| Oft discussed, oft erudite, oft esoteric, oft arcane, paradoxes at the edge of infinity and infinitesimal oft exposing flaws in linguistic analysis, thinkabouts with answers not otherwise attainable... |
[See also uniform convergence and 'infinity comp' and sometimes mathoughts-notes 'til reedited]
#1. The edge of infinitesimal can be illustrated by the following comparison:
{y = x ; y = 0};
{y = x2 ; y = 0} .
Both systems, pairs of equations, have an intercept at [x y] = [0 0]. However, the first system instantly diverges while the second does not: For x = ∂, infinitesimal not identically zero, the first diverges to [∂ ∂] on a slope of δ=1; while in the second [∂ ∂²] the y-value is an infinitesimal of the x-value itself, on a slope of δ = 0,—challenging the notion that the two curves in the second system intercept at only one, point, when a slope of 0 is satisfied by an adjacent y-value actually =0, (x≠0,y=0), but-yet which slope lasts only x-distance-length =0 which is-not a slope but a point, so its slope=0 is only in-the-limit, as the slope on either side is negative or positive and only zero can lie between, so a lone point would not have a single-slope but a discontinuous pair of slopes = 0± = ±1/∞ ≠ 0, (the usual proof of derivative-slope is that the point is midway on a curve with chord-slope=0 and therefore there exists a tangent-slope=0). The finite interpretation is that a difference in x results in no difference in y for zero slope, and infinitesimal times zero should likewise not—not if 0.999... = 1.000.... But eventually y diverges infinitesimally when x exceeds infinitesimal, to the square-root of infinitesimal in this case, though not-yet finitesimal. Thus there is a tower structure in the transfinitesimal, qualification, range, and infinity is not the direct extension of arbitrarily large, and induction has a ways further to go—never crossing the chasm between finity and infinity, though mathematicians had long presumed.
Let's consider more closely:
y = e−1/x : x>0 ; y=0 ;
its first y equivalent to:
y = e-n : n=1/x : x>0 ,
like the number 1.000...-0.999... at its n-th fraction-digit: this system converges slower-still near x=0,
And is discontinuous there: Its first y-derivative is e−1/x/x² : x>0
which is about e-n n² but even as n→∞ and nears x=0 it
never reaches y=0 nor slope=0: For n to reach infinity -its all- is no sooner than 1/x and its discontinuity...
Thus implicating a class principle of continuity, versus a distinction between 0.999... and 1.000...,
and a regradation among finitely vs. infinitely, Any, Each, Every, All.
#2. Integration of derivative leftovers: (something that bothered thought since pre-freshman college)
One of the early calculus paradoxes was why we'd differentiate ('take the derivative') without keeping infinitesimal excesses (and doubly-infinitesimal terms ∂x2 etc.) somewhere in the result for later re-integration (and double-integration ∫∫∂x2 etc.) to recover exactly the original rather than always arbitrate with some additive constant—especially when there was more information originally... cf antiderivatives add 'C' which cancels in the evaluation (∫ab0 = C(b) - C(a) = 0)...
This is like keeping information-more-than-common-arithmetic through processes e.g. √(−1)2 = −1... Or like keeping more physics-information of perpendicularity-units in foot-pounds of torque vs foot-pounds of work...
#3. Is 1/n : n→∞ closed or open:
As n→∞, 1/n→0 but does it = 0 or = 0+ when n goes infinite (albeit finite never does)...
(cf e−1/z = supra-∞-pole at z = 0-, = 1 at z = 0 along the y-axis, = 0 at z = 0+)...
1/0+ = +∞,
1/0- = -∞,
but 1/0 is undefined, not = ±∞
(except 1/0 is-defined symbolically-logically as used in projective geometry, but standalone, not
positive-or-negative, infinity)...
Infinitesmal, infinitesimal, infinite-small, is represented numerically as 0+, 0−, ∂, and differentially (taking a derivative) as dx, ∂x, and 1/0+ = +∞, 1/0− = −∞, and conversely 1/+∞ = 0+, 1/−∞ = 0−, but 1/0 is undefined (cf +∞ ≠ −∞) and so, in the geometry postulate of unique parallel lines in flat space, there is a line tilted 0+° and a line tilted 0−° on opposite sides of the unique parallel line except, crossing through the defining point, and, they 'intercept, one at positive-infinity, the other at negative-infinity', (the undefined 1/0 case may be handled as an analytic continuation in complex number theory)...
#4. A simple infinity-race-paradox:
A and B play infinity-paradox games: In the simplest, A and B both, iteratively pick successive numbers one higher than the other—so ultimately which of A or B picks the highest number, Does an ultimate exist, Does it change by who-goes-first (equality of plays), Do we call it a tie because neither succeeds ahead of the other...
#5. Zeno time-forced-decision paradox:
Suppose they play faster with practice, and each pick takes 10% less time ergo the total time is 1/(1-90%) = 10 units, and the last play is done, so there must be an answer; Do they tie-win as neither can finally pick the largest, or more specifically did they transition into infinite-counting, so they infinity-tie, But, how do they transition to 10 units of time: they take one-last 'final' infinitesimal time-step (which for Zeno was like 1/10n n→∞); But when, did finite-counting transition to infinite-open-count; And, does countable-infinity transition to uncountable-infinity (supra-infinity) infinite-closed-count some kind of infinity-plus-infinity where All is infinite but infinite is not all...
Between the finitesimals (the infinite-set-of, as there is no smallest nonzero finitesimal) and, infinitesimal, is an infinitesimal step, And, likewise-between infinitesimal and zero, is an infinitesimal step, (two, infinitesimals, sum, to infinitesimal, but in this case each is distinct—much like we compute derivatives right-down-to-infinitesimal-divided-by-infinitesimal-being-a-ratio-slope, retains distinction)...
#6. Advanced game where the first player accumulates a list of picks:
The same goes for the game of picking a list of fractions to include all-fractions-not-previously-included, Does A's list include B's last exceptional single, or does B ultimately get the last not in A's all-inclusive list... and is custodian C's sweep of everything B discards, as large as A's list except for the first-play B never picked....
Consider where A picks the very-number B picked, to be the next-number in A's list—does B ultimately pick the infinite-last not-in-the-list, or does A ultimately include B's last into the list, (same race as above)...
(Note that this implicates the number of B-picks is also countable, so it's probably not the case that B had more.)
For if-so then the number of irrationals is countable for the list of all fractions generated high-digits-first must include every single fraction of eventually in-the-time-constrained-sense infinitely many digits...
But whereas the number of steps is deemed countable, does countable become uncountable when it goes infinite—(is infinity countable)...
(We also note that the whole process of generating outpicks is a finite—ergo countable—process [construction] so the count of outpicks cannot be 'uncountable' as so many mathematicians have supposed... Also one might -try to- find a procedure for generating 'all' irrationals into a 'superdiagonal' list—for starters n√m/k which already include, all rationals, and expand that by periodically replacing natural numbers by other such algebraically generated rationals and irrationals—but then there'd be transcendentals to generate, too, and, super-'esoteric'-transcendentals that aren't generated by any elementary function....)
Note that this 'argument, walks-right-into-the-rational-saloon' where the last digit of e.g. 1/7th is unknown though time-runs-out and must be determinable-if-and-only-if 'the-last-digit' exists, in-the-limit... but then so-likewise the last-digit of 0.999... is-not-determinable though 'we-think-we-know'—'logically-demanding'—it must be a '9'—the infinity-fact is that there is no-last in an infinite, sequence,—no matter what its value is supposed-to-be, logically, symbolically, 'intimidatingly' ('proof-by-intimidation evoking sidebar giggling—because-I-said-so')...
#7. Ultimate Zeno:
What is the value of n immediately preceding Achilles = Tortoise... and, the value immediately preceding that, i.e. if infinite (for an infinitesimal distance-difference) then when did n transition from finite to infinite—who, takes, those very-last-one-more-steps to the limit from open-set toward closed-set, if mathematical numbers cannot... and, how many infinitesimal last-steps are there, once the first infinitesimal is reached (an infinitesimal of an infinitesimal is infinitesimal, nonzero)....
As the difference in the last-few-steps becomes (1/∞)² for 1/∞ (or smaller for 1/10∞), does the one-last-step make one-larger-than-countable-infinity enough to reach the limit, or is it a logically separable process [regime]...
Counting infinity is already larger than infinity in that infinity always has room for infinite more than counted...
We say ∞ = ∞+1 which is true in the finite context, but unless infinitesimal = zero the points thereat must be orderable in a Zenoic utility sense—the space between successive infinitesimals is infinitesimal-squared O(1/∞)²...
#8. Uncountable is not-much larger than countable:
If the step from 'Zenoic' infinite-open (1/∞) to infinite-closed (= 0) is one, step, then the number of irrationals must be uncountable in an expanded-property-sense (cf quaternions lose commutativity but as anti-commutative), and these are thus only indirectly-countable associated one-for-one with a countable... e.g. in the Lebesgue paradox which is merely Zeno's paradox applied to the full interval of rationals (m/n rather than 1/n), the rationals are zero-width points while the fill-in-closures indirectly-countable-one-each between the rationals have length-measure where point-to-point-spacing becomes 'closed' (ergo integrable)....
#9. 'All digits' (open-infinite) of a fraction:
Specifying a fraction by its infinitely-many digits must have the same infinite-open constraint—that it's not-closed though infinitesimally-close to the number-intented, because it has a leftover infinitesimal-value-last-step-to-closure, even under the Zenoic-time-constraint that leaves closure to one-last-infinitesimal-time-step....
This, is how, indirectly-countable-fill-in-closure-irrationals are infinitesimal-intervals, not zero-width points such as rationals (m/n), but between points... in fact any number defined by asymptotic infinite series, such as also repeating-decimal form rationals, is an infinitesimal-interval (having Lebesgue measure)—which is why we say the irrationals are included in the top-down-digit-construction of decimals... (Note that points also include all the definitively constructible irrationals such as m√n/k and as well limits, definitively one-step-beyond nonterminating, one-step-beyond Open definition, etc...).
#10. Zeno-type relative-race decisions:
#10a. velocity as speed coming vs going—
A line of movie-goers with tickets proceeds to the open entrance door and without stopping each flashes their ticket at the usher and continuing walks through into the theatre... Assuming the velocity is constant for each goer at entry, we consider the relative-velocity of approaching the door as positive and receding as negative... But is their velocity right-in-the-doorway, positive, negative, zero (like an average), all velocities within those limits, no velocity... certainly the velocity is bounded within the positive and negative limits-values...
But more reasonably, like (#3) 1/0 is undefined but 1/0± = ±∞, velocity is undefined, at a single, zero-width, point, (it's not-directly-defined except by including 'in-the-limit' in the definition, translating from logical to arithmetical)—ergo it takes an infinitesimal (or finitesimal) step on the abscissa, to derive a slope, a velocity... so, velocity=0 in a tangential sense (symmetrically equal infinitesimal time at equal instantaneous acceleration on either side of zero yields equal distance, cf the y=x² case of #1 but squarely)...
In a sense this is the Cartesian-to-Polar transformation problem with infinitesimals...
But it also 'sneaks' into math-problems such as the search-for-principal-roots of x↑x (exponential tower of x^x^x^...) where mathematicians chortle over x↑x = 2,4, (but consider logx ↑x = ↑x yet x=1 has no derivable-convergeable answer-path)...
#10b. removing a zero-width point, from a line or boundary—
Can you 'remove' a single, point, where there's always ~infinitely many~ more zero-width points at zero-distance, arithmetically right-there at/in the same point-place... ergo, we must remove 'all' points, at a point, but which implicates removal of an infinitesimal, interval, around the point, leaving an infinitesimal gap—as there must be no, zero-width, points remaining, once all such are removed...
Logically, removing the point, is the usual method used in set theory, but—logic requires fancy arithmetic of removing an infinitesimal interval, around that point...
#10c. 'division' aka the ratio of intervals closed vs half-closed—
Mathematicians always had a problem in defining π (pi) where the circumference is half-closed-half-open but the radius is both-ends-closed and the ratio π (pi) is off-by-an-infinitesimal-interval and not-a-zero-width-point... or... the circumference must be one-point-redoubled with its infinitesimal-extra added-on... either way an infinitesimal difference, this time in a division...
Note, in order for π (pi) to be a zero-width-point we'd take its limit, as with lim 1/(n→∞) = 0 but which, zero, prexists, whereas lim π (pi) would be circular-reasoning 'the-limit-is-what-it-equals' (another 'proof-by-intimidation')...
The problem also existed in the original definition of 'division' in rationals, as, taken-to-mean both numerator and denominator were both-ends-closed which resulted in the midpoints being half-closed-half-open belonging to one-sub-side or the-other, else, they'd be both-infinitesimally-off when ratioed, which factors-in to fractionalize the ratio of their infinitesimal distinctions thus remaining infinitesimally-off e.g. 3-∂/2-∂ ≈ (3/2)+(∂/4)+O(∂²)... except for n/n = n-∂/n-∂ = 1 but which result lacks, its infinitesimal, contradicting casewise...
(This is probably-why point-set-topology is deemed unuseful, and instead, mathematicians do algebraic, topology...)
#10d. Half-points by ordinary division—
In the general case of numbers, compared with intervals, we actually learned to work with half-points, e.g. dividing numbers, e.g. 1 ÷ 2, must start and end with the same objects, numbers per se but we mean here well-defined numbers: if closed intervals, e.g. [0,1] ÷ 2, the result must be closed intervals, e.g. = [0,½] + [½,1], but then without duplication of endpoints—ergo half, endpoints, in the sense of one-sided so that the resulting equal subintervals abut closed without sharing: resulting in half--or-partial-endpoints and intervals closed 'ambi-clusively'...
This is Interval Theory, instead of (Point) Set Theory which cannot subdivide zero-width points...
But nevertheless, an amusing alternative, is, to refine line-set-theory for left-leaning and right-leaning intervals, wherefor integer division the splitting-points do the same lean... but however, this means that the divisor is not a point itself but the 'edge-of-a-point' separating an infinitesimal span from its closure point... But, which would mean we'd invented numbers that are 'not-a-number' ('NaN'), with a slew of new-type operational details... Or, the next-best choice is, the divisor is an infinitesimal-off but the absolutely-nearest, infinitesimal, to the divisor, so we're inventing a new number sense more logical than arithmetic: the A.N.I. number is, by comparison with any other infinitesimal, the-nearest...
#10e. 1/0 is undefined, NaN, but, like, a number—
e.g. 1/0 ≡ lim 1/Δ : Δ→0 (whereas infinity-itself is-not-limited)
e.g. 1/0 = 1/(−0) ergo sin(1/0) = −sin(1/(−0)) = 0, ∈ [−1,+1] (closed)
e.g. cos(1/0) = +1 (cf #3: e−1/z = +1 ≥ 0 along the x-axis)
e.g. 1/0 [all points] • 0 [point width] = 1/∞ [no zero widths left] = 0/0
e.g. 0! = lim (0)(−1)(−2)(−3)(−4)… = +1 yet 0•∞ = 0, (‘lim infinity' is even)
i.e. watch for implicit limits... (the 'lim∞' of the formal-math-world)...
#11. Moving within a bin: (listing vs. finding in infinite sets)
If a sample is tossed back into its original infinite source set, will it ever be found again... p = 1/∞ →0+.
E.g. if we have a list of all rationals on the interval (0,1) = {m/n : m<n, ∈N}, and toss a number, an irrational, into the middle of that interval-list, it'll never be found: there are infinitely many elements in-between before-and-aft, though the total-infinity is the same and removing all will have found it...
#12. Set subtraction: infinite sets are by rule, not by roster...
If 2 are added and 1 is removed, iteratedly infinitely, does it fill, or does it empty because each added is eventually removed... Differentially it fills... Does the outcome change by subprocessing, renaming, renumbering the entries or exits (reducing the probability of finding each; is infinity dependent on probability), And how-about by regrouping/reordering the summations e.g 1+(1−1)+(1−1)+(1−1)+... vs 1+1−(1−1)−(1−1)−(1−1)−....
In the Zenoic time-forced-decision, sense, there never is a number which is itself infinity, but, when stepped to the limit, beyond-infinity, infinity must've been achieved (and we symbolically represent this as '∞'), but its value is different in summation, depending on its correlation-dependency, as defined by the process; e.g. the stepping-rule '+2−1' sets a correlation-dependency such that the set is always increasing during the process, and, though stepping never ceases in the finite context, nevertheless the forced-decision forces stepping cessation, and, cessation of the dependency... (thus allowing that ∞ + 1 > ∞ iff/if-and-only-if the specified ∞ is autocorrelated as 'equal')... whereas, if the removal-of-one were specifically independent, then 'after' the fill, independent removal would clean-it-out...
#13. Knock-off paradoxes:
#13a. Infinite outer cosmos, finite inner cosmos—
where infinitesimal processes integrated over infinite-time-preceding and infinite-space-to-occur resulted in finitesimal particles and objectifications, e.g. the cosmos-we-know-inside and infinitely-many-preceding and-to-come-after outside but the travel-distance to the next exo-cosmos may exceed the existence time of either, whence there is no intercommunication in the present tense but for occupants who've preexisted in some-other long-long-long,-ago....
#13b. Particles—
are energy convolved on itself at the local speed-of-energy (i.e. light speed) no faster nor slower but for
exponentially-diminishing terms, and so steering-diverting to new course by absorbing energy (e.g. a photon)...
#13c. Time travel—
means being aware that your future-self is attempting to communicate back to you now vs. being aware of
cause-and-effect now... alternatively would need FTL-travel and cosmic synchronization to reach time-forward or
time-past cosmic situations at distances ahead or behind, within story-time, or, would have zero-mass (as being
developed in my Plan of Time screenplay)—or a cosmic-aether-mass-energy-resource but n.b.
gravity-potential-energy is already its extended mass-energy 'halo'... (or, wackier, e.g.
coming-from-behind-to-arrive-before as in Einsteinian Relativity).
#14. Stability, convergent differences:
#14a. series Σ∞1/n total = odds + (evens = 1/2 total) so odds = 1/2 total but odds - evens = (1/2) + (1/12) + (1/30) +... → ln 2, finite difference;
Unless, half-speed sampling of evens 'gains at half-speed' and comes-up with a 'smaller' infinity—the issue of infinite countability must include Zeno-speed-rates....
#14b. Riemann analytic continuation is pointwise convergent not uniformly, convergent....
#15. Projective infinity algebra:
Contemporary algebra/group theory tells us that integers do not have reciprocals but, modulo arithmetic/finite groups, of integers, may e.g. 1 ≅ 2×4 ≅ 3×5 mod 7, So, We consider integer reciprocals modulo primal infinity, 1 ≅ p×[-∞′/p] mod ∞′ where ∞′ = Π∞p + 1 the product of all primes p, plus one, which includes itself, and though not directly-finite, arithmetic, we can do symbolic-infinity algebra, (note the correlation-dependent use of ∞, it's not a problem of ∞-independency); This may be like but not-exactly the subject of projective geometry and its 1/0...
#16. Implicit definitions: (an interesting instance of proof)
An irrational arithmetic sequence, an = a×n n∈N≥0, is closed at n=0 but open as n→∞, But, wrapped around a unit circumference, implicitly-defines its opposite arithmetic sequence, (⌈a⌉-a)n , and, welded together at a0 = (⌈a⌉-a)0 the compound-definition is thus both 'ends' open, stepwise 'continuous' at n=0 but its stepwise 'derivative' has a discontinuity there...
CLOSING SIDEBAR COMMENT:
'Linguistics, type-one intelligence language, is sufficiently concise for defining exact sciences and mathematics.'
A premise discovery under the title,