Like you, I am a firm believer in the old “absolute continuum” idea championed since the time of the ancient mathematicians, and argued for, quite compellingly, I believe, by many mathematicians and philosophers, at least since the time of Charles Sanders Peirce, if not much before. I have always found it almost incomprehensible that while Cantor took on the enormous task of attempting to legitimate the transfinites to the mathematical community, he staunchly opposed the equal legitimacy of the infintesimals, which are, after all, merely their reciprocals!

One very simple agument I sometimes use to attempt to convince those not yet initiated into mathematical Necromancy that the infintesimals do indeed actualy exist is as follows. I will be curious to know whether any of you readers, and especially you, Rudy, believe this example actually does make it foolish to deny their existence.

Step one: Take the unit interval and mark its left side ‘0″ and its right side “1” as usual.

Step two: Make the unit line twice as long; and continue to mark its left side “0,” but now mark its right side “1/2.”

Step three: Double the line’s length again, keeping the “0” at the right side, but now placing “1/4” on the left side.

Step four: Continue doing this doubling of the line’s length, and halving of portion displayed, over and over.

Thus after “n” recursions one will have a line that is 2^n long, but which covers only the 0 to 1/[2^n] portion of the whole original line.

[In essence, then, one can think of this procedure as simply one of “magnifying” the original line by 4 with each iteration, since one is always doubling the length and halving the value of the left limit of what is displayed.]

Now we come to the point of all this!! Imagine now that “n goes to infinity.”

Our line thus becomes 2^omega units long, i.e. vastly longer than the entire Real line! But the marking that would be rightly placed out at 2^omega point on the line would be 1/[2^omega]!

We would thus be left looking at something which looks just like the Real line we all know so well, but which is composed only of numbers, — or better locations — which are LESS than 1/[2^omega] i.e. infintesimals tiny even by the tiny standards of ordinary infintesimals!

Nor would there be ANY so-called Real numbers on the entire, infinitely long line we now have drawn, since for something to be a Real number it must have at least one digit in its base 10 expression which is non-0, and as soon as that digit shows up it becomes only a very small FINITE quantity, and as such is far too large to appear anywhere on our line containing only INFINTESIMAL quantities!

To my mind this little exercise demonstrates nicely that even within the Real number “contiuum,” as it is generally conceived, there exist an uncountable number of infintesimal quantities between each and every Real number, — although tht stronger claim has not here been proved, only the case for the origin has. This reality of the infintesimal plenum seems even more amazing when we consider that even the always still finite Real numbers themselves can only be designated finitely, and thus almost universally merely indicate a linearly tiny, but full cardinality preserving, uncountable set of possible distinct extensions, and that even if their finite designation is carried out to the trillionth, or googelplex-eth, digit!

Anyway, I will be interested in seeing whether readers of this agree with me that this recursive “proof” seals the deal for the reality of the infintesimals, and hardly merely “under non-standard models,” but under the more emphatic standards of common sense!!

]]>https://www.amazon.com/Infinity-Mind-Philosophy-Infinite-Princeton-ebook/dp/B07SSTRVGY

I see “This title is not currently available for purchase.”

Perhaps the ebook is not sold in Europe?

I just don’t think that the absolute continuum gets past countable; it just gets into paradox. So the surreals, to the extent that they’re definite, are countable; past that is the fuzz inevitable in a continuum.

I do like that in a surreal universe we are infinite beings. But by the same logic, we are also infinitesimal beings.

]]>down with heaven and hell

and all the religious fuss

infinity pleased our parents

one inch looks good to us

He meant it as mockery of modernity; but I take it as praise. One inch? How modest of us, compared to our diva parents.

I say that a universe countably infinite in extent but finite in depth fulfills anyone’s need for mystery. Such a universe is really a multiverse, with this branch repeated infinitely often, and infinite variants of this branch repeated infinitely often. You call it waste; I call it redundancy, a safety factor. Uniqueness is over-rated.

It is also a completeness principle; all variants of you are realized in the multiverse. What’s more, those variants are all quantum-entangled at the Big Bang. So though you are inwardly finite at the Planck scale, it is through that inward finitude that you connect to infinity without.

A googolplex lightyears past Polaris, I married Naomi; a googolplex lightyears past Sirius, I married Shelley; a googolplex lightyears past Algol, I married Margo; and here I married Sherri. So Team Me has all bases covered; and looking within, I see that we all have the same heart.

As for transfinity; that is a failed attempt to impose binary logic on the infinite, which is inherently paradoxical. So I say infinity is its own power set; Cantor’s antidiagonal merely finds a buzz-bit. Omega is countable but paradoxical.

Even more radical than the Reflection Principle is the Lowenheim-Skolem Theorem, which says that for any countable language, and any model of set theory, the set of all truths about the model in the language has a countable model. So countable infinity replicates all transfinites.

I hear that when Cantor heard of Russell’s Paradox, he rejoiced that his monster, the set-of-all-sets, does not exist. I say, it’s nonexistent to binary mind. Admit paradox, and the set-of-all-sets is merely the truth constant:

for all x, (x is in the set-of-all-sets) equals True.

Russell’s set-of-non-self-containing-sets has the equation:

for all x, (x is in R) equals (x is not in x).

But then

(R is in R) equals (R is not in R).

A paradox! This freaked out Russell, Cantor, and many others. I say; calm down, it’s just tertium datur. Logic contains a grey area, a boundary value. Is dawn night or day?

In paradox logic, Russell’s Paradox is no threat; and Cantor’s antidiagonalization just detects boundary points. 0.111111… = 1.00000…; this number has two opposite digits at every place; so it could fit as an antidiagonal anywhere. So paradox logic allows a countable – though paradoxical – continuum.

down with aleph zillion

and all the transfinite spree

omega terrified Cantor

countable’s fine for me