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Kaph

Or, How I Discovered an Uncountable Infinity and Learned to Love Set Theory (Redux)



As the first letter of Kether, it is appropriate that the letter path of Kaph takes pride of place at the head of the center column connecting Kether and Tiphareth. As such it is also the path deemed to be the one "crossing the Abyss" upon the Lightning Path of descent, the zig-zag of the creative spark from the most High to his Beloved. That which bridges the ineffable heights of the Emanated and Creative worlds with that of Formation and Action. The connection between the absolute Unity represented by Kether and the Duality/Multiplicity which is the dominant aspect of a Created world. And does not AaShRIM (which means Twenty) equal 620 = Kether (KThR)?

As 20, Kaph is a visual representation of the Baphometric adage 2 = 0 and the sum of the first four Triangle Numbers. As the 11th letter it is representative of the doubled One, that which follows the complete cycle of a countable 1 (10 being the fullness of a cycle). In it's expansion of 210, Kaph once again is a visual glyph of the nature of the universal paradox. For it is only in the subtle play between the many and the none that the one can be apprehended.

Picture if you will (or at least try) a totality... a totality so pervasive, that literally nothing can be said about it. It isn't anything that might be described of it, for it has nothing to which it can be compared, it isn't nothing, it isn't something. Of it the seers of the Kabbalah only give it the appellation AIN (61), "Without", for nothing can be said or even conceptualized of Ain. However, as it is in the nature of the Jewish mind to kibbitz, for conversation sake they came up with an abstraction of that which cannot be abstracted from - AIN SVP (207), meaning "Without Limit", or more plainly - Infinity. (It should be noted here, that Ain Soph can equally mean "The Limit of Ain"). In Kabbalah, Ain Soph is equatable with God, though not really the one one would pray to - this is the unthinkable described in a thinkable, but uncomprehendable manner. An infinity so... infinite, that it comprises an infinitely larger totality than what could be quantified.. an infinity that is uncountable.

What is a young Kabbalist to do with an uncountable and uncomprehensible totality that masks an even deeper and utterly more profoundly unknowable... whatever? Like anyone else, they of course try to make sense of the senseless... they start to name.

Say you have a circle, a circle with an infinite circumference. Say that along this circumference is the sum totality of all that is, will and can be. Say you are standing on this circumference. You look one way, and the line goes on forever. You look the other way, and the line recedes to infinity in that direction as well. Nothing to distinguish one from the other. You are at a point... a point on an infinite line... but you could not say where your are on that line. You might have bisected infinity, but each half is still infinite. So, you make a mark where you are, and you move off around the circumference a little ways and you make another mark. Now you have two points.

... and like the dawning of some heretofore unknowable sun, you now have something you can point out. You have a segment. You have both a beginning and an end. Say you wanted to define "Blue". You go along the line until you get to a point where you decide that here, here is definitely blue. So you make a mark. You might think that is enough, that you have found the beginning of blue - so you have blue. But you are on a circumference, an infinite one, but still a circle - if you go far enough you will return from whence you came. So a start point is simply not enough to distinguish anything. For all you know blue just keeps tapering off all the way around. So you set off an a journey of discovery. You keep going along the line until you reach another point where you are quite sure that things are no longer blue. You make another mark. Now you can point out blue... it is distinguishable. Using just two points on a line you have taken the infinite and broken it. Now you have something that you can call "blue", something that is, on its own, apart and different from the indistinguishable totality.

Flush and giddy with your success, you start placing points willy-nilly - here is the beginning of horse and here is its end, here is tree, here is your Aunt Sally, here is love... you keep going till you can distinguish no more. Now you have a multitude. An infinite multidue of distinct things (for our line is by definition, infinite), of distinct "ones", separated on the line from one onther by two yawning gulfs of... nothing. Points on a line. Dimensionless points (for if they took up "space" then that space would be something too.). To get a one, a countable one, it takes two slices of nothing. Two zeros, 20. With our now countable one, this can be pictured as 0-1-0, or in my conceit as 210.

(Though this begs the question, if you have marked off and accounted for everything on that line.. where then are you? The answer to that is the only thing that comprises real Knowledge.)

((What is the sound of one Kaph clapping?))

In set theory, and especially in the work of Georg Cantor, the notion of countable and uncountable infinities (or an uncountably infinite set to use the proper lingo), is deemed intrinsic to the nature of reality (or at least mathematical reality). A set is deemed countable if there can be a one to one correspondence with a series of natural numbers - like one, two, three, four etc., or in essence; first, second, third, fourth and so on. An infinite set of this nature is said to have a cardinality (number of elements in the set) of "Aleph-Null". Written as ℵ‎０. (Despite the last name, and the usage of Hebrew symbols for his mathematical constructs, Cantor was apparently not Jewish... ). Cantor, through his "diagonal argument" demonstrated that Aleph-Null, while infinite, could not actually contain all numbers - there would always be numbers that are inexpressible in a infinitely countable set no matter how.. infinite it is. That is: a countably infinite set, was smaller (had a lower cardinality) than an uncountably infinite set. This makes good common sense, until you realize you are talking about the notion of an infinity that is objectively larger than another type of infinity... in the context of Cantor's thought experiment here, the uncountable set is the set of Real Numbers.. this includes all numbers, not just rational numbers or ordinal numbers. So it includes things like pi and the square root of two and whatnot. The cardinality (number of elements) of the set of Real Numbers is called the "Cardinality of the Continuum".

In set theory as well, there is a notion called the Power Set. (Metal much \m/?) Basically this is what you get when you take all of the elements of a set and describe the number of combinations of those elements you can have - the set of all the subsets of a given set. For example: the cardinality (number of elements) of the Power Set of a set containing three elements (for instance red, blue and green) is 8. You can have red, blue, or green, red and blue, red and green, blue and green, red and blue and green and last and entirely least you can have noting at all. Nothing, in this context is known as the "Empty Set"... all sets, by definition, always contain the empty set. So that makes 8 distinct subsets from an original set containing three elements. Pretty apparently, a Power Set is always larger (has a greater cardinality) than the set from which it was derived. In fact it is always larger in the same manner - by 2 raised to the power of the cardinality of the original set. (23 = 8).

Then Cantor started to wonder about the Power set of a countably infinite set (Aleph-Null). By the above mentioned formula, the cardinality of that Power Set would be 2 raised to the power of Aleph-Null: 2ℵ0. (2-Aleph(1)-Null(0), 2-1-0... 210... get it? Get it?) Cantor got it into his head that this value would equal the Cardinality of the Continuum. He called this Aleph-One. 2ℵ0 = ℵ‎¹. Well done Georg! We have arrived at what I consider an excellent depiction of the nature of the relationship of number and letter (all numbers are infinite sez Uncle Al.. and the first number-letter is Aleph).

The Universe is infinite and at it's heart (LB 32) absolutely unquantifiable. Yet the Universe is also patently present, immediate and countable. What we experience is a dance - a dance of that which can not be conceptualized becoming that which can - named and identifiable, a one that is defined by two bits of nothingness which in turn can only be defined by a countable one. All we see, all we experience is that bridge that carries us across the abyss....

In its final form Kaph is 500, The Indwelling of God and The Scapegoat banished to the Wilderness. The jealous Demiurge and the Throne of Knowledge. And are the four not again 1000? The Aleph once more? Oh Prince! Oh Holy King! Are you not the figure of Man? Defiled.... she dreams...

The Ur number of Kaph indicates the beginning of a new cycle of totality. The Ur number of Kaph (final) is 528 and as such symbolizes the Baphometric Tree as a whole, holding the 32 paths within itself as it begins anew. In each and every instance Kaph is the representative letter of the hidden sephiroth of Da'ath, that which makes the leap across the abyss of unreality, the unity separated only to advance the totality in union. It is that which Has Come To Be, the Dream that offers its Palm upturned in welcome.... remember, a circle has no end...