, worth looking at but still unpolished

I've noticed that the translation process from WORD sometimes does not map characters properly, like the ones used in mathematical formulae. I have tried to catch all these and “put ’em back the way they wuz”, but...

SECTIONS

The Bijection Permutation Paradox in Set Theory — Intro

A Review Of the Use of Induction, Finite and Transfinite

Bijection Permutation Theorems, Paradox, and The Continuum Hypothesis

“The Bijection Permutation Paradox”, The Continuum Hypothesis and The Inconsistency of Set Theory

The Bijection Permutation Paradox in Set Theory — Intro

Set Theory has at its foundations the paradoxical result that a transfinite set can be bijected with a proper subset of itself. This result is fundamental to transfinite counting and transfinite arithmetic. The ability to construct a bijection from a set onto another set has come to be the fundamental paradigm for counting — whether of finite or transfinite sets of elements — in Set Theory, and the foundation stone for transfinite arithmetic as distinct from finite arithmetic.

That a set can be bijected with a proper subset of itself is especially paradoxical in Set Theory since set subtraction says that subtracting the subset from the set yields a non-empty set, seeming to indicate that they have different numbers of elements. Many beginning mathematicians have expressed “concern” over this issue.

Set Theory has constructed “proof(s)” that a transfinite set (but not a finite set) can be formally bijected with a proper subset of itself. This is done by constructing a bijective mapping from one onto the other. The classic variant is the construction of a bijection from Á ~ {0} onto Á, where Á 4 {1,2,3...}, i.e. from {0,1,2,3...} onto {1,2,3...}. Every number n in {0,1,2,3...} has — or seems to have — 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...}. (There is a flaw in this reasoning, that this set of web pages is attempting to examine and analyze in detail.)

But the set {0,1,2,3...} obviously has one more element than {1,2,3...}, so its cardinality — the “number” of elements in a set — 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 also have the same cardinality, so, paradoxically... a₀ + 1 = a₀. (This is perhaps the most basic expression of the ancient paradoxes that e.g. there seems to be the same number of even natural numbers as natural numbers (even and odd), or the same number of squares or cubes as numbers.) This is perhaps the most paradigmatic of the fundamental transfinite arithmetic results in Set Theory.

SECTIONS

The Bijection Permutation Paradox in Set Theory — Intro

A Review Of the Use of Induction, Finite and Transfinite

Bijection Permutation Theorems, Paradox, and The Continuum Hypothesis

“The Bijection Permutation Paradox”, The Continuum Hypothesis and The Inconsistency of Set Theory

A Review Of the Use of Induction,
Finite and Transfinite

Since finite induction is important in what follows, and since many mathematicians are unaware of certain important ramifications of finite induction, a review is in order.

Standard finite induction comes from the postulates of Peano (Giuseppe, 1858-1932) that define the system of natural numbers. (Some like to note that they originated with Dedekind.) But many mathematicians mistake the meaning of the “finite” of “finite induction” and sometimes think that one can only prove theorems about finite sets of natural numbers. (An example of this will be offered with regard to Theorem 8 and Theorem 10.) So a review of standard finite induction is in order.

Finite Induction:

if it can be shown for propositions P(n), where n is a natural number, that:

1)      P(1) is true (the base clause); and that

2)      if P(n) then P(n + 1) (the recursion clause)

then it has been proven that P(n) for all natural numbers n.

Finite induction, which is integral to set theory, proves a transfinite (countably infinite/­denumerable) number of propositions, one for each natural number. (It is sometimes generalized to allow starting at n ¹ 1.)

To help counter confusion with regard to finite induction, the following theorem is offered, which to some may seem to be equivalent to general/second kind induction, but which has a clearer statement with regard to the treatment in this paper. It is offered here for purposes of clarity and cogency, and is not considered a main result of this paper. NOTE that this theorem is completely isomorphic to standard finite induction, i.e. that it is another way of saying that the set S of natural numbers n for which standard finite induction proves the propositions P(n) is the set Á = {1,2,3...} of all natural numbers, a transfinite set; i.e. S 4 Á.

Theorem 1: “Transfinite Case” Finite Induction Theorem:

“Transfinite Case” Finite Induction:

if it can be shown for propositions P’(S), where S is a set of natural numbers, that:

1)      P’({1}) is true; and that

2)      if P’({1…n}) then P’({1…n + 1})

then it has been proven that P’(Á), the “transfinite case”, as well as P’({1…n}) for every natural number n.

We can also say that: if P’({1…n}) is true “for all finite n”, then it is true in the “transfinite case”, P’(Á).

Further, “Transfinite Case” Finite Induction is logically equivalent to standard finite induction.

Proof: There are two obvious approaches, either one of which should suffice.

There are two obvious approaches, either one of which should suffice.
The first approach is to apply standard finite induction as follows: let P(n) = P´({1…n}), which we can also write as P(n) = P´({i | 1 £ i £ n}). (NOTE that this is different from some statements of general induction which refer only to strictly proper subsets, i.e. i < n.) By condition 1) we trivially have P(1) = P´({1}) and by condition 2) we trivially have P(n) ® P(n + 1); therefore, by standard finite induction, P(n) holds for all n Î Á, and therefore P´({n | n Î Á}) holds in addition to P´({1…n}) for all n Î Á; but {n | n Î Á} 4 Á, therefore P´(Á).  NOTE that this first approach demonstrates that standard finite induction implies “Transfinite Case” Finite Induction.

Logical equivalency follows trivially by showing that “Transfinite Case” Finite Induction implies standard finite induction, as follows: let P´(S) be interpreted as meaning that for all elements n of the set S (of natural numbers), P(n) holds. If P´({1}) is true, this implies the truth of P(1). And if P´({1...n}) is true, this implies the truth of P(1), P(2),... and P(n), and therefore implies the truth of P(n), which is merely one of those propositions whose truth is implied by P´({1...n}). Similarly for P´({1...n + 1}). So if we have P´({1}) being true, and if we have that if P´({1...n}) then P´({1...n + 1}), then we also have that P(1), and that if P is true for n, then P is true for n + 1. Thus “Transfinite Case” Finite Induction implies standard finite induction, and they are logically equivalent.

A second, corroborative approach is to notice that, if we are constructing a set S, 1 Î S and 1…n Î S ® 1…n + 1 Î S (and no other elements in this set S) is logically equivalent to the standard definition of Á, i.e. S 4 Á. If that isn’t sufficient, consider the following: Conditions 1) and 2) construct such a set S with property P´. Condition 2) guarantees the property P´ is inherited by all thus constructed supersets of {1} Í S, which latter is condition 1); thus P´ is inherited by S 4 Á, therefore P´(Á). The argument that S ¹ Á fails, since in that case S must differ from Á by some least element m; m cannot be 1 by condition 1; therefore m – 1 Î S; by condition 2, m – 1 + 1 Î S, i.e. m Î S, which contradicts the assumption of non-equality, thus standardly accepted mathematical reasoning gives us S = Á.

Mathematicians usually accept that one can prove inductively that a proposition P’({1…n}) can be true “for any/all finite n”, but for some strange reason they sometimes object to the necessarily consequent truth of P’(Á), even though it follows not only from the standard reasoning given above, but from standard general or second kind induction.

Thus it was felt that this review of finite induction was in order here.

It is also worth taking a quick look at transfinite induction, a form of induction on ordinal numbers as opposed to natural numbers. (See the entry for transfinite induction in The HarperCollins Dictionary of Mathematics.) It is considered equivalent to the Axiom of Choice, the well-ordering theorem (that any set can be well-ordered), and Zorn’s Lemma.

Transfinite Induction:

if it can be shown that if P(α) holds for all ordinal numbers α < β, then P(β) holds (a combination of base and recursion clauses), one may conclude that P(α) holds for all ordinal numbers α.

NOTE: it is common to use transfinite induction to prove theorems for not all transfinite ordinals, but only those associated with a given well-ordering of a given transfinite set. In other words, the terms of use are understood implicitly to mean all ordinals associated with that particular well-ordering of the elements of that particular set, not “all” transfinite ordinals. When it is used in conjunction with a well-ordering of a transfinite set, the propositions P(α) that seem to be propositions about ordinal numbers are actually propositions about the elements of the set.

Also NOTE: transfinite induction works just as well for finite sets.

Finite and transfinite induction have something in common that is relevant to our situation here: if P(x) can be proven for an arbitrary x independently of the truth/proof of any other P(z), then the base clause and the recursion clause (in either finite or transfinite induction) always trivially hold, and the proof by either kind of induction is trivially valid. An extremely simple example: if we have a bijection from a set S onto itself, and we can show that for an arbitrary element x in S that x is bijected onto itself, then we have trivially shown that all the elements of the set are bijected onto themselves, and thus that we have an identity bijection.

SECTIONS

TThe Bijection Permutation Paradox in Set Theory — Intro

A Review Of the Use of Induction, Finite and Transfinite

Bijection Permutation Theorems, Paradox, and The Continuum Hypothesis

“The Bijection Permutation Paradox”, The Continuum Hypothesis and The Inconsistency of Set Theory

Bijection Permutation Theorems,
Paradox, and The Continuum Hypothesis

Bijections are now the standard method for counting, i.e. for putting the elements of one set into a 1‑to‑1 correspondence with the elements of another. Set theory and its transfinite arithmetic are based on bijections that are paradoxical (using the word in its now usual informal sense), where a set is bijected with a proper subset of itself. Peirce (Charles Sanders, 1839‑1914), and later (but much more famously) Dedekind (Julius Wilhelm Richard, 1831‑1916), suggested that the existence of a bijection of a set with a proper subset was definitional of transfinite sets. (This is now accepted as standard). The reason such seems paradoxical is that the sets are held to have “the same number of elements”, i.e. “the same cardinality”, even though set subtraction yields a non-empty set. But it has been overlooked that it is quite possible to derive, from the standard axioms and rules of inference of set theory, a counterpart to set subtraction for bijections, specifically for the “bijections” from sets onto proper subsets of themselves that are fundamental to transfinite counting and arithmetic.

We will forgo completeness in favor of adequacy, and try to restrict definitions, theorems, proofs, and observations to those that are critical or potentially paradigmatic. The preparatory definitions and theorems are obvious enough that the reader may wish to proceed quickly to Theorem 9.

Definition 1: Disjoint Bijections:
2 bijections, B from set S1 onto set S2 and B’ from set S1’ onto set S2’, are “disjoint bijections” if (and only if) S1 and S1’ are disjoint, and S2 and S2’ are disjoint.

Definition 2: Subbijection:
a bijection B’ from set S1’ onto set S2’ is a (proper) “subbijection” of a bijection B from set S1 onto set S2 if (and only if) S1’ is a (proper) subset of S1, S2’ is a (proper) subset of S2, and every element of S1’ has the same image (in S2’) under B’ as it has (in S2) under B.

Theorem 2: Subsets Define Subbijections and Subbijections Define Subsets:
if we have a bijection B from S1 onto S2, then any (proper) subset S1’ of S1 or S2’ of S2 defines a (proper) subbijection B’ of B. Likewise, any (proper) subbijection B’ of B defines (proper) subsets S1’ of S1 and S2’ of S2.

Proof: Any such S1’ has an image (that we might as well call S2’) under B in S2. Let B’ be the bijection from S1’ onto S2’ in which every element’s image under B’ is the same as under B. B’ is the subset (here S1’) defined subbijection of B. Likewise any subset S2’ of S2 is the image of some pre-image subset (that we might as well call S1’) of S1. As previously, let B’ be the bijection from S1’ onto S2’ in which every image under B’ is the same as under B. Again, B’ is the subset (here S2’) defined subbijection of B. Subbijections define subsets similarly by definition.

Definition 3: Union of Bijections:
if we have 2 bijections, B from S1 onto S2 and B’ from S1’ onto S2’, the “union of the bijections”, B ~ B’, is the mapping M from S1 ~ S1’ onto S2 ~ S2’ in which the image of every element of S1 ~ S1’ is the same under M as under B, B’, or both if either of the sets S1 ? S1’ or S2 ? S2’ is non-empty, i.e. of elements common to both S1 and S1’. Notationally, we can also write B + B’.

We will forgo the usual theorems about commutivity, associativity, etc.

Theorem 3: Bijection Preserving Union of Bijections:
the union of 2 bijections, B from S1 onto S2 and B’ from S1’ onto S2’, is itself a bijection:

1) if B and B’ are disjoint bijections,

2) if B and B’ are both subbijections of a third bijection,

3) or in general if the subbijection of B from S1 ? S1’ onto S2 ? S2’ is (identically) equal to the subbijection of B’ from S1 ? S1’ onto S2 ? S2’. Stated without proof.

Definition 4: Partition of a Bijection:
if we have a bijection B from S1 onto S2, a (proper) partition of B is a collection of disjoint (proper) subbijections of B such that their union is B.

Theorem 4: Subbijection Partition (-ing) of a Bijection:
any (proper) subbijection B’ from S1’ onto S2’ of a bijection B from S1 onto S2 partitions B into 2 (proper) subbijections, the first B’ itself, and the second B’’, the subbijection (of B) from S1 – S1’ onto S2 – S2’. Notationally, we can write B’’ = B – B’ and B = B’ + B’’. Stated without proof.

The above are pretty much just straightforward counterparts of similar definitions and theorems for sets. The following are specific to bijections.

Definition 5: n-Element (Pre-) Image Swapping Permutation of a Bijection:
an “n‑element (pre-) image swapping (or switching) permutation of a bijection” B from S1 onto S2 is a bijection B’ from S1 onto S2 such that the images of n elements of S1 in S2 (or pre-images of n elements of S2 in S1) are different under B’ than they are under B. An “identity image swapping permutation of a bijection” is an 0‑element (pre-) image swapping permutation of a Bijection, i.e. involving the swapping of n = 0 elements, and thus the permuted bijection B’ is identically equal to the original bijection B. Further, the bijective mapping (of a subbijection) from one element onto another can be referred to as a “link”, and one can refer to “switching links” to mean “switching (pre-) image elements”. As long as the sets of the initial bijection have no orderings that can be adversely affected (the standard definition of “bijection” takes no notice of any orderings of its sets), image swapping can be equivalently thought of as pre-image swapping.

Theorem 5: 2-Element Image Swapping Permutation of a Bijection:
if we have a bijection B from S1 onto S2 (|S1| = |S2| ≥ 2, where |S| is the cardinality of S), there exists a bijection B’ from S1 onto S2 such that the images in S2 of each of 2 and only 2 arbitrarily chosen elements of S1 are different under B’ than they are under B; i.e. these elements are “swapped” under B’, as per Definition 5. The smallest (non-zero) number of image elements that can be permuted in a non-identity image swapping permutation of B such that the resulting mapping B’ remains a bijection from S1 onto S2 is n = 2.

Proof: Let j and k be any 2 elements of S1. Along with their images in S2 under B, they form a subbijection of B, B’’, from {i,j} onto {B(i),B(j)}. B’’ partitions B into B’’ and B’’’. If we permute B’’ into B’’’’ by swapping the images of i and j under B’’, i.e. making the image of i under B’’’’ equal to B(j), and the image of j under B’’’’ equal to B(i), B’’’’ is a bijection, now from {i,j} onto {B(j),B(i)} instead of onto {B(i),B(j)} (again, the notation should be sufficiently clear). The subbijection B’’’ of B remains unchanged by this element swapping permutation of B’’ into B’’’’. The union of B’’’’ and B’’’ is a bijection from S1 onto S2, per Theorem 3. The bijection formed by this union B’ = B’’’’ + B’’’ is the bijection B’ of the first part of the theorem. The second part of the theorem follows trivially, since a non-identity permutation of B must involve more than 0 elements of S1, and if it involves only 1 such element i, then we have a subbijection from {i} onto {B(i)} that cannot be non-trivially (non-identity) permuted so as to remain a subbijection from {i} onto {B(i)}, and thus so as to allow the overall bijection, the 1 element counterpart to B’’, to remain a bijection from S1 onto S2 (i.e. in the sense of Definition 5).

This last definition and theorem, though seemingly trivial, are given because they are paradigmatic both for general permutations of bijections, and eventually for a new set theory. A simple but paradigmatic example of its use suggests itself.

Theorem 6: 2-Common Element Image Swapping Permutation of a Bijection:
if we have an arbitrary bijection B from S1 onto S2 (|S1| = |S2| ≥ 2), and S1 and S2 have at least 1 element in common, and we have an element n arbitrarily chosen from all the common elements, and this n in S1 is not mapped onto itself in S2 under B, (i.e. n in S2 is not the image of n in S1) then there exists a bijection B’ from S1 onto S2 such that the images in S2 of each of 2 and only 2 elements of S1, i.e. n and 1 other element j, are different under B’ than they are under B; i.e. n in S1 is mapped onto n in S2 and j in S1 is mapped onto some k in S2; i.e. these elements are “swapped” under B’, as per Definition 5 and Theorem 5 so that n is bijectively mapped onto itself. If n is initially already mapped onto itself under B, we can alternatively speak of the swapping permutation as an identity permutation or swap. If any element m in S1 is already mapped under B onto itself in S2, then it will remain mapped onto itself under B’ And for future reference, this gives us idempotence for this class of permutations as operators or functions.

Proof: By Theorem 5 we can swap the images in S2 of 2 arbitrarily chosen elements of S1. Let 1 of those arbitrarily chosen elements of S1 be n, the image of which in S2 is k, and let the other be the pre-image (counterimage or inverse image) j in S1 of n in S2. After applying Theorem 5, n in S2 will be the image of n in S1, and k in S2 will be the image of j in S1. Any element m already mapped onto itself cannot be mapped either to or from n, j or k, so the m onto m submapping or subbijection remains invariant under this permutation of B, as does every other subbijection not involving n and j in S1 or n and k in S2 (also as per Theorem 5).

If we think of ISP as a class of 2‑common element image swapping permutations/­permutation operators ISP(n) that can be applied to an arbitrary bijection B giving us a bijection B’ such that: if n is not an element common to both sets of the bijection B (perhaps belonging to neither S1 nor S2), then applying the ISP(n) operator to B gives us B’ = B; if the element n is common to both sets S1 and S2 but n is already bijectively mapped onto itself under B, then applying the ISP(n) operator again gives us B’ = B; if the element n is common to both sets S1 and S2 but n is not already bijectively mapped onto itself under B, then applying the ISP(n) operator will give us B’ ≠ B; if we let N be the set of elements n such that the operator ISP(n) has been applied at least once in the cumulative sequence of permutations of the initial bijection, any further application of ISP(n) such that n Î N will always give us B’ = B (where B’ represents the current cumulative permutation of the initial bijection, and B here represents the immediately previous bijection in the cumulative sequence as opposed to the initial bijection). This is an interesting generalization of idempotence for operators, and it is included here for future reference since it is felt that 2‑common element image swapping permutations are an paradigmatically essential for studying bijections.

This last theorem is deceptively simple. It suggests a simple but paradigmatic example of its use.

Theorem 7: Permutation of a Bijection From a Set Onto Itself Into the Identity Bijection By Means of 2‑Element Image Swapping Permutations: if we have an arbitrary bijection B from S1 onto S2 4 S1 (|S1| 4 |S2| ≥ 2), it can be permuted in a bijection preserving fashion into the identity bijection by swapping only 2 image elements at a time, with at most one non-identity swap or permutation per element.

Proof: by the well-ordering theorem a well-ordering W of S1 must exist. Starting with the least element of S1 under W, chose each element i of S1 in order in accordance with W. If i is not already mapped onto itself under B, apply Theorem 6 so as to map it onto itself. The rest is a trivial proof by induction, finite or transfinite because the recursion clause is trivially proven for either finite or transfinite induction since an arbitrarily chosen element is either already mapped onto itself, or can be made to be so independently of any other element by applying Theorem 6. I.e., for finite induction, by Theorem 6 P(1) is true and also by Theorem 6 P(n + 1) is trivially true independently of P(n); and for transfinite induction, by Theorem 6 P(1^st) is true (i.e. the 1^st element of the well-ordering of S1) and, independently of whether P(α) is true for all ordinals α < β, by Theorem 6 P(β) is trivially true for the β^th element of the well-ordered S1. Since each element, once bijected/mapped onto itself, remains invariantly so bijected/mapped, only 1 permutation or swap is needed per element (because of the idempotence referred to in Theorem 6).

This last theorem seems highly intuitively obvious, and it is. It is vaguely a variant of a software bubble sort algorithm. But for some mathematicians it ceases to be obvious when it is noticed that it does not depend on the bijection being from a set onto itself, but only on the sets having elements in common, such as when a set is bijected with a proper subset of itself. (See Theorem 8, which is generalized so as to allow application to such situations.)

Theorem 8: Permutation of a Bijection So That an Identity Subbijection Is Constructed From the Subset of Elements Common to Both Sets Onto Itself By Means of 2‑Element Image Swapping Permutations: if we have an arbitrary bijection B from a set S1 onto a set S2 = S1 (|S1| = |S2| ≥ 2) with a set of elements SC in common, B can be permuted in a bijection preserving fashion, by swapping only 2 image elements at a time, so that its composite permutation remains a bijection and has as a subbijection B’, the identity bijection between SC in S1 and SC in S2.

Proof: by Theorem 6, as was noted in the proof for Theorem 7, the proof of the recursion clause for either finite or transfinite induction trivially does not depend on the conditional since it is true for every element of SC independent of every other element of SC for an arbitrary well-ordering of SC.

Along with Theorem 10, this is where some mathematicians have tried to object that the theorem can only be proven for finite numbers of common elements. But Theorem 1 clearly proves that this objection fails, as do the arguments presented in Theorem 6 and Theorem 7 concerning the independence of each element as far as permuting the bijection so as to bring that element into an identity subbijection with itself.

This last theorem is really all we need for a crucially important further result, i.e. applying to sets bijected with proper subsets of themselves, but the approach that will be taken in Theorem 9 and Theorem 10 seems more compelling.

There are many other important such theorems, but the topic of permutations of bijections, paradigmatic in general, is extremely complex. We will forgo further general development at this time.

Now we get to the most compelling of the essential results.

Theorem 9: Single Common Element Subtraction From a Bijection: if there exists a bijection B from set S1 onto set S2, where S1 and S2 have an element n in common, then there also exists a bijection B’ from S1 – {n} onto S2 – {n}.

Proof: because B is a bijection there are only 2 cases:

1) if n is already bijected onto itself under B, the result follows trivially since the bijection from {n} onto {n} constitutes a proper subbijection of B that partitions B as per Theorem 3;

2) otherwise, we could resort to Theorem 6, choosing n of S1 as the arbitrary common element and the element j of S1 whose image under B in S2 is n as the other arbitrary element of S1; when we do this, we have a new permuted bijection B’ from set S1 onto set S2 with n bijected onto itself, and we can proceed as in part 1), just above. To be safe, though, we will again go onto the critical details: there exists a subbijection B’’ (of B) from {n,j} onto {k,n} for some n and j in S1 and n and some k in S2 that partitions B into B’’ and another subbijection B’’’ (improper for |S1| = 2); B’’ can be permuted into 2 disjoint bijections, Bn from {n} onto {n} and Bjk from {j} onto {k} (and thus trivially preserving bijection); if we take the union of the disjoint bijections Bjk and B’’’, we get the needed bijection B’ = Bjk + B’’’ from S1 – {n} onto S2 – {n}.

Theorem 10: All Common Elements Subtraction From a Bijection:
if there exists a bijection B from a set S1 onto a set S2, where S1 and S2 have a set SC = S1 ? S2 of elements in common, then there also exists a bijection B’’ from S1 – SC onto S2 – SC.

Proof: Theorem 9 can be applied to all the elements of SC, in any order. “Choosing” elements should not be a problem, but since it is sometimes a major stumbling block, we can note that the application of induction, finite or transfinite, to any well-ordering suffices. (See comment at end of section “A Review Of the Use of Induction, Finite and Transfinite”, above.) By Theorem 6 it needs to be applied at most once for each element. So at most we need transfinite induction and, by implication, the Axiom of Choice. If we are dealing with natural numbers, we only need finite induction, since they are well-ordered by construction and we don’t need to invoke the Axiom of Choice. Again we can note that the proof is trivial for either finite or transfinite induction since Theorem 6 and Theorem 9 show that the proof of the inductive clause trivially does not depend on the conditional since it is true for each and every element of SC independent of every other element of SC, and independent of any ordering of SC.

Along with Theorem 8, this is where some mathematicians have tried to object that the theorem can only be proven for finite numbers of common elements. It must be emphasized that Theorem 1 clearly proves that this objection fails, as do the arguments presented in Theorem 6 and Theorem 7 concerning the independence of each element as far as permuting the bijection so as to bring that element into an identity subbijection with itself.

Note the important similarity between Theorem 10 and Theorem 8 with regard to the standard bijections from sets to proper subsets of themselves.

As often happens in mathematics, though it seems trivially obvious, it is essential to observe and keep in mind that because of the precise 1‑to‑1 nature of bijections:

1) partitioning a bijection (by definition into disjoint subbijections) preserves bijections,

2) taking the union of disjoint bijections preserves bijections,

3) permuting a bijection by taking 2 of the elements in the from set and switching their images in the onto set preserves bijections. (A standard theory of such permutations is still lacking, which is not surprising since it quickly leads to Theorem 8 and Theorem 10, and then to Theorem 11 and Theorem 12.)

Taken together these guarantee the validity of the proof of Theorem 9, and thus the proof of Theorem 10.

SECTIONS

The Bijection Permutation Paradox in Set Theory — Intro

A Review Of the Use of Induction, Finite and Transfinite

Bijection Permutation Theorems, Paradox, and The Continuum Hypothesis

“The Bijection Permutation Paradox”, The Continuum Hypothesis and The Inconsistency of Set Theory

“The Bijection Permutation Paradox”,
The Continuum Hypothesis and
The Inconsistency of Set Theory

The paradoxical but standard bijection from Á ~ {0} onto Á (so fundamentally paradigmatic in Cantorian set theory) is usually proven to exist roughly as follows: since for every n in Á ~ {0} there (apparently) exists a unique corresponding n + 1 in Á, and vice versa, this (ostensibly) gives us a bijection between them. (This n ↔ n + 1 mapping is an example of Cantor’s concept of “reordering”, that he held demonstrates that these 2 sets are of the same “cardinality”. This concept of “reordering” pervades the bijections — the modern term for the earlier “1‑to‑1 and onto functions” — that underpin Cantorian transfinite cardinal arithmetic.) The cardinality of Á ~ {0} is  a₀ + 1, since it has 1 more element than Á 4 {1,2,3…}, and the cardinality of Á is a₀ by definition. Together with the standard bijection between them they give us the paradoxical yet fundamental theorem for standard transfinite cardinal arithmetic: a₀ + 1 = a₀. (Remember, this fundamental theorem, and by implication a₀ + 1 = a₀ since we have the implicit assumption of the consistency of set theory along with its lack of paraconsistency, is essential to the Continuum Hypothesis, though not strictly the other way round.)

But Cantor et al didn’t explore far enough.

Theorem 11: “Bijection Permutation Paradox”:
The Paradoxical Bijection From {0} Onto @:
if there exists a bijection from Á ~ {0} onto Á, there must also exist a bijection from {0} onto the empty set, @ 4 { }.

Proof: it suffices to apply Theorem 10 to the standardly derived bijection from Á ~ {0} onto Á. Note that it would also be easy to use Theorem 8 to achieve this same result, one of the reasons that Theorem 7 and Theorem 8 are paradigmatic.

This should be considered a new paradox in set theory, and we can call it the “Bijection Permutation Paradox” (or perhaps paradoxes, since it is a transfinite schema of such), but it should not be considered “mere paradox”, as most of set theory’s paradoxes — mostly paradoxes of infinity — have so far. Another useful (buzz) term is “Paradoxical Bijections”, corresponding to the theoretically related “Paradoxical Measure”, now considered an orthodox branch of Measure Theory. Unlike most paradoxes, it points in the direction of resolution, in this case less of the paradox and more of the system that gave rise to it. As might be expected, there are more paradoxes related to this one, and some of them will be examined in sections that follow.

Theorem 12: A New Resolution of the Continuum Hypothesis Question:
there does not exist a bijection from Á ~ {0} onto Á, and necessarily a₀ + 1 > a₀, giving us a new answer to the Continuum Hypothesis question. Since Theorem 10 applies to sets of any cardinality, finite or transfinite, this result is general, especially paradoxically applying even to Cantor’s absolutely infinite sets and their cardinality. In particular, one does not need to resort to power sets to increase transfinite cardinalities; simple succession by adding 1 will suffice.

Proof: this result follows from Theorem 10 and Theorem 11, especially if we wish to avoid the formally provable existence of bijections from {0} — and from many other non-empty sets — onto the empty set. I.e. we use the contradiction found by Theorem 10 and Theorem 11 to “prove by contradiction” that the assumption that there exists a bijection from Á ~ {0} onto Á is false. Remember, it has been shown in a very clear and cogent manner that if a bijection exists from Á ~ {0} onto Á, then there must also exist a bijection from a non-empty set onto the empty set, a contradiction that not even making set theory paraconsistent is likely to be able to handle. The n ↔ n + 1 mapping that is usually used to construct the “bijection” from Á ~ {0} onto Á will be analyzed in detail in the next section, “‘Counting’ and Cantorian ‘Reordering’”, and the flaw whose existence is suggested by the above results will be demonstrated.

This result is fundamentally paradigmatic for a new, non-classical non-Cantorian set theory and any theory which uses it as a foundation. Along with Theorem 9 and Theorem 11, it shows that one needs much more of a very different kind of foundation than set theory can supply to allow any kind of infinity + 1 = infinity result.

The last two bijection-cardinality results perfectly match up with standard set subtraction for Á ~ {0} – Á, since in set subtraction each common element automatically gets matched up with itself as it gets subtracted, which the standard formal existence of a bijection from Á ~ {0} onto Á and thus their having the same cardinality do not. They also make clear how replacing the standard a₀ + 1 = a₀ with a₀ + 1 > a₀ could eventually lead to a satisfactory resolution of the renormalization problem that plagues physics, notably quantum mechanics, and also of the Banach-Tarski Paradox.

The Continuum Hypothesis question really depended on finding the least rapidly increasing successor function for transfinite numbers that yielded an obviously larger transfinite number (or cardinality). It was historical happenstance that lead to the power set seeming to be that function. But Cantor et al had a psychological need to have an infinity that could not be made larger by adding 1. This is hard to understand, and it is even harder to understand why they didn’t analyze that concept using at least something like the permutations of bijections presented above. It is now rather clear that it is the assumption (ostensibly proven) that transfinite sets can be bijected with proper subsets of themselves that leads visibly to the Continuum Hypothesis, and invisibly to “problematic” paradox, i.e. inconsistency of the fatal sort since set theory is not paraconsistent, and probably cannot be made so successfully.

For completeness we should make formal:

Theorem 13: The Inconsistency of Set Theory: set theory, which holds that there exist bijections between sets and proper subsets of themselves, is inconsistent.

Proof: this result follows from Theorem 10 and Theorem 12.

We must remember that a theory is standardly inconsistent if it is possible to derive a contradiction, both a theorem and its negation. Theorem 9 and Theorem 12 take us beyond “mere paradox” in this regard. They demonstrate the strong desirability of a new, non-classical non-Cantorian set theory.