The Good Shepherd’s Paradox

A New Paradox of Infinity

In Set Theory

(cont.)

Analysis of The Golfer’s Paradox

(If, at any time, you find yourself saying “you can't do that BECAUSE you get a contradiction or something undefined or [any variant thereof]...”, please refer to Fundamental... Oversights.)

One of the most crucial observations our Golfer — a canny Scotsman — makes is before the 19th hole, during the Game of Golf. He gets our Mathematician to admit that a derivation cannot be considered invalid just because a contradiction is obtained, because it might be that the theory itself is inconsistent, in which case one should be able to — and, we add here, should also wish to — derive a contradiction. More on this and its psychology in Fundamental... Oversights.

At the 19th hole, our Golfer melodramatically presents a single golf ball in a single “wee small glass” and a single lone golf ball, the paired ball and glass unarguably bijected, i.e. paired 1-to-1, the lone golf ball unarguably totally unpaired and unbijected. This beginning, humble though it is, sounds the death knell of set theory, or at least of its concept of infinity that includes a₀ + 1 = a₀ (etc). To help us judge the importance of this question, our Mathematician, himself, remarked that 2/3 of modern mathematics depends on set theory, and most of that 2/3 has that concept of infinity with its a₀ + 1 = a₀ pumping its transfinite heart.

(See Figure 1 in The Golfer’s Paradox.)

At first our Mathematician sees no problem, the lone ball and paired ball and “wee small glass” cannot possibly be manipulated so that they are all 3 paired...  uhhh... “1-to-1”, “1-ball-to-1-glass”. It even sounds silly putting it that way. But that, as it turns out, is just the point.  Willie asks a seemingly frivolous question: would switching the lone ball for the 1 paired ball in the 1 “wee small glass” help? or switching it many, even an infinite number of times?? Our Mathematician says “no, of course not”. In fact, he goes on, not even an absolutely infinite number of switches would suffice. Silly, perhaps, but this question is crucial to our Golfer’s Paradox, as we will see later.

It is clear that with 1 lone ball and 1 paired ball and “wee small glass” we would have to cheat, either by adding an unpaired “wee small glass” or by disappearing a lone ball (which our Banker would refer to as... “embezzlement”).  Just adding yet another paired ball and “wee small glass” will not enable even our Mathematician to construct a completely 1-to-1 pairing of balls and glasses, or adding yet another, or yet another...

Willie’s seemingly frivolous question about switching the lone ball with the paired ball an infinite number of times now comes to the fore. If switching the lone ball with the ball in the first paired ball-and-glass an infinite number of times won’t help, it is that much more obvious that switching it or any other lone ball with a ball in some other glass won’t help, either. Each paired ball-and-glass is “abstractly equivalent” with regard to such switches in our case, here. Unless the balls and/or glasses were numbered, we wouldn't even notice the difference between switching the lone ball with this or that paired ball-and-glass. We would always have a lone ball left over, no matter how many switches were performed, and no matter with which paired ball-and-glass.

So even adding an absolutely infinite number of paired balls and “wee small glasses” would not suffice. And worse, as our Banker notes, any attempts along these lines — i.e. any combination of adding or subtracting of balls and/or glasses beyond the first “countably infinite” set of them, the one we are concerned with here for our Golfer’s Paradox — would make the original “countably infinite” set both non-self-equal and non-self-equivalent, both concepts as yet “not fully appreciated” in set theory.

(See Figure 2 in The Golfer’s Paradox.)

But, back to our Mathematician. Our Mathematician starts to see the light when Willie, our Golfer, offers the second paired ball-and-glass. I.e. he sees that Willie  is doing an “n + 1”, just as one does in the Recursion Clause, the second and final stage, of a proof by Finite Induction. In this stage one proves that if a given proposition, p(n) is true (i.e. for n), then it follows that p(n + 1) is also true. Here this means that if using n paired balls and “wee small glasses” is not adequate to match the lone ball and all the paired balls and glasses 1-to-1, then adding 1 more to make n + 1 pairs won’t be adequate, either. The first stage was proven when Willie started his demonstration with the lone ball and 1 paired ball and “wee small glass”, i.e. the Base Clause for “n = 1” (i.e. 1 paired ball and glass). Our Mathematician quickly realizes that Willie has just shown, by the canny Scotsman’s version of Finite Induction, that not even adding an infinity of paired balls and glasses — i.e. 1 pair for every Natural Number — will do this.

The other inescapable horn of our Mathematician’s dilemma is that (any current, i.e. 20th Century) standard set theory “proves” that “infinity + 1” set elements (commonly, the “infinity + 1” elements of {0,1,2,3...}) can be bijected, paired 1-to-1, with “infinity” set elements (commonly, the “infinity” elements of Á 4 {1,2,3...}). (This 1-to-1 pairing or “mapping” is formally called a “bijection”.) Standard set theory proves it in the following way: since, by definition of the Natural Numbers, for every natural number n there exists the natural number n + 1, pair the 0 of {0,1,2,3...} with the 1 of {1,2,3...}, pair the 1 of {0,1,2,3...} with the 2 of {1,2,3...}, “and so on”... n ↔ n + 1 for all n. There is “always” an n + 1 for every n, so all the numbers of {0,1,2,3...} have unique — it is essential to emphasize the uniqueness of the paired number to ensure a completely 1-to-1 pairing — numbers in {1,2,3...} for them to be paired with. “QED.”

But, when we think in terms of the golf balls and “wee small glasses”, we get an essentially different feel for the situation. We see the fallacy in the purely formal mathematical approach. True, for every ball, including the original lone ball, there seems to be a “glass holding the next ball over” (in our case “to the right”), but that is the fallacy! The “glass holding the next ball over” is already doing just that, holding a ball. When we switch balls, we get another lone ball, but the same “infinity” of golf balls paired 1-to-1 with “wee small glasses” (“same” in the sense of abstract equivalence in our situation). It can be seen to make no difference if we pick a different “wee small glass” to switch the lone ball into, no difference at all with regard to eventually pairing-bijecting the “infinity + 1” balls 1-to-1 with the “infinity” “wee small glasses”.

(See Figure 3 in The Golfer’s Paradox.)

It is obvious how to make this argument formal. We notice that when we construct the 1-to-1 pairings from all the ns to the all n + 1s, which we must do in accordance with the original definition of the Natural Numbers (see Fundamental... Oversights), we have at least 1 number on the “infinity + 1” side of the pairings that is not in a 1-to-1 pairing. We can prove by Finite Induction that this is true for all n in {0,1,2,3...} where the n in {0,1,2,3...} corresponds to “the lone ball being switched into a glass”.

We also start to notice some of the... oversights of Cantor and company. They can be a little tricky to describe, but they are very obvious once seen. Cantor wanted a “completed infinity”, not a “potential infinity”; this is part of the... oversight(s). When a set has been “completely” constructed, something happens that doesn’t happen in the intermediate/intermediary sets: the “completely” constructed set must be “self-equal” and “self-equivalent”, both predicates that are not made use of in standard set theory in the way we are using them here. (E.g. we are not talking about the usual reflexivity of a set, s, “s = s”, or some such.) The intermediary sets keep changing, getting larger, but the final set must be “completely”... uhhh... “settled”, “defined”, after which it can never again change. It is at this stage of completion that self-equality and self-equivalence make sense, and are completely necessary.

Digression: mathematics in general, and set theory in particular, have yet to come to terms with those entities that in computer programming are ordinary, every day entities, “variables”, and their import for e.g. set theory. Every programmer knows you need to be careful about when a variable has not yet been completely defined, when it has reached its proper value, and to not let it be changed thereafter, except when it gets redefined to yet another proper value. When set theorists biject {0,1,2,3...} and {1,2,3...}, they unconsciously, invisibly redefine {1,2,3...}. It is not necessarily bad to redefine {1,2,3...}, but it is bad to do so without being aware and formally acknowledging that that is what we are doing.

When we created each of the various successor sets of {1} in the first place, we formally went through a very special process (that only validly applies to “sets” still under construction, i.e., not yet truly sets which never thereafter change, which remain forever self-equal) that can be informally described as constructing yet another empty “wee small glass” and populating it to its maximum capacity with a single golf ball. When Cantor (and others) conceived of the n ↔ n + 1 mapping, they... oversighted the fact that the + 1 was by its very nature a successor-set operator. If we apply it to e.g. n - 1 in the set {1,2,3...n}, we get the number n, still in the set. But if we apply it to every number in the set, we also get n + 1 as a number and the successor set, {1,2,3...n + 1}, as a set. Cantor thought that this would be different with the set of all natural numbers {1,2,3...}, because it was ostensibly defined by not having any limit to the number of times the successor operation had already been applied, so they would all be there in advance, so to speak... He even defined a₀ as the “first cardinal (number) that can no longer be made larger by adding 1”. If you have been paying close attention, you should be saying something like “so far, so iffy...”

But Cantor also had the... contradictory idea of the transfinite ordinals, where the first transfinite ordinal (a set), ω, had a successor (also a set) not identically equal to itself, ω + 1. (For Cantor, ω + 1 is still of cardinality a₀ by a process of “reordering” of the same kind as Willie is demonstrating with his golf balls and “wee small glasses”; i.e. switch the lone ω + 1 — lone since it’s not paired with a natural number — with 1, 1 with 2, etc., supposedly pairing them all 1-to-1.) By a variant of circular reasoning, Cantor... oversighted the relationship between the transfinite ordinals and the transfinite cardinals when applying the n ↔ n + 1 mapping. The n ↔ n + 1 mapping for all n is an inherently ordinal thing, producing a different successor set. The “completed infinity” of the set of “all” natural numbers becomes no longer “complete”. The “self-equality” of {1,2,3...} is violated, and its “self-equivalence” is violated, as well.

Our Golfer used the physical analogy of balls already placed in “wee small glasses”, an infinite number of them. By doing so, he made the self-equality and self-equivalence of {1,2,3...} much harder to violate unknowingly. When we construct the natural numbers, we have heretofore treated it like an infinite sequence of golf balls that can be shifted 1 golf ball to the right any time we want. (I.e. we would have “all the balls in the air at once”, much harder to keep track of.) But, like Willie demonstrates, the natural numbers are actually like an infinite sequence of “wee small glasses” each with precisely 1 golf ball already in it, and the cheating becomes more obvious.

In addition, after Willie’s demonstration it becomes even more completely obvious that, to put a lone ball in any glass whatsoever, one must first remove the paired ball from that glass, switching them. And switching balls in glasses doesn’t help a bit. The lone ball changes but never goes away, not if it is switched an infinite number of times in any 1 glass, or even in every glass, not even an absolutely infinite number of times.

Digression: we can also note that Willie’s demonstration could have been to just humbly place the lone ball on the table, close his eyes meditatively, and silently stare down our Mathematician — in pure Zen and The Art of 19th Hole None-Up-Man-Ship style. What we get from adding the first paired ball-and-glass is actually the second stage of the finite induction proof that can be started with just the lone ball, which cannot be paired 1-to-1 with Zen paired no-balls-and-no-glasses. But Willie probably figured that the high road would take too long, so he took the low road, and he was in Scotland well before everyone else. Actually, that isn’t it. Humility and silence just aren’t Willie’s long suits.

Of course, someone will probably still want to take the handwaving — “all the balls in the air at once” — approach and shift all the golf balls in the glasses “simultaneously”, since it’s obvious it can’t be done any other way. In that case, the situation-problem becomes clearer if we realize that we must be able to replace “the set of all natural numbers” with its definition, i.e. by its serial-sequential construction. When we do that, then the handwaving we do when we “simultaneously shift the infinity of golf balls 1 golf ball to the right” is seen for just that, since as we are reconstructing the set as we did in the first place, we always get a new “wee small glass” out there to the right, even at infinity, just as Cantor got new ordinals out past the first transfinite ordinal. This violates the self-equality and self-equivalence of the set of all natural numbers {1,2,3...}. Since that new glass must come with its own ball, it’s obvious that somewhere we need an extra empty glass, further violating the self-equality and self-equivalence of {1,2,3...}.

Although it is strictly illegitimate (see Fundamental... Oversights), some mathematicians dismiss formal arguments that refer to the original definition or construction of the entities involved, and themselves present arguments that refer to the “simultaneous existence” of the already serially-sequentially constructed-defined entities. In other words, these... mathematicians argue to the effect that:

• “Now that the entities in question have been defined or constructed sequentially-serially, they have a ‘simultaneous existence’, and no longer depend on their humble origins.”

Note the logical similarity to:

• “We are out on a limb, and we no longer depend on the tree we climbed to get out on the limb.”

One example of this ‘simultaneous existence’ is the map, or mappings, from all the ns to the all n + 1s, in order to avoid the reasoning given above.

But in our case here, even this “simultaneous existence” argument is fatally flawed. In all previous constructions, the “+ 1” was an action which created a new element and added it to a new “intermediate set” of the set under construction. Although we can argue that for any single n that there must exist an n + 1 in the set of all natural numbers, when we apply the “+ 1” to every element in the set of all natural numbers, we are effectively doing what we did when we extended {1,2,3...n} to {1,2,3...n,n + 1}, adding a new element to the set which was ostensibly already completely defined or constructed. (NOTE: the set {1,2,3...n,n + 1} has as elements all the elements of {1,2,3...n} and all those same elements “+ 1”. The duplicates are invisibly turned into singletons due to our definition of set.) This is also a standard contradiction, although noticeably more difficult to see when presented merely formally. This is why The Golfers Paradox is so important, since it makes not only the fatal flaw per se “intuitively obvious”, but it also makes obvious how to make the demonstration formally rigorous within set theory.

Actually, let us partly retract that last. Let’s also examine an argument that would have been a little difficult to put in the mouth of Willie, our old Pro Golfer. It relates to the “all the balls in the air at once” approach. (It's always good to carefully analyze even handwaving arguments before rejecting them out of hand. Ya never know...) We can show that if there exists a bijection between a given set and the set of all natural numbers, it can be constructed without having “all the balls in the air at once”.

To show this, we jump ahead to the Banker’s demonstration and use numbered balls and glasses. Let’s (mix metaphors and) assume that there exists a bijection B between the set {0,1,2,3...} and {1,2,3...}, and each ball b_n of B will be paired with glass n. (The reader may have noted that this is not the most general case, which is left as... ye olde exercise for the reader.) But let’s also assume that we start with every ball n paired with glass n, plus a lone ball 0, like Willie did. We want to prove that we can permute the starting (identity) bijection, as represented by the starting pairings of each ball n in each glass n, to the desired bijection B, with each ball b_n of B paired with glass n, and in such a way that we don’t need to have “all, or even an infinite number, of the balls in the air at once”. We will do so by FINITE INDUCTION:

• BASE CLAUSE: if (the correct for the new bijection, B) ball b₁ = n is not in glass 1, then we find it and switch it with ball 1 (which must be in glass 1). If b₁ is not the lone ball, we put ball 1 in glass n; if b₁ is the lone ball, we leave ball 1 as the new lone ball.

• RECURSION CLAUSE: if we have so paired (the correct for the new bijection, B) balls b_i with glasses i, for i = 1 to n, then we can pair ball b_n+1 by the same means. I.e., if b_n+1 is not in glass n + 1, then it cannot be in any glass preceding it. It is either in some glass m > n + 1, or it is the lone ball. If it is some glass m > n + 1, then we switch it with the ball in glass n + 1. If it is the lone ball we switch it with the ball in glass n + 1. But either way, the correct (for the new bijection, B) ball winds up in glass n + 1. This completes the proof by FINITE INDUCTION.

So there is no need to have all, or even an infinite number, of the balls in the air at once.

In set theory, heretofore, no one has worried about any variant of the halting problem of automata theory. But they will have to start. Let us note that the method just given above must terminate in the transfinite sense, since it uses a finite procedure that need only be done a limited, even finite, number of times (once in our case here) for each natural number. Since we can e.g. construct the natural numbers in theoretical-zero time, we multiply a finite number by that theoretical-zero and still get a theoretical-zero time. If we knew that we were concerned with only e.g. a₀ finite procedure applications for each natural number, we would still have an acceptable transfinite halting time (i.e. we can still make it a theoretical-zero time). BUT... if we proceed on the basis of switching balls in glasses till they are all paired 1-to-1, then we have a “halting problem”, a non-halting problem. Not even an absolutely infinite number of switches will accomplish the objective, so there is not even an absolutely infinite halting time.

Even if the inconsistency of set theory is already accepted, we will still want to ask and answer the question “what if a bijection between {0,1,2,3...} and {1,2,3...} exists, not by the use of any standard method of construction (since they have all been effectively ruled out), but by magic, by luck, by supernatural means, or by any means whatsoever?”...

This is why, in addition to The Golfer’s Paradox,  The Banker’s Paradox is so important (instead of merely redundant), since it likewise makes obvious how to make the demonstration formally rigorous within set theory that, assuming a bijection somehow exists between {0,1,2,3...} and {1,2,3...}, one can still easily produce a standard contradiction, verifying the death knell sounded by Willie’s 19th hole demonstration with golf balls and “wee small glasses”.