, worth looking at, but still in early stages

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More Real Number Theory Paradoxes

The Vanishing Remainder Paradoxes are not the only paradoxes in Real Number Theory. The are many other paradoxes, some that are fundamentally variants of the Vanishing Remainders Paradoxes, concern Cauchy sequences, Dedekind cuts, and related topics. They can be difficult to classify as to whether they are properly paradoxes in Real Number Theory, Set Theory, Measure Theory, or some other theory.

There are further paradoxes that have been standardly overlooked. The “Countable Reals Paradox” and the “Non-Denumerable Rationals Paradox” (neither, again, widely recognized nor having official names) straddle set theory and real number theory.

If we look at the “limit” of the closed interval [0,1/n] “as n goes to T (infinity)”, by standard theory we get [0], an interval/set of “measure zero”. Standard real number theory and standard measure theory have both ignored the “vanishing remainder”.

When we ignore the vanishing remainder we get the paradox that, if we look at the countable closed cover (not a partition since the intervals are not disjoint) of the unit interval [0,1/n],[1/n,2/n],…,[(n-1)/n,1], we find that “as n goes to T”, “in the limit” we get a countably infinite/­denumerable closed cover for the unit interval, each closed interval of which has at most 1 real number in it. The paradox is that this means there can be at most countable real numbers in the unit interval (easily extendable to the entire real number line), as opposed to the standardly accepted non-denumerable real numbers. This is merely the initial aspect of the “countable reals paradox” overlooked so far by the mathematical community.

The situation gets worse. Each closed interval in the cover shares precisely 1 point with each of its neighbors, and this gives us a real mess. Each closed interval has only 1 number to share. By induction we not only get that the unit interval can only have 1 real number in it, but that there are at least 2 obvious candidates, 0 and 1. The situation gets worse yet if we look, not at a closed cover of the unit interval, but at a semi-open cover, actually a partition since the intervals that cover the unit interval are disjoint. What is the “limit of [0,1/n) as n goes to T”?! How many points are there in the semi-open interval [0)?! The “countable reals paradox”…

• Countable Real Numbers Paradox: the real numbers must be countable (as seen above), perhaps even… “finite”.

This denumerable partition of the unit interval is yet more paradoxical. The partition intervals are each associated with a natural number. The number 1 associates with the paradoxical interval [0), and the “number” a₀ associates with the “last” interval, the one “closest to [1)”, none of which can exist, yet also must exist, overlooked by standard theory. All the standard natural numbers, the “finite” ones, are associated with intervals “close” to [0). But the intervals not close to [0) but not next to [1) either, what numbers do they associate with?! They are distinctly greater than “finite”, but also distinctly less than “a₀”, another Continuum Hypothesis paradox. Informally, at this stage, we seem to be able to speak e.g. of “a₀/2”, “3a₀/2”(out past 1), etc. A new, non-classical/non-standard non-Cantorian set theory would need to live with such entities non-paradoxically, perhaps by conceiving of “infinity” (of the “neo-natural” numbers) as “fuzzy”, or “relativistic”.

Earlier (Theorem 12 in Bijection Paradoxes) we saw:

• Continuum Hypothesis Paradox 1:   a₀ + 1 > a₀.

And now we find:

• Continuum Hypothesis Paradox 2:
  there also seem to be infinities less than a₀.

Mild Digression: One question that comes up is “what infinity does n go to?!” (A similar question arises in automata theory, where there is also the question of the halting of a Turing machine after it has written an “infinite” number of ones, so as to correspond to Cantor’s idea of “completed infinity”. Mathematicians have never been as careful as they should be when speaking of “infinity” and “infinite” to distinguish the denumerable and the non-denumerable.) If instead of the usually implied a₀ it goes to , we can fend off some of the paradox for a while. But in the long run, it will still catch up with us.

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Standard theory holds that between any 2 rational numbers there exists a real number. But it also holds that between any 2 reals there exists a rational. The “non-denumerable rationals paradox” has been overlooked.

Since there are non-denumerable reals in e.g. the unit interval, and each of these distinct reals is either greater than or less than all the others, there exists a non-denumerable partition (very many such, actually) of the unit interval, each subinterval of which has non-rational endpoints (and non-denumerable reals between, although neither of these points is strictly essential to the argument), and paradoxically, at least 1 rational number. I.e. by standard theory, even though overlooked, there must be non-denumerable rationals, in counterpoint to having only countable reals.

• Non-Denumerable Rational Numbers Paradox:
  the rational numbers must be non-denumerable (as seen above).

The argument is informal for reasons of space, but the “vanishing remainder”, “countable reals”, and “non-denumerable rationals” paradoxes combine to give us the “Cauchy-Dedekind paradox” (again not yet widely recognized as a paradox and having no official name). I.e. they tell us that Cauchy sequences of rationals and Dedekind cuts (using 2 sequences of rationals, 1 greater and 1 less than the irrational number being “uniquely” defined) cannot successfully define all real numbers. E.g. a sequence of rational numbers has 1 as its smallest non-zero numerator and n “as n goes to T” as its largest denominator/divisor, necessarily combined with the smallest non-zero “vanishing remainder” of 1, giving us the “smallest distance between rationals distinct from absolute zero”.

We can informally say that this smallest distance, the “granularity” of the rationals, is (approximately, for many reasons) “1/a₀”, as when we constructed a denumerable closed cover for the unit interval. If we assume that we don’t really want non-denumerable rationals, then we have rational granularity intervals containing  reals but no rational numbers. Not even an infinite sequence of rationals can get us unambiguously and indefinitely closer to precisely 1 (arbitrarily chosen) real number in that interval of  reals than that “granularity” suggests. E.g., between 0 and “1/a₀” there must be  “quantinuous” reals with no rationals, but a sequence of rationals can only “converge” to either the 0 or the “1/a₀”, nowhere in between.

“Information theoretically”, infinite sequences of rationals should allow us to encode  real numbers (with something akin to the topological discontinuities we get when projecting spaces onto lower dimensional subspaces), but the arithmetic convergence requirement takes away that ability. The reals (will) have their own granularity, and one of the casualties there will be the Bolzano-Weierstrass theorem with its “cluster points”, since the granularity that derives from the actually non-vanishing remainder ensures that we can always find a non-null punctured neighborhood around any point with no numbers/points in it whatsoever.

• Cauchy-Dedekind Paradox: the real numbers cannot be defined by Cauchy sequences of rational numbers nor by Dedekind cuts.

This set of issues also relates to the Archimedean property, a standard property of the real numbers. By the Archimedean property “for any positive reals a and b, there exists an n such that n · a > b.” (All 3 numbers are standardly assumed to be “finite”.) It is well-known that the Archimedean property is inconsistent with the existence of infinitesimals (since n must be “finite”), and overlooked that it is inconsistent with the existence of infinite/­transfinite quantities, and effectively prevents the field of the reals from being extended to the transfinites (which many have thought they would like to do). One can also derive the “countable reals paradox” directly from the Archimedean property, since we can rewrite n · a > b as n > b / a, and note that n is obviously an upper bound on the number of points between 0 and b that are separated by a minimum distance a; since it retains this property for all a > 0, the reals, which standardly have the Archimedean property, are paradoxically also provably countable. The existence of  reals evenly distributed in e.g. the unit interval should have given us the sense that there must be non-zero differences among them as small as 1 (> 0 since we don’t want the remainder to vanish) part in , a distinctly non-Archimedean value.

If we take into account the Real Numbers Paradoxes 1, 2 and 3 (as seen in a previous section on the Vanishing Remainders Paradoxes) and the Cauchy-Dedekind Paradox (as seen just above), then we have the:

• Real Numbers Existence Paradox:
  the real numbers can be seen not to exist by any of the standard definitions, in our current real number theory based on Cantorian set theory.

and the:

• Continuum Existence Paradox:
  a “true continuum” would require a “granularity” (as informally defined above) of absolute zero, with no “vanishing remainder paradox” and the “quantinua” that come with it.

Reminder: once we speak of the “continuum” as a “number” of points-numbers, even an “absolutely infinite” number of points-numbers, we actually have a “quantinuum”. By Theorem 12 (in Bijection Paradoxes) even absolute infinities can be made greater by adding 1, and even absolute infinities used as a divisor leave that strictly non-vanishing remainder of 1 that gives us a non-null punctured neighborhood around any point that cannot have any numbers-points in it, characteristic of a “quantinuum”. A true continuum could have a transfinite number structure placed on top of it, leaving vast empty spaces capable of holding transfinitely many, transfinitely vaster such structures. We mistook our actually “quantinuous” structures for a “continuum”.

Mild digression: a₀ is not considered to be an integer in the field of the reals, even though it is the cardinality of the integers greater than 0 (etc.), even though it is the ultimate successor cardinal of 1, the “limit of n + 1 as n ‘approaches’ infinity”. It is worth noting that a field has much the same definition as the set of all natural numbers, since if a and b are field elements then a “+” b is also a field element. Mathematicians try to back away by insisting that only “all finite” combinations­successions are being considered, but the definition­construction of the natural numbers and finite induction also give us “all finite” combinations­successions. Since finite induction gives us Á it should also give us a₀, but by standard theory it doesn’t. If pressed, mathematicians will try to insist this is mere “paradox”, not “inconsistency”.

Real number theory has some serious paradoxes in need of public recognition (as do measure theory, analysis, et al). We need to accent that both real number theory and these paradoxes derive from Cantorian set theory, and relate to the results in Bijection Paradoxes Theorems 9‑12. It becomes obvious why we should start to develop non-Cantorian variants of set theory, real number theory, measure and integration theory, analysis, topology (some), etc.