, worth looking at, but still less than fully mature

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Paradox... Oversights

Real Number Theory Paradoxes

Set Theory Paradoxes

General Introduction to Paradox in Modern Math

Paradox... Oversights

(If you like you can peruse a General Introduction to Paradox in Modern Math before proceeding.)

Set Theory stands out for having “paradox” built into its foundations, especially since the heretofore overlooked paradoxes that can be found there seem to underlie paradoxes throughout mathematics, both older paradoxes thought to have been resolved satisfactorily, and new paradoxes, as yet unrecognized as such and unevaluated not to mention “unresolved” by the community.

But Real Number Theory, too, has many interesting new paradoxes that are well worth examining, many of which can be linked back to Set Theory. And there are many other paradoxes hard to classify as to whether they properly lie in Measure Theory, Real Number Theory, Set Theory, or whatever.

Mathematics has many paradoxes in its foundations that have been overlooked for more than a century, paradoxes that have not yet been publicly recognized, perhaps because of cognitive dissonance. That is, these paradoxes may be too paradoxical. They get too close to what people will accept as mathematical inconsistency of the unacceptable kind... eventually.

Paradox has been popular since ancient times, but... not always popular in the best sense of the word. The ancient Greek philosophers known as Sophists presented the community with paradoxes that showed that the abstract reasoning that the Greeks had come to pride themselves on was fatally flawed. Many Sophists were actually put to death for doing so, much like Socrates, by the Inquisition of that day. Unfortunately the spirit of the Inquisition is still with us, though we notice it more in places like the Middle East.

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Paradox... Oversights

Real Number Theory Paradoxes

Set Theory Paradoxes

General Introduction to Paradox in Modern Math

Real Number Theory Paradoxes

The many paradoxes in Real Number Theory stand out mainly because they should have been recognized long since. An important example is/are the “Vanishing Remainders Paradoxes” that arise when one constructs the infinite decimal expansions of e.g. 1/3 = 0.333..., 2/3 = 0.666..., 1/7 = 0.142857..., or of any rational number that is relatively prime to the base of the representation. The Vanishing Remainder Paradoxes are very important since these infinite base expansions are held to be equivalent to the real numbers, i.e. to be all the real numbers and only the real numbers. Out of this examination of vanishing remainders and related oversights we get the Real Numbers Paradoxes 1, 2 and 3, all concerning the ability of infinite decimal (or other base) expansions to faithfully represent all the rational numbers, all the irrational numbers, and even all the real numbers.

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Paradox... Oversights

Real Number Theory Paradoxes

Set Theory Paradoxes

General Introduction to Paradox in Modern Math

Set Theory Paradoxes

Set Theory has paradoxes that have not yet been officially recognized. They seem to almost all derive from the Bijection Permutation Paradox that is a consequence of Set Theory’s use of bijections from transfinite sets onto proper subsets of them selves. (See an entertainingly informal account in The Good Shepherd’s Paradox.)

Charles Sanders Peirce (1839‑1914), and later (but much more famously) Julius Wilhelm Richard Dedekind (1831‑1916), suggested that the existence of a bijection of a set with a proper subset was in fact definitional of transfinite sets, and this is now accepted as standardly fundamental. The reason this is considered to be 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 easy to derive, from the standard axioms and rules of inference of set theory, a bijection counterpart to set subtraction, specifically for the “bijections” from sets onto proper subsets of themselves that are fundamental to transfinite counting and arithmetic. I.e. we find the Bijection Permutation Paradox that a bijection must formally exist between a non-empty set and the empty set.

And even mathematics greats like Bertrand Russell and Ernst Zermelo made significant oversights, Russell in his Russell’s Paradox (see Russell’s Great... Oversights), and Zermelo with regard to his Axiom of Separation, a paradox that we have given the name of “Axiom of Separation Paradox” (see Zermelo’s Great... Oversight).

General Introduction to Paradox in Modern Math

The set theory developed by Georg Ferdinand Ludwig Philip Cantor (1845-1918) has been paradoxical since its inception. In fact, with the acceptance of Cantor’s set theory, paradox, which had earlier meant inconsistency in a mathematically unacceptable sense, has come to be considered not merely acceptable, but conventional. The acceptance by mathematicians of this marriage of paradox and consistency in the two thirds of modern mathematics that has Cantor’s set theory — along with logic — as an essential part of its foundations can be summed up in David Hilbert’s rhetorical question: “What mathematician would want to be expelled from the paradise which Cantor created?” In order not to be “expelled” , i.e. in order to avoid turning acceptable paradox into unacceptable inconsistency, we even selectively abandon fundamental principles of mathematics. A standard example is the avoidance of the standard application of the standard rule of inference that equal quantities can be subtracted from both sides of an equation to the fundamental equation of transfinite cardinal arithmetic: a₀ + 1 = a₀. There are more.

This result — that a₀ + 1 = a₀ — is often proven by means of constructing a bijection from Á ~ {0} onto Á, demonstrating that |Á ~ {0}| = |Á| (where |S| is the cardinality of the set S, and Á 4 {1,2,3…} is the set of all natural numbers, and thus the set union of Á and {0} has 1 more element in it than Á). (It can be noted in passing that modernly some people have started defining Á as Á 4 {0,1,2,3…}, but this should not be a problem here.) Since |Á ~ {0}| = |Á| 4 a₀, and |Á ~ {0}| = |Á| + |1| = a₀ + 1, the equation a₀ + 1 = a₀ is considered to follow immediately. (See the section “‘Counting’ and Cantorian ‘Reordering’”, below, for an analysis of the ostensibly bijective mapping from Á ~ {0} = {0,1,2,3…} onto Á 4 {1,2,3…}.)

In the face of a₀ + 1 = a₀, since the theorem 1 = 0 would be considered not merely paradox but inconsistency (unacceptable since e.g. all natural numbers would become equal), we back off from the usual definition of a mathematical theory being all the theorems that can possibly be derived from the axioms and rules of inference, and from the usual definition of the inconsistency of a theory being the possibility of formally deriving both a theorem and its negation from those same axioms and rules of inference. Relatedly, we also back off from the fundamental principle of always being able to replace a defined/constructed entity/symbol by its initial definition/­construction. E.g. once we define/construct transfinite arithmetic from finite arithmetic, we allow proofs using a “simultaneous” mapping from n to n + 1 for all natural numbers n in Á 4 {1,2,3…} without further reference to the serial definition/construction of the natural numbers, and we get very different results.

With regard to the fundamental theorem of set theory that a₀ + 1 = a₀ we have selectively, in a dubious context sensitive fashion, deliberately failed to apply a standard rule of inference merely because it would yield a standard inconsistency (in a theory not noted for its paraconsistency). We can speak of hidden “Rules of Deference” in such situations, where we quasi-formally refrain from performing a standard derivation, e.g. applying a standard rule of inference, which derivation/application, by definition of a mathematical “theory”, gives us a valid theorem of the theory, and gives us a contradiction if and only if the theory is inconsistent. Any mathematical theory will be “consistent” if we proceed on such a basis.

Paradox in set theory is usually associated with the Axiom of Choice, the use of which seems to lead to rather more paradox than many mathematicians are comfortable with. But, in fact, unacceptable paradox in set theory will be seen to derive from the Continuum Hypothesis, or rather from the fundamentals that give rise to it. 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.

Besides “paradoxes” (such as above) that are known but not yet adequately appreciated, there are many new paradoxes that are as yet unrecognized in set theory and related theories such as real number theory. These paradoxes, some of which will be studied in later sections, also suggest that a new, non-classical kind of non-Cantorian set theory is needed for transfinite sets and their arithmetic, at least one that satisfactorily resolves all known paradoxes, especially the more embarrassing ones, in particular the new ones presented in this paper.

“Non-Cantorian set theory” (see definition) has classically been defined, with great lack of imagination concerning the entire range of possibilities, as an otherwise standard set theory that axiomatically rejects (as opposed to deriving a falsification of) the Continuum Hypothesis, which has so far seemed to be an independent axiom (e.g. Gödel and Cohen). However the Continuum Hypothesis will be found to be not only falsifiable, but provably false (see Theorem 12), and a new set theory will be needed that more than just rejects it, or even falsifies it. An alternative to the Continuum Hypothesis, a “Quantinuum Hypothesis”, will be seen to derive naturally from all this, and to suggest paradigms for a new theory.

The desirability of developing a Quantinuum Hypothesis and a new and non-standardly non-Cantorian set theory (or theories) will be proposed in this paper, but in the sense of necessity, not just in the sense of the explorations of new systems for their own sake as with non-Euclidean geometries. A new theory is needed with a whole new concept of infinity or “transfinity”, one from which e.g. “quantinuous” infinitesimals/“transfinitesimals” derive naturally. An example of a possible benefit of such a new non-Cantorian theory is that it might offer a resolution to the problem of renormalization that we find in e.g. quantum mechanics, where, related to the a₀ + 1 = a₀ paradox, infinite quantities are subtracted — with a hope and a prayer — from both sides of various equations to get equations of finite quantities that are theoretically useful.