| Finite-small equality does not imply infinite-small equality |
[See also transfinitesimal qualification and 'infinity comp']
[Expanded from my sci.math post, 2/13/96 - nota bene: I am its original and sole author]
We learned in elementary school that 0.999... = 1 , as 9.999... - 0.999... = 9 . Well and good finitely it is, but that equality is merely valumetric: we say, 0.999 = 0.999999, as well, in our valid mathematical notations of significant figures, because any difference sufficiently small is beyond our arithmetic ability to distinguish it, and so we determine it is, equal... [In some cases a dot is put over the equals sign, 0.99 ≐ 0.999]
* (It is an 'oddity' perhaps not even mathematical that any might esteem that a decimal fraction of infinitely-many digits is 'obviously' not-finitely-expressed while yet an infinite-sequence of progressively-finer finitely-expressed fractions should have none with infinitely-many digits... a 'Zenoic-infinity-of-infinitesmals-never-achieved' paradox.)
* (The 'trick' aligns each irrational number into a list and invents one that cannot be in the list because it is constructed of digits differing in some corresponding digit place with each number considered in the list... it assumes that ∞+1 > ∞ in certain cases, or even 10∞ > ∞ , while yet agreeing ∞2 = ∞ when enumerating infinitely many pairs-of-counting-numbers for rationals and 2D-spaces and-so-on....)
* (The 'trick' is likenable to 'proving' the counting numbers uncountable because a number can be invented, picked, larger-than each number in an ascending list and pretended it is not any in the list at-least-stepwise none-so-far-considered... the invention for counting numbers is just as rigorous and just as specious, though lacking finitesmal convergence giving it an 'air' of realism, in using scalar addition rather than digit assignment, polynomial addition, yet requiring just as countably-infinite-many digits specified, to be, equal, or not.... The fact is, such an invention is the definition of 'infinity', not of 'uncountability' as that applies to infinity... and 1 is rather a discontinuity for 0.999... → 1.000..., comparable to 1/0 being neither ±∞ yet 1/0± is-equal.)
* (The 'trick' also relies on an assumption, misguided, that, a list of all rationals is all-inclusive-'closed' against finding more-to-be counted than have-been counted, by some finitely-constructed counting-technique, whereas countably-infinite lists do-find more already, easily including twice-as-many numbers even infinitely-many-more, or ten-times-as-many for decimals, as can ever-infinitely be examined... so any decimal of choice can be in the list yet nowhere found, because a countably infinite list is open, not-closed, to more than ever found...)
The correlative trouble is that the trick asserts that decimal fractions are unique by their digits—yet contradicting the claim that 0.4999... = 0.5000.... It is counter-intuitive to suppose that ending-in-nines collapses onto the adjacent ending-in-zeros, and-or that the collapse happens at infinity which is never-ending.... It may be notable that this situation was discovered in rational-division where in fact the borrow-bit was never fully resolved but neglected, tossed onward, as-if-under-the-never-finished-infinite 'carpet'—there was always a leftover finitesimal undiscussed, uncounted.... 0.999... = 1.000... only on any 'valumetric' finite, scale, not on any infinitesimal-scale such as tangential-adjacency-of-points, especially as points are zero-width-not-merely-infinitesimal-width....
The second correlative trouble is that if numbers are deletable by such equationing, then in different radixes it's different numbers, and, different numbers-of numbers....
The second trouble is that the student must ponder whether there is an answer, or a paradox, for the construction really only regards priors, listed, (the same trouble finding larger counting numbers: you just haven't gotten to it yet in the longer list), meanwhile the choice could be anywhere-next in the infinite-list, even far-down-the-line so long as it's 'denumerable', e.g. the list {0,0.1,,,0.9,0.11,,,0.99,0.111,,,..., or cleaner without the ...0-duplicates} has all decimal-fractions on the real unit interval [0,1) to infinitely precise, however long it takes to get there: any 'choice' is somewhere among the infinitely many choices next, and, subdividing infinitely many choices by successively selecting digits, in nowise reduces infinity, nor in anywise reduces the infinitesimal probability of finding it, to zero; and, all, listed, means all, digits—this listing method is merely an exponential construction of an infinite-square-area list, no-less-countable than a spiral or diagonal raster-scan construction... there is nothing special about exponentially, large counts or, constructions, over power-law large, in finite arithmetic... and the simplest construction is to specify every-and-all digits, and, likewise, every-and-all constructions are included,—exponentially farther along if non blank-terminating ('0...'-terminating too) because it's a 'list' of fractions constructed by digits (and-or blank-terminators)....
A correlative trouble is that the cumulating exception could-be (no smallness of possibility can prevent) the very-next-number in the list... (countability is not the 'end-all' to infinity—a 'proto-Zenoic' paradox—more mathematical definition to figure out)....
(It is remarkable that the method generates an infinite sublist among fractions where well-ordering and partial-ordering may be useful methods in certain applications such as for the notion of enumerating-all-of-them-after-enumerating-half-of-them such as finding ways to find numbers thrown-back into an infinite heap, among endless mathematical studies... When applied to counting numbers the methods are at-their-simplest the induction successor function, n+1 , or more-general induction enclosure or hull function, max(n:prior)+1 , or, if anticipating the next possible digit, as was done for fractions, max(n:prior)+10i at the ith step, radix 10 , etc....)
Such irrationals are thus countable, all, (not just the algebraic, irrationals, having finite rational constructions)....
The third trouble is that a counter argument can be constructed to force inclusion of those very 'different' numbers, by herding; and there are only countably many [rational] constructions of such 'different' numbers: And therefore it is a paradox, not an answer.
And its correlative trouble is that the 'all' declaration of digits is specious: Infinite all, is not the same-as nor extension-of, finite all: There are infinite-many digits that can never be reached, -to be specified,- by a finite all, process... (Even as infinite all rational, numbers, in a list, has infinite many numbers that can never be reached by finite progression of all digits, Yet, only by specification of infinite all, digits, can any finite repeating-digits 'rational' number be specified... even 1.000... Exception by specification to a rule that of-itself can only ever specify a finite, number of nonexceptions, is shy of the infinite, all, rule, -whatever that may be in the case:- the finite rule supposes-upon infinity, but is incomplete as a rule... nevertheless finite-covering, infinity, is still an appropriate process—if, it is 'uniform'...)
The fourth trouble is that the suggestion that a number can be specified for all its digits to be sufficiently different, is intrinsically a Zenoic paradox: At every stage in the development its 'discovered' number may be further in the list... Merely to assume the infinite all can compass itself to exclusion to break-through or-not in the infinite-all case, is not a proof but a self-paradoxic assumption: For example of the paradox itself consider the infinite sequence, 0.0, 0.10, 0.110, 0.1110, 0.11110,... and 'different' number 0.11111..., and the test-question is whether the sequence includes it:- Certainly no finite position in the sequence is it, but the sequence is infinite and has just as many numbers as there are digits in 0.11111..., whence if 0.11111... can be specified 'for-all-digits' then that sequence can be specified 'for-all-numbers', which means that there are numbers in the infinite sequence that are as long as the sequence is long—ergo the sequence must include 0.11111... in its 'infinite all' numbers as surely as 0.11111... exists only in all, its digits... It is simply 'in, there'....
The fifth trouble is deep in the notion of countably infinite progressions in the context of pointwise—in that it is statistically not possible to prove that a number given of the proper 'countable' property exists in the list—you can't really declare that you have, such, a list.... To put it simply, the likelihood of finding a given number that by definition exists in the list, say for example the numbers reciprocal of Natural numbers, in the list of such by the same property—the likelihood of turning up that particular number by constructive method, is (1/∞) per try, and, when you run-out the entire countably infinite list, your likelihood of finding the same is 1-(1-(1/∞))∞ which is only 1-(1/e), Not guaranteed, not even finitely close, to unit-probability... So, apparently, countably infinite is (e/e-1)× larger than itself—and 37% of those numbers existing by the rule among countably infinitely many, will never actually exist in their own list... So, it's not surprising, that, some one, surd, being by definition, different, is not-going to exist in a countably infinite list,—when 37% of infinitely many being similar and so belonging won't be anyway—probabilistically... (probabilities and deterministics are somewhat at odds)....
A sixth trouble is a related problem the 'size' of infinity, countable vs uncountable, in particular the number of fractions 0.ddd... (decimal, binary, autc.) by digit-combinations on the real interval [0,1), = 10ℵ₀ = ℵ₁ supposedly the number of irrationals as-above but is included in the countable-infinite by mathematicians (reputedly Hilbert lecturing a century ago) wherein the number of mapped indice-doublets 2i3j or primes pij (var ref) is countably infinite but fastens the purportedly uncountable irrational infinite many unmapped values within the same map-space range... n.b. there exist maps of indice-pairs without, exponentiation, e.g. the standard example diagonal-counter j+½(i+j)(i+j+1) ...
A seventh trouble is the similar process of constructing such an exception-finder in scalar-arithmetic rather than in polynomial-arithmetic (digit selection), for example, in a list of purportedly all, rationals n/m ∈ R (or a certain interval), m.n ∈ Zradix , an exception-finder could be the scalar-average of nearest rationals, i.e. (nq+mp)/2mq for n/m ≈ p/q , which, like the digit exception-finder, narrows infinity, to infinity, by a factor at each step, but, as a finite and-ever-rational construction it never really constructs a rational-not-in-its-list-following of all rationals (or interval thereof) as enumerated by the countably-infinite spiral or diagonal method which 'obviously' does-include every, 'exception'....
NB. This technique also points to another discontinuity: that the calculation of 0ρNNN... = 1 , holds in all radixes until ρ=1, where N=0, where it does-not hold even in the limit, but would hold for ρ=1+ð, ð infinitesimal, if ð and the count of digits were inversely relatable.
In summarial remedy, It is probably best were mathematicians to distinguish, Any, and, All,
and not-identicably, in the Infinite case.
__
See also a discussion of transfinitesimal qualification.
* [Uniform Convergence is usually presented in Real Analysis as a zero-convergent
sequence of integrable functions squeezing standup area along the functions' pointwise
infinite mesh into the interval boundary: never reaching 'all-zero'; here squeezing to
infinitesimal]
* [Recognize that these are as proofs-by-contradiction, that, Uniformly handling all,
digit places, does not yield uniform convergence]
* [I argued this out before I ever posted 2/13/96 on the Internet where it has been
argued repeatedly—even after I gave my proof]
Topic note: The meaning of "real" number has principally the mathematical sense here, and whence can deal with infinitesimals,—as the cosmological sense of "real" is lower-bounded by aether-noodle granulation at its 'next-level' of 'under-standing' and 'oper-ation'.
A premise discovery under the title,