, worth looking at, but still in early stages

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Real Number Theory Paradoxes

Vanishing Remainder Paradoxes

Real Numbers Paradox 1

Real Numbers Paradoxes 2 and 3

A “Quantinuum Hypothesis”

Real Number Theory Paradoxes

There are a number of paradoxes in real number theory heretofore unrecognized by the community.

Some of these paradoxes concern infinite decimal (or other base) expansions of real numbers. (See the Vanishing Remainder Paradoxes and, related but not strictly a Vanishing Remainder paradox, the Real Square Root of 2 Paradox.) These 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 real  numbers. Out of this examination of vanishing remainders 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.

Other paradoxes, some that are fundamentally variants of the Vanishing Remainders Paradoxes, concern Cauchy sequences, Dedekind cuts, and related topics.

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Real Number Theory Paradoxes

Vanishing Remainder Paradoxes

Real Numbers Paradoxes 1 and 2

Real Numbers Paradox 3

A “Quantinuum Hypothesis”

Vanishing Remainder Paradoxes

A set of interrelated paradoxes come up when we reexamine the well-known paradoxical result that all real numbers (usually based implicitly on the popular ZFC variant of Cantorian set theory) have a unique infinite decimal representation, except the integers. E.g. 2.999... = 3.000... are ostensibly 2 distinct but numerically equivalent infinite decimal expansion representations of the integer 3. Key to this reexamination is to study how the infinite decimal representation of the rational number 1/3 is constructed.

The zeroth decimal place of 1/3 is “0”, i.e. the quotient of 1/3. The first decimal place is obtained by multiplying the remainder of 1/3 by 10 (decimal) and again dividing by 3, giving us so far “0.3” (the “3” being the quotient of 10/3). The succeeding decimal places are obtained in the same way, giving us “0.333...”.

But certain questions, that can be considered paradoxes, were never asked (nor answered), such as:

1)      In constructing 1/3 = 0.333... or 1/9 = 0.111..., what happens to that strictly non-zero remainder of 1?!

2)      Does it vanish just like the 1 in a₀ + 1 = a₀ does on the right hand side of the equation?

3)      If it becomes a strict zero, “vanishing”, as real number theory assumes by ignoring the issue, how does it do so?!

The questions sound satirical, which they are, but they also have very serious import:

4)      If we divide 1 by n enough times, does the remainder of 1 become 0?!

5)      If we divide 1 by n, does the remainder of 1 become 0 if n is big enough?! (More on this in a later section.)

We should also ask:

6)      What happens to the remainder of 2 when constructing 2/3 = 0.666...?

7)      What happens to the repeating sequence of non-zero remainders (3, 2, 6, 4, 5, 1,...) when we construct the infinite decimal expansion of e.g. 1/7 = 0.142857142857...?

If the remainder doesn’t become a strict zero, then there is more to a real number than heretofore suspected. If it does become a strict zero, there is more to real number theory than heretofore suspected.

E.g. “1/3 = 0.333...” must have a remainder of 1 ≠ 0 in the “last” decimal place, which by standard theory doesn’t exist, allowing us to overlook and/or ignore the problem easily. (This 1 ≠ 0 links us back to the 1 = 0 question that is raised by the paradoxical a₀ + 1 = a₀ and its reexamination above.) This remainder could conceivably be either 1, 2, or the 0 usually implicitly assumed, since we have a modulus 3 situation. When we multiply “1/3 = 0.333...” by 3, we actually get:

“1 = 0.999... + 3 × the remainder of 1 in the ‘last’ decimal place, which last, when divided by the divisor of 3, gives 1; this last 1 then needs to be added to the quotient of 10 / 3 = 3 (i.e. the value in the ‘last’ decimal place) × 3 (the divisor again) = 9; and when that 1 is added to that 9 we get 10, i.e. a 0 in the ‘last’ decimal place with the 1 carrying into the ‘next to the last’ decimal place, which also doesn’t exist, and so on ripple-carrying all the way back to and just past the decimal point giving us 1.000...”.

But if 0.999... is distinct from 1.000... it raises the spectre of infinitesimals, the existence of which is still denied by standard theory. (They are included in non-standard analysis, but forcibly; they are not naturally derived.)

We can also note that the standard proof that e.g. 2.999... = 3.000... involves multiplying 2.999... by 10 and then subtracting the original 2.999... (ostensibly) giving us 27.000... which we then divide by 9 giving us the 3.000... we saw above. The multiplying by 10 has an essential correspondence to the n → n + 1 mapping we saw above since it maps the nth decimal place onto the (n – 1)^st decimal place, for all n.

The “vanishing remainder paradox” gets more complicated when we consider two more things. The first is that if we had started with 1/9 instead of 1/3, the possible remainders for the “last” decimal place would obviously be 0 through 8, but we would still have “0.999...” as the representation of both 3 × 0.333... and 9 × 0.111.... Just looking at “0.999...” doesn’t tell us whether there should be a remainder in the 0 to 2 range or the 0 to 8 range. We could also try to carry along information about the divisor, but... if we studied all the rational arithmetic expressions which were arithmetically equal to 1, we would get more than just this simple “vanishing reminder paradox”. The vanishing numbers we would have to take into account would probably require a Galois to unravel.

The second thing is that if we look at the infinite ternary (base 3) representation of 1/3, we find that the vanishing remainder problem itself vanishes, at least temporarily. The rational number 1/3 can be described in an infinite ternary expansion as 0.1000..., i.e. with not even the “infinitesimal” loss of accuracy associated with a “vanishing remainder”. The real numbers, as defined by infinite base expansions, are highly dependent on the base used in a way obviously relating to the prime decomposition of the base.

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Real Number Theory Paradoxes

Vanishing Remainder Paradoxes

Real Numbers Paradox 1 and 2

Real Numbers Paradox 3

A “Quantinuum Hypothesis”

Real Numbers Paradoxes 1 and 2

The vanishing remainder paradox gives the even more paradoxical result that:

Real Numbers Paradox 1:
the real numbers as defined by infinite decimal (or any base) expansions do not contain (all) the rationals.

E.g. we cannot successfully represent either 1/3 or 1/9 as a infinite decimal expansion real number since that remainder of 1 does not truly vanish, nor is it properly taken into account.

We might as well quickly look at the “Real Square Root of 2 Paradox” that generalizes to the “Un-Real Irrationals Paradox”. If we look at an extended rational approximation of the square root of 2 in decimal representation we get:

which we can rewrite as:

Number theory tells us that both sides of the equation must have the same prime decomposition, and also must be perfect squares since the right side of the equation is a perfect square. Except for the first 2, both sides of the equation are perfect squares, but the extra 2 on the left hand side of the equation means that 2 cannot have an even exponent on the left hand side unless 2 is a perfect square. So number theory is telling us is that:

Un-Real Irrationals Paradox: when we try to turn the rational approximation of e.g. the square root of 2 into a real number in an infinite decimal (or other base expansion), we need enough decimal places to give the prime number 2 a non-trivial prime decomposition, and a square one at that. I.e. the irrational number that is the square root of 2 cannot be a real number as defined by infinite decimal (or other base) expansion.

Notice the similarity between this proof and that of proving that the square root of 2 cannot be a rational number (proven by the Greeks over 2000 years ago).

The above means that not even Cantor’s absolute infinity can give us enough decimal (or any other base) places to represent the square root of 2. Although it is not strictly a vanishing remainder paradox:

Real Numbers Paradox 2:
the real numbers as defined by infinite decimal (or other base) expansions do not contain (all) the irrationals.

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Real Number Theory... Oversights

Vanishing Remainder Paradoxes

Real Numbers Paradoxes 1 and 2

Real Numbers Paradox 3

A “Quantinuum Hypothesis”

Real Numbers Paradox 3

We can add to that a combination paradox, for good measure (informally stated, but recognizably relating to the problem that different bases can give us “incommensurate” infinite base expansions, depending on the prime decompositions of the bases; e.g. there are ternary reals that are not contained in the decimal reals, and vice versa):

Real Numbers Paradox 3:
the real numbers as defined by infinite decimal (or other base) expansions do not even contain (all) the reals.

We are also reminded, since  here (as opposed to standard theory) seems to be necessarily different from , that it was never really that obvious why  was the “continuum”, and not e.g. .

SECTIONS

Real Number Theory... Oversights

Vanishing Remainder Paradoxes

Real Numbers Paradoxes 1 and 2

Real Numbers Paradox 3

A “Quantinuum Hypothesis”

A “Quantinuum Hypothesis”

A “Quantinuum Hypothesis” as an alternative to the Continuum Hypothesis will be seen to be inevitable since the Continuum Hypothesis really depends on those remainders vanishing at an infinity of decimal places. In a new, non-classical non-Cantorian set theory we will need to develop the concepts of “quantinuum”, or rather of many “incommensurate” “quantinua”, “quantinuities”, “disquantinuities”, etc. The concept of “continuum” as we know it will be a casualty, since once we speak of the “continuum” as a “number” of points/_­numbers we actually have a “quantinuum”. And a non-vanishing remainder per force gives us not only a quantinuum, but naturally derivable infinitesimals, or to put Cantor’s terminology to new use, “transfinitesimals”. As with quantum mechanics, we will need to distinguish macro-properties (the “classical” ones that we are more familiar with) from the micro-properties, e.g. the “local quantum micro-topologies”.