Cantorian Diagonal

Discussion of the nature of Ultimate Reality and the path to Enlightenment.
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windhawk
Posts: 83
Joined: Thu Aug 23, 2007 4:47 am
Location: Michigan

Cantorian Diagonal

Post by windhawk »

Most of us here have an appreciation of the infinite as it relates to our personal sense of meaning, the lack thereof, the necessity of constructing such meaning within ourselves, the lack of objective meaning, etc. You understand what I'm saying, everybody has, or more importantly, should have some sort of relation to the infinite, a sense of God, your own personal Jesus, a nihilistic sense of doom. You name it.

Mathematics has its own sense of infinite, which may just give you pause.

Georg Cantor is rightly famous within mathematics for his application of set theory as it applies to infinite sets. Think a set of all numbers, one to ...

Ya' gotta love those three dots. They go out forever.

In any event, what Cantor realized was that once he established an infinite set 1, 2, 3, ... infinity, he could apply a simple function to an infinite set, and produce an uncountable set larger than the set of all numbers (bigger than infinity).

How does this work? Think again of the set

1,2,3...
2,3,4... (This set contains all countable numbers. Think Really Big.)

If I apply the operation of adding 1 to the first number, and then adding 1 to the second number of the second number (the 3,in line two), and I do the same, e.g. that is adding one to each successive number on each successive "line," I will create a number that is different from the first number, different from the second, third and so on. It's simply is not in the original, as it differs by one from every number. This is Cantor's application of function to an infinite set.

The result of applying a simple function to an infinite list is that I create a number that is different from every number by 1 (or whatever).

Now, think of a half byte computer program where the bits are 0101, this is equal to the number 5 base ten. What if we write a computer program of a half byte in length as such: 1010. In base ten, 1010 is equal to the number 10. Clearly, if we examine every extant program, we will be able to translate the binary code (number) of the program into a base 10 number. In other words, every program that has ever been written is simply a number.

Some numbers are "good code" in that they produce a working program. Think Windows 7, it is in fact simply a number, although a very large one. Some numbers are "bad code," that is, they don't produce a workable algorithm.

Here comes the fun. Every program ever written is reducible to a number that exists on the number line, but there are necessarily numbers (programs) that exist only through the application of an infinite number of functions (n + 1), (n + 2) to an infinite list of whole numbers. You can write it, but you can't generate it.

God, has thus, hidden an infinite number of programs that solve an infinite number of problems in such a manner that we must experience a solution that fixes the problem to hand, as the problem represents itself.

Wow, we thought simple existence was odd.
windhawk
Posts: 83
Joined: Thu Aug 23, 2007 4:47 am
Location: Michigan

Re: Cantorian Diagonal

Post by windhawk »

More to this.

Free will not only exists, it is demanded by the fact of our own existence. We are literally thrust into a reality in which choice is paramount. It is our essence. For some, a curse. But transcendence is of not only the number line, but of ourselves.

Choices:

I was riding down the highway as the Sun was going down,
and I'm sick and tired of living in your crazy mixed up town.
You keep your fucking Cadillac, and penthouse in the sky,
While I keep my wheels a humming as the miles go rolling by.

North, South, East and West it's all the same to me,
As busted steel and twisted wheels remain my one ambition.

--John Prine
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ardy
Posts: 341
Joined: Tue Oct 29, 2013 6:44 am

Re: Cantorian Diagonal

Post by ardy »

Love the idea of all programs being a number set. Can't get the connection to the infinite and numbers. Are you referring to the number of combinations that our life [and all life] and decisions can take?

Or are you referring to something else?

This is closer to my view.

The Ocean by Pablo Naruda [2nd verse]

Not the final breaker, heavy with brine,
that thunders onshore, and creates
the silence of sand, that encircles the world,
but the inner spaces of force,
the naked power of the waters,
the immoveable solitude, brimming with lives.
It is Time perhaps, or the vessel filled
with all motion, pure Oneness,
that death cannot touch, the visceral green
of consuming totality.
windhawk
Posts: 83
Joined: Thu Aug 23, 2007 4:47 am
Location: Michigan

Re: Cantorian Diagonal

Post by windhawk »

What a stunningly beautiful poem that grasps the meaning of a "purely mathematical" idea of transcendence.

The concept of infinity in the argument of every writable program being necessarily reducible to a number, is that there exist numbers (programs), that lie outside of an infinite list. An infinite list of infinite lists is required.

From wiki: "The diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers."

By the application of a simple function to a set, an infinite series of sets (uncountable sets) of sets (sets of sets) become accessible, but not countable.

If we were to try to reduce consciousness to an equivalent computer program, or a "number," we find that the idea the human mind is a program (so to speak) is untenable. Thought is unbounded, and inexpressible by a computer program.

Free will reigns within the context of an inexpressible universe of posibility.
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