could be both, but can’t be 0 — is matched with itself.
After such a switch, we never again unpair them if they are properly paired.
With me so far?!” He was beginning to enjoy this.
The Mathematician looked a
little sick, but the others smiled their agreement.
“So we can use the Professor’s
finite induction to... do I need to go on?!”
The Mathematician looked even
sicker, but the two others smiled even bigger smiles as they shook their heads
“no!”
“So — in the same time it took
to create the set of natural numbers — we can ‘repair’ each and every original
pair, pair up every ball and wee small glass with its original counterpart with
the same number, and in such a way that we cannot complain that the ball 0 was
unfairly deprived — or even deprived at all — of a wee small glass. Excuse me,
make that coins and slots. So, coin 0 must be in a slot, but it can’t be
in slot 1, and it can’t be in slot 2, and it can’t be in slot 3... I’m sure
I don’t need to go on!
Figure 7. Many Golf Balls in Wee Small Glasses,
All Matched Up... Except 0?
“I was going to object” said our Man of the Cloth
“that you weren’t reaching a proper limit, but then I caught on that that’s
precisely the failure you wanted to point out, and that it’s a failure of
set theory itself, not of your procedure or derivation or whatever.”
Everyone looked at our Mathematician who nodded
mutely.
“Willie showed that you can’t biject aleph‑null + 1
golf balls with aleph‑null wee small glasses in the first place, and I have just
shown the other side of that ‘coin’, that if you assume that you have done so —
with coins and slots — then you can switch the coins in their slots in a
completely bijection-preserving fashion, and repair all the original coins with
their corresponding slots, leaving coin 0... well, I think it’s safe to say that
you can find it in the pocket of the desk clerk at the Hilbert Hotel, which, by
the way, to a Banker’s eyes resembles nothing so much as that classic swindle, a
pyramid scheme. If you don’t mind my expressing it this way, there is an “infinite”
difference between installing aleph-null arriving guests in aleph-null empty
hotel rooms, and installing them in aleph-null occupied rooms. You will
have to keep juggling them just like swindlers do their victims.
“We all know you mathematicians don’t like to argue
from physical analogies, and especially not to use them to ‘prove’ your
theorems. Perhaps you can tell us why you don’t think this analogy of coins and
slots — or golf balls and wee small glasses — reflects the formal essence of
transfinite set theory’s infinity + 1 equals infinity.” He paused, looking at
the Mathematician who just barely looked back.
“And now I see the wisdom of Willie’s questions on
the 17th green! If a theory is inconsistent, then we must be
able to derive a contradiction, so we can’t say a derivation is invalid
merely because we derive a contradiction! Willie, you are a wonder!”
“So, ‘reductio ad absurdum’ doesn’t apply!” said
our Man of the Cloth. “I was wondering if the Professor was going to say that
the assumption that you could apply standard rules of inference in the standard
way you were applying them was a bad assumption, as demonstrated by your ending
up with a contradiction, or that you were somehow invoking the Axiom of Choice,
or perhaps accuse us all of rabid Platonism. That’s what I would have done. But
Willie preempted him! Well, in the future one will certainly need to be more careful
when trying to apply proof by contradiction — or should I say
dis-proof by contradiction — because the assumption you are
disproving might be provable within the theory, and you might overlook that you
were actually proving that
the theory was inconsistent!
“By the way, this Auditing reminds me of what the
Professor called set subtraction. As the coins and the slots get paired up, they
are effectively subtracted out, leaving... us a present!”
You’re right! It is just
like set subtraction!”, our Banker agreed, and paused thoughtfully. “And that’s
as it should be! There’s nothing — that is,
there should be nothing — more
certainly in a 1-to-1 correspondence with a set than the set itself.”
“And, too,” added our Man of the
Cloth, “I think the similarity of those n times infinity equals infinity
equations to paradoxical measure is more than merely coincidental, that and that
childhood trick of proving that 1 = 2.” The others
looked at him, and nodded.
“Makes more than good sense”
said our Banker.
“That childhood trick
of proving that 1 equals 2 by multiplying and dividing by 0. How did it go?”
Our Banker remembered, “I think
you play with x2 - x2.
You can algebraically manipulate it into both x·(x -
x) and (x + x)·(x - x). Then divide both sides
by (x - x) and you get x = x + x = 2x. Yes!
yes! Paradoxical Measure!”
Our Man of the Cloth then added,
“I also thought it was a bit ironic that aleph-null to the aleph-null was
so much less than two to the aleph-null.” The others looked at
him, somewhat startled, with new eyes, even with new respect.
“Hoot, man! Good un’! I missed that un’ misself.”
said Willie.
Our Banker exchanged amazed but
approving glances with Willie, tried to with our Mathematician, who seemed lost
in thought, and then proceeded.
“Well, it looks like it’s
my turn to one up again. It’s a bit complex, but here goes: I just hate
to mention it...” he continued.
“I’m
sure you do!” muttered our Mathematician.
“... but I also find a problem
with your Axiom of Infinity. By finite induction — and
let me emphasize that that means by using only the standard axioms and rules of
inference of set theory, or results derived from them — every successor set of
{1} must have a maximum element, and
Á is a
successor set of {1}. But the Axiom of Infinity seems to say that there must
exist a successor set of {1} — you said
Á is the
usual
example — that has no maximum element. This is standardly inconsistent. Your
Á can’t
be a ‘set’ — at least not a successor set of {1} — if you want that kind of
consistency, which you say you do.”
“You’ll have to sharpen your
Axiom!” put in our Man of the Cloth, “and stop sittin’ on it!” added Willie.
“That was actually just Part 1.
Here’s Part 2, which synergizes with Part 1: Professor, in your proof that a0 + 1 = a0,
you use a variant of circular reasoning, but it’s
hidden somewhat when you use the ‘mapping’ from n ↔ n + 1
for all n in {0,1,2,3...} to
all n in {1,2,3...}.
You have been assuming you could do it
quote ‘for all the members of the set’
end-quote without
having to add a new member to the set, just because you could do it for an
arbitrary member with no apparent ill-consequences. But the ‘all members’
implicitly makes it the set you are acting on, like taking the successor set,
linking back to Part 1.
“The reason I call this circular
reasoning is because there is another way of describing it: you
assume that you can map all
n in {0,1,2,3...} to all
n + 1 in {1,2,3...} with the
n ↔ n + 1
mapping (without extending the second set with a new element, which would be...
cheating, or worse), but you do it by first
implicitly assuming that you can do it for all
n in {1,2,3...} to all
n + 1 in {2,3,...}, then
trivially mapping 0 ↔ 1
to fill out the result.
That this is an assumption is hard to see because you
‘prove’-as-if-intuitively-obvious-and-do
all the n ↔ n + 1
mappings ‘simultaneously’,
as it were, instead of sequentially or serially like when you
defined the set {1,2,3...} in the first
place using the Axiom of Infinity, thus failing to notice Willie’s paradox, which
shows that you must extend the set with an empty glass to find a home for the lone
ball. I want to accent that ‘defined’ because, aren’t you required to
be able to substitute the original definition in place of the defined entity at
any point?! If you say you don’t, it’s a lot like:
‘We are out on a limb, and we no longer depend on the tree we climbed to
get out on the limb.’”
Our Banker paused as if
wondering which of many alternatives to proceed with. “Anyway,
I would like to call it ‘recursively circular reasoning’, or perhaps ‘infinitely
recursively circular reasoning’ because it recurses off in the direction of
infinity.”
“Where
it’s very hard to figure out what really happens when you get
there!” added our Man of the Cloth. The others nodded thoughtfully. Our Banker
continued.
“The
synergy with Part 1 makes it hard to see because, when you created the Axiom of
Infinity, you only bothered to see the consequence that there was provably no
maximum element. You didn't bother to also prove — as is trivial to
do — the consequence that there also
must be a maximum element, in our case
for the set of all natural numbers {1,2,3...}, the ultimate successor set of
{1}. Apparently you assumed that if
you had already proven the former true, its negation was thereby proven false
and could therefore not
also be proven true!
Bad Assumption! (Especially after your
friend Gödel raked theoretical completeness and
consistency over the coals.) It can... if
set theory is inconsistent!”
“Though I’m
sure we all still pray tha’ it is
na’!” litanied Willie.
“Amen!” responsed our Man of the Cloth.
“Professor,”
went on our Banker, “you mentioned the
infamous ‘Axiom of Choice’, and how you mathematicians invented it because you
found that when you constructed certain infinite sets, it seemed to produce
nasty ‘paradoxical’... situations. I think I know why. Your Axiom of Choice allows
you to construct infinite sets in certain ways, but you already have infinite
sets that you constructed without using the Axiom of Choice, and then you
construct more sets from them and get... ‘paradox’ of some sort.
“I called my process ‘Auditing’
because I’m a Banker. I liken this situation to a Bank having a set of Books,
but when the Auditor comes to town and wants to examine the Books by
constructing another set of Auditing Books — to keep track of and compare
certain key pieces of information — this new set of Books, constructed
from the old, frequently causes just the same kind of problems you have!
Actually, your Axiom of Choice is not problematic by itself, but because it
allows you to conduct Auditing procedures on your preexisting sets of Books! But
in your case you questioned the Auditing, or rather the ‘assumption’ that it’s
allowable to construct the necessary Auditing Books. You failed to question the
preexisting Books and the system that produced them. The Auditing Books
merely showed up the pre-existing... oversights.
“And Willie strikes again, since
you are not allowed to avoid such derivations or constructions just because they
produce contradictions! But that’s just what you did when you invented the Axiom
of Choice. You decided that — only sometimes, to be sure — Auditing causes
Embezzlement, so you created an ‘Axiom of Auditing Allowed’, inescapably
infamous, in order to be able to dis‑allow Auditing if it became too...
troublesome.
“Professor,” our Banker went on, “maybe you can
tell us if Willie and I have avoided implicitly invoking an Axiom of Choice-like
substance. I believe we have, but it doesn’t make that much difference either
way. You now have a ‘0’ that will refuse to just be ‘1’ of the crowd, and a ‘1’
— as in aleph‑null + 1 — that will refuse to be a ‘0’ — as in aleph‑null + 0.
Or as in 1/n = 0
but with a remainder of 1, where
n is one of your infinities — make that
‘transfinities’ — since the remainder of 1 will never go away, no matter how
big the transfinity, and no matter how many times you divide that remainder of 1
by that transfinity. By
the way, this all means ‘infinitesimals’ — make that ‘transfinitesimals’ — are natural — contradicting your
Archimedean axiom, unless you extend it to transfinite numbers — and don’t need to be postulated, and your concept of
‘continuum’ will have to give way to a concept of ‘quantinuum’, well, actually
multiple ‘quantinua’, since they will mostly be incommensurate.
“There’s more. Your infinity
can’t be approached gradually; you have to make a big leap to get there. But if
infinity + 1 is always greater than infinity — and it seems it must be — then
you naturally get an infinity that can be approached and exceeded gradually, a
‘fuzzy’ — or perhaps a ‘relativistic’ — infinity. (Wasn’t someone looking for a
‘quantum relativity’ back a while ago?!) At least you
should be able to develop a theory of such, one that would even allow a
solution to your renormalization problem, maybe even a consistent one!
“But in any case, proceeding to
the bitter end... paradoxically, coin 0 both must be paired and
can’t be paired with a wee small glass! Slot! Abstractly equivalent to being
between the traditional rock and the traditional hard place! So I call this...”
“The Bean Counter’s Paradox!”
the other two
shouted out in unison. Our Mathematician looked a little green.
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