Palo Alto Institute for Advanced Study
2007-12-18
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Palo Alto Institute for Advanced Study 2007-12-18
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The Good Shepherd’s Paradox A Game of Golf(If you are familiar with mathematics, especially set theory, you may wish to skip straight to The Golfer’s Paradox.) One day, a Mathematician was golfing with three good friends, an Old — and very Scottish — Pro Golfer, a Banker, and a... the old-fashioned term is “Man of the Cloth”, which we might as well use here since he was... well, old-fashioned. As they played, the Mathematician launched a discussion of mathematics that would turn out to be more thought provoking than usual, especially with regard to set theory. As they played, the Mathematician filled their ears with the marvels of mathematics, especially the paradoxical mathematics of infinity, and of the infinity of infinities of modern transfinite set theory, and how it was the foundation of 2/3 of modern mathematics. If you are familiar with this mathematics, you may wish to skip straight to the The Golfer’s Paradox. Warning: it is assumed throughout that you are at least somewhat familiar with mathematics in general — for example, finite induction — and the transfinite part of set theory in particular. In particular, our Mathematician talked about a0 (aleph‑null), which is the first and only “countable” infinity of set theory, and which is the “cardinality” (roughly, the size) of the set of all the natural numbers, Á 4 {1,2,3...}. He told how it is constructed by starting with the number “1” and the set “{1}”, and then constructing the “successor” to 1, the number “2”, and the successor to the set {1}, the set “{1,2}”; you keep doing this so that, if the number n is in the set, Á, then the number n + 1 is also in Á; this is all done an “infinite” number of times... but in absolutely no time so that even though it is “infinite” — which in standard English seems to mean “unfinished” — the set is “complete”, and never has any more numbers added to it. “Finite induction” is a method of proof that derives from this manner of defining the set Á. This is all made possible by standard set theory’s “Axiom of Infinity”. “Transfinite induction” was also described. Our Mathematician rhapsodized
about the paradoxical “Hilbert Hotel”, which holds an infinite number of guests,
but can be made to hold infinities of infinities more guests by moving guests
from room to room. (Sound familiar?!) He then went into rapture about
He spoke of the transfinite
ordinal numbers, e.g. ω, ω
+1, and ωω and how their arithmetic was different from that of the transfinite
cardinals, e.g. a0 and
By definition of
Á, the
ultimate successor set of {1}, if the number n is in the set
Á, then
n + 1 is also in
Á. This
means that if we construct the set {0,1,2,3...}, we can show that it is the same
cardinality or size as {1,2,3...}. We do this by constructing a bijection between
the two sets. Every number n in {0,1,2,3...} has a unique counterpart
n + 1
in the set {1,2,3...}, n ↔ n + 1, so we have that each
n in
{0,1,2,3...} maps or bijects onto, i.e. to-from, each n + 1 in {1,2,3...}. Since there
exists a bijection, the sets are the same cardinality, i.e. are the same size.
But the set {0,1,2,3...} obviously has one more element than {1,2,3...}, so its
cardinality must be 1 greater than that of the set {1,2,3...}, i.e. its
cardinality must be a0 + 1 since the cardinality of {1,2,3...}
is a0.
But, since we just saw that there exists a bijection between
Á ~ {0} = {0,1,2,3...}
and Á 4 {1,2,3...},
they must have the same cardinality, so, paradoxically...
a0 + 1 = a0.
A comparable argument can be given for the other, uncountable infinities, e.g.
Our mathematician mentioned some of the other similarities between
transfinite sets and finite arithmetic, for example that one could exponentiate
2 sets, a and
b as ab
, and describing the arguments for fundamental set exponentiation theorem that
gives the cardinality of ab:
|ab| = |a||b|. In fact, our Mathematician went on,
all these arguments could easily be extended to
prove that n·a0 = a0,
These “equations” are paradoxical since each seems to imply that 1 = 0 (or even that a0 = 0). In standard arithmetic, any equation that has the form n + 1 = n can not be a valid equation since one is, and must be, able to subtract equal quantities from both sides of any valid equation and still get a valid equation. [The inequality n ≠ n might occur to some, but it, too, would standardly be considered invalid. In any case, it was not mentioned.] The “equation” 1 = 0 is therefore not considered to be valid; in fact, it is considered to “contradict” the standard inequality 1 ≠ 0. This is, or would give rise to a... “contradiction”, so in general one is not allowed to subtract infinity from both sides of a transfinite equation. (Our mathematician was also inspired to relatedly mention the all too frequent problem of renormalization in physics, at least its similarity to the problem of subtracting infinity from both sides of a transfinite equation, but he took it no further than that.) If an invalid equation or any other contradiction could be derived — validly, of course — from the axioms and rules of inference of arithmetic, or set theory, it would mean that arithmetic, or set theory, was “inconsistent”. Inconsistency would not only be considered bad; it would be... Apocalypse. And that would be... Bad. The Scotsman interjected a quick question: “if set theory is inconsistent — and I’m sure we all pray tha’ it is na’ — then we must be able to derive an invalid equation or other contradiction, is tha’ na’ so?” The Mathematician concurred, “yes, at least if all the possible theorems of the theory are derived. If you could derive a contradiction but fail to do so, the theory would still be formally inconsistent but you just wouldn’t realize it.” “Then we can na’ say a derivation is invalid just because we obtain a contradiction, is tha’ na’ so?” The Mathematician hesitated, but finally concurred. He seemed to have been affected by the Scotsman’s questions, because he then expanded on his earlier remarks: so, although the normal rules of finite arithmetic are used to develop transfinite arithmetic, once that transfinite arithmetic is developed, one then needs to abandon some of the normal rules so as not to wind up with... Apocalypse. Infinity is full of paradoxes, but that doesn’t make set theory... inconsistent. The Mathematician reassured them that far from being there any problem, mathematics was doing wonderfully. Infinity and paradox have been inseparable for millennia, but in the latter half of the 19th Century Georg Cantor and others succeeded in incorporating the paradoxes of infinity into modern mathematical theory consistently. The “success” of this marriage can be summed up in David Hilbert’s rhetorical question:
Yes, a marriage made in heaven. He never even heard the muttered “or some such place...” Their talk, which had been all but one sided, had all but stopped as they approached the 19th hole, their favorite...
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(ever so much larger than 
,
etc., but ω +1 ≠
ω, rather
ω +1 > ω, more
like we are used to for finite arithmetic. He first explained that two sets are the same cardinality, or size, if and
only if there existed a “bijection” between the two sets. A bijection is a
special case of a “map” or “mapping” — a connecting of elements — between two
sets, one in which each element of both sets is “mapped” to and from, i.e.
paired with, one and only one element of the other set. 
= 
= 
...
and so on. (He had an inspiration and also mentioned the interesting similarity of some of these last
transfinite equations to “paradoxical measure”, in which one can take a ball,
divide it up into a small number of pieces, and then rearrange the pieces so
that they form a ball that is twice the volume. But, he added, this similarity was “merely
coincidental, uhhh... probably.”) 
