I is for IDIC

[This is the 9th of my April A-Z Challenge series of posts on Symbols, Glyphs, and Sigils. Each day I'll try to include some material that old-school role-playing gamers will find useful, but I can't guarantee that there won't also just be a few posts filled with weirdness for the sake of weirdness....]

Known by the most unbridled denizens of geekdom such as myself, IDIC is an acronym from Star Trek that stands for "Infinite Diversity in Infinite Combinations."  It's a core principle of the logical philosophy of the Vulcans, and it also comes in nifty pendant form:

In reality, the IDIC pendants seem to have been a crass marketing ploy by Trek creator Gene Roddenberry... so crass that even Leonard Nimoy needed lots of convincing to wear it on the show.  But the idea of it really resonated for me back in the day, so I'll keep my fond memories of it anyway.  :-)

Many fans embraced "infinite diversity" because of its resonance with tolerance for alternate ways of living.  I always thought that trying to fold your mind around "infinite combinations" gets quite close to the sacramental core of the Glass Bead Game (see collected quotes here).  It's also reminiscent of the ideas in my letter E post of a few days ago.

One great way to stretch your mental muscles with IDIC is to think about the various degrees of infinity that mathematicians have come up with.  Georg Cantor got the ball rolling in the late 1800s with transfinite numbers, but that was only the start.  Want to take a ride?

• First up is the idea of a "countable infinity."  You know:  1, 2, 3, 4, ...  The "dot dot dot" just means to keep on going forever.  Cantor proved that if you can write down an algorithm that accounts for all the members of a set by counting (even if it goes on forever), then that set has essentially the same number of members as any other set that satisfies this criterion.  This can be a bit counter-intuitive, since it means the set of all even (or odd) integers is the same size as the set of ALL integers.  The set of all rational numbers (integer divided by an integer) is also countable, and thus the same "size" as the set of all integers, too.  This degree of infinity is called Aleph-Zero.
• Are there sets bigger than Aleph-Zero?  Yes!  The set of all real numbers (including all those pesky irrational numbers like pi and the square root of 2) contains an "uncountably" large number of members.  Unlike, say, the set of all integers, there is no way to say what the "next" real number is that follows any given real number. There are always infinitely more reals squeezed in between any two that you can write down.  This degree of infinity is called (by some) Aleph-One.
• Are there sets bigger than Aleph-One?  You bet!  As big as it is, something even bigger is the set of all possible SUBSETS of Aleph-One.  In other words, the set of all possible "infinite combinations" of the members of Aleph-One.  There's Spock and his pendant again!  This degree of infinity is called Aleph-Two.
• You can keep on going and going, with Aleph-(N+1) being the set of all possible combinations of the members of Aleph-N.  You can even substitute another "Aleph number" in for N.  But some mathematicians take the next leap to be the following: Consider an infinite set so big that it's the SAME size as the set of all of its possible subsets.  It's so expansive that it is its own subset.  Your mind has to really stretch to see how that could be possible.  Reading the mathematics literature starts to get hard to follow from here on, but I think these "Inaccessible Cardinal numbers" are often given the symbol theta.  (And there's the Ecology symbol again?)
  
• The mathematicians go on, with hyper-inaccessible cardinals, indescribable cardinals, ineffable cardinals, and so on...  I can't keep them straight!

Anyway, I highly recommend Rudy Rucker's classic book Infinity and the Mind for those who want to dig deeper -- or climb higher up that infinitely tall mountain!