The Good Shepherd’s Paradox

A New Paradox of Infinity

In Set Theory

(cont.)

Analysis of The Banker’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.)

Our Banker is even more careful than Willie. He thinks in terms of... auditing. If balls — or coins as he has it — can be switched one way, they can be switched back in such a way that no fiscal harm can be done to the 1-to-1 relationships, i.e. to the bijection. He starts out with a lone ball 0 — coin 0 — and paired coins and slots with numbers 1,2,3...

(See Figure 4 in The Banker’s Paradox.)

Our Banker then randomly matches coins and slots “1-to-1”. He points out the totally obvious, that either a coin — and each coin has a unique “serial” number — is paired with a slot with that same serial number, or it’s not. If it is, nothing further need be done with it. If it’s not, then the coin, let’s say with serial number n, can only be paired with precisely 1 slot, with a different number, and the slot with serial number n can only be paired with precisely 1 other coin, again with a different number. The 1-to-1 nature of the pairing — formally called a “bijection” — guarantees this. It also allows the guarantee that our Banker’s auditing procedure can be made perfectly valid.

The elemental transaction in our Banker’s auditing procedure is as follows: our Banker — or a machine could do it — switches the coins in their slots so that coin n is in slot n, and the other coin is in the other slot (not worrying about their numbers). This transaction is guaranteed to work — with only two conditions — for any coin n and slot n, i.e. to get coin n and slot n paired, at the same time preserving the 1-to-1 relationships, not only among the 2 coins and 2 slots switched, but among all the other coins and slots as well. The conditions are that first, — without cheating — there is strict 1-to-1-ness, and that second, — without cheating —  there must be both a coin n and a slot n, so it won't work for coin 0. (Remember, our Banker is at the 19th hole, using the golf balls as coins and the “wee small glasses” as slots.)

It seems trivially obvious, but it is important to note that this coin-slot switch transaction:

• cannot deprive a coin of a slot — or, less importantly, a slot of a coin

• cannot unpair an already paired coin and slot (that have the same number)

• cannot cheat, e.g. by magically “disappearing” a coin or “appearing” a slot

With this simple coin-slot switch transaction, our Banker also sounds the death knell of set theory, the other side of the “coin” that Willie, our old Pro Golfer, presented. It is the bridge — the essential relationship — between set subtraction and cardinality as determined by bijections (1-to-1 pairings), even for transfinite sets and cardinalities.

(See Figure 5 in The Banker’s Paradox.)

Our Banker’s full auditing procedure starts as follows: if coin 1 is not in slot 1, then use the coin-slot switch transaction for coin n and slot n given above to switch coins and slots so that they are paired. Reminder: it is guaranteed that one both may and can do this by the original “guaranteed” (i.e. assumed, but about to be proven false) 1-to-1 pairing of all the coins numbered 0,1,2,3... with all the slots numbered 1,2,3...

(See Figure 6 in The Banker’s Paradox.)

Our Banker points out the above mentioned guarantee for coin 1 and slot 1, the first stage in a proof by Finite Induction. He then points out that the same can be done for any other coin and slot, any other, that is, except coin 0, which has no corresponding slot. And since it can be done for any coin n and slot n, by finite induction it can be done for all coins and slots that have the same numbers.

This last point can be shown quite easily. If we have all coins and slots 1,2,3... up to n paired, then we can pair coin n + 1 and slot n + 1. The proof of this is trivial since we don’t even really need the condition that it has been accomplished for all numbers up to n to prove that it can be done for n + 1. So the proof by Finite Induction is trivially established.

The essential relationship between our Banker’s auditing procedure and set subtraction is overwhelmingly obvious. Since any element can be bijected with only 1 other element, that other element might as well be itself. We can take any given element that is common to both sets of the original (ostensible) bijection, and work with its subbijection, one that involves at most 2 pairs of elements. If necessary, we switch the elements in this subbijection so that the given element common to both sets is bijected onto itself, then we subtract out this identity sub-subbijection that consists of the given element paired with itself. To help make everything painfully clear, we can create 2 new identically equal sets and an identity bijection from one onto the other,  starting with this first given element identity sub-subbijected onto itself. Working 1 common element at a time, we switch elements, if necessary, and subtract out the identity sub-subbijection of the common element we are working with, and add it to the  developing identity bijection, taking it out of further consideration. We wind up with a separate new identity bijection between 2 new sets that each contain all of and only the elements common to both sets of the original (ostensible) bijection, and what is left of the original (ostensible) bijection. When all common elements and their identity sub-subbijections are subtracted out of the original (ostensible) bijection in this manner,  in our case here this leaves... the set theory equivalent of the original ball 0, all alone, obviously invalidly bijected, i.e. bijected onto “an element of the empty set”.

When it is done for all n, the auditing procedure is over... except for one little detail: the Final Report.

The Final Report:

(See Figure 7 in The Banker’s Paradox.)

Coin 0 originally had no slot 0 to be paired with, so it was paired with some slot n. And it still has no slot 0 to be paired with. But... every slot n is paired with coin n, for n = 1,2,3... so there is no “wee small glass” — slot — for ball 0 — coin 0 — to be in. BUT... the Banker’s auditing procedure cannot possibly deprive any ball of a “wee small glass”, so ball 0 — coin 0 — could not have been deprived of a glass — slot.

This is a standard proof by “reductio ad absurdum”, proof by contradiction, that the numbers in the set {0,1,2,3...} and the numbers in the set {1,2,3...} cannot possibly be paired 1-to-1. Or, in the more formal language of modern set theory,  Á ~ {0} = {0,1,2,3...} and Á 4 {1,2,3...} cannot possibly be bijected, and the standard fundamental theorem of transfinite cardinal arithmetic, a₀ + 1 = a₀ , is false. And since that bijection is a theorem, albeit of suspicious parentage, of the theory...

By the way, our Man of the Cloth’s comments about needing to be careful with proof by contradiction, which should actually be called dis-proof by contradiction. One can easily... oversight the fact that the theorem dis-proven is, or may be, actually a theorem of the theory, if one is not extremely careful. If a theory is inconsistent, one should not pass up the opportunity to detect and communicate that fact to the community at large.

As incredible as it may seem, some mathematicians try to object that this sequence of switches that our Banker uses — in his... auditing — “does not have a (valid) limit”, and therefore (implying) that the contradiction derived from doing this sequence of switches is an “invalid contradiction”, and therefore doesn’t prove that set theory is inconsistent. But the assumed bijection between the 2 sets, the bijection-preserving nature of the switch, and the rules of inference of set theory, require that it have a valid limit, as is much more obvious with the physical analogy of our Banker’s coins and slots (or our Golfer’s balls and glasses). Even in a completely formal setting the rejoinder would be: it can easily be done for 1, with a valid bijection resulting (and if a mathematician objects to this, he objects to the fundamentals of set theory); and if it can be done for n, then it can easily be done for n + 1, with a valid bijection resulting (and if a mathematician objects to this, he again objects to the fundamentals of set theory); so, by finite induction, the derivation is valid, and must therefore yield a valid bijection. To object at any stage is to reject at least some of the fundamentals of set theory, effectively proving the inconsistency of the standard combination of the axioms and rules of inference of set theory.

Speaking of “as incredible as it may seem”: some professional mathematicians have objected to the Banker’s Paradox by saying that it requires an infinite number of operations (to switch each coin into its own slot) and it is impossible to perform an infinite number of operations.

Concerning the idea of the “limit” of the sequence of switches, we can say further: as we proceed from 1 to infinity, each pair from 1 to n has settled down, i.e. it never changes thereafter. (A formal argument would speak of the “idempotence” of the operation in question.) After we have made the full sequence of switches, no further changes are possible, since the settling down has taken place for all natural numbers. It is impossible for this to happen and simultaneously for the sequence not to have a limit. It is the validity of the value of the limit that is in question, not the validity of the procedure for obtaining it. It is obvious that the Banker’s auditing procedure must yield a valid limit if the bijection was valid to begin with. It is also obvious that the limit it reaches does not have ball 0 paired with any natural number, since they must all be paired with themselves by the auditing procedure. The lack of a “valid” limit in this case means that it is set theory itself that is... “invalid”.

One of the “signs” of inconsistency is that one can easily prove, by the standard theorem concerning the cardinality of set exponentiation applied to the transfinite ordinal (set) ω^ω, the rather too paradoxical inequality:

•   <<< (since the cardinality of the first term is countably infinite, i.e. = a₀, and a₀ is very much less than )

Once pointed out, other such too paradoxical results become easy to find, e.g.:

• The real numbers can be proven to be countably infinite instead of the standard uncountably infinite since sub-intervals of length 1/n can be used to create a countable cover for the unit interval, [0,1], each sub-interval of which — it doesn't matter for this whether they are open or closed intervals; it also doesn’t matter if they overlap or not — can have at most 1 point in it when n -> infinity; countable infinity, to be sure.

Our Banker’s remarks about “recursively circular reasoning” strike at the heart of the fatal flaw in set theory, or flaws since the Axiom of Infinity has its own flaw that synergizes with the “(infinitely) recursively circular reasoning” to create this... oversight.

• We should note that the use of circular reasoning is merely (standardly) invalid, and its use in a (standard) proof doesn’t make that proof’s (standard) theorem false any more than it makes it true.

The Axiom of Infinity does not merely define an entity, in our case an infinite set, which could be defined but not have any instances, thus allowing consistency. The Axiom of Infinity formally posits the existence of an “inconsistent entity” (in our case {1,2,3...} which both “must have no maximum element” and simultaneously “must have a maximum element”), one whose formal existence means that the theory in question is thereby formally inconsistent. This is compounded by the circular reasoning in a standard and extremely fundamental theorem-proof.

In the — or a — standard proof that  a₀ + 1 = a₀, there is a hidden assumption, one that turns out to be false. There is the assumption that there exists a valid bijective mapping from a set to a proper subset of itself, the n ↔ n + 1 mapping. Peirce (and later, and more famously, Dedekind ) even put that forward as characteristic of “infinite sets”. They failed to notice the consequent inevitability of... The Golfer’s Paradox, and its partner-in-paradox, The Banker’s Paradox. The recursive nature of the circular reasoning is much easier to appreciate when one examines the nature of the definition-construction of the set of all the natural numbers Á 4 {1,2,3...}, i.e. the Axiom of Infinity with its flaw of provable “must have no maximum element” and simultaneously provable “must have a maximum element”. One must be able to replace a defined entity with its definition-construction, and when one does this in the case of {1,2,3...}, one precludes the possibility of the handwaving argument that all the  n ↔ n + 1 sub-mappings happen “simultaneously”, or in Willie’s case, that “all the balls are in the air at once” when we shift them over to the next glass to the right.

If necessary, it is simple to prove that each coin-ball can be moved to its final slot-glass in the bijection that demonstrates the cardinal equivalence while having at most 1 coin-ball “in the air” (between switches), again by Finite Induction: if the lone coin-ball (initially 0) is the coin-ball that will finally be in slot-glass 1, we simply switch it with the coin-ball in slot-glass 1; if some other coin-ball in some other slot-glass will finally be in slot-glass 1, we can switch the lone coin-ball with it, and then switch the new-temporary lone coin-ball into slot-glass 1; this is the BASE CLAUSE. The RECURSION CLAUSE works in the same way, since either the lone coin-ball needs to go in slot-glass n + 1, or it is first switched with the coin-ball that needs to be in slot-glass n + 1, and that new-temporary lone coin-ball is then switched into slot-glass n + 1 (QED).

Our Banker has important remarks about the Axiom of Choice and how its origin and reputation derive from the fact that it effectively allowed variants of... auditing. He also pointed out the bitter truth, that the situation in set theory with the Axiom of Choice is akin to the satirically humorous “auditing causes embezzlement”. Since the Axiom of Choice allows us to take an embarrassingly closer look at what we have constructed, we get... paradoxical results that are uncomfortably close to... inconsistency.

Our Banker refers to the future when he points out that:

• Infinitesimals — transfinitesimals — must exist
  since 1/n can never become 0 with a remainder of 0
  for any finite or even transfinite number n.

This means that “the continuum” will need to be superseded by the concept of “the quantinuum”, or rather of many — and often incommensurate — “quantinua”. I.e. since infinity can always be made larger by adding 1, and the remainder of 1 when integer divided by that larger infinity steadfastly remains 1, our quantal transfinitesimals can always be subdivided ever closer to absolute 0.  One also wonders why more people didn't question why was considered the “continuum” instead of e.g. ₂ , etc. And we will, more or less of necessity, have an infinity that can be approached and surpassed gradually, a “fuzzy infinity”. These concepts need much more attention than can be afforded in this context.

And, another Final Report: no mere paradox, standard set theory is thus standardly inconsistent.

And the foundations of the 2/3 of modern mathematics that depend on variants of  a₀ + 1 = a₀ — from real number theory and analysis to measure and integration theory — fall.

The Good Shepherd...