Thursday, March 08, 2007

Aleph-Plexity & Transinfinity

י"ח באדר תשס"ז

In my previous post I noted the connection of the Hebrew letter ayin ע with both infinity and the sefirah Hod (the sefirah denoting the Temple). Mathematically, infinity is usually represented by an infinity symbol (∞) , the lemniscate. Infinity, in complex analysis and calculus, denotes an unbounded limit. On unbounded mathematical limits (wiki):

In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either gets "close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements, as their index increases indefinitely. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity.

Another type of infinity involves transfinite numbers and mathematical set theory. This branch of mathematical science is characterized by the aleph-numbers. On transfinite numbers (wiki):

Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite.

On aleph-numbers (wiki):

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line, or an extremal point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

Taking all this together, it can be seen that the Hebrew letter ayin, connected to both absolute infinity (∞) and the Holy Eight (a "place" number among the "countable" "sets" of the sefirot), represents a "transfinite" synthesis of absolute infinity and quasi-finite, quasi-infinite reality (aleph numbers), where Hod might be likened to an aleph-number set.

Certainly, I'm no mathematician. Consequently, if any mathematicians read this post and see any error in my logic or can offer any further insight on the matter, please comment.

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