Dear Thomas; Cause of Causality; In the Beginning

[yppop]

– continued –

Cantor hypothesized a hierarchy of alephs based on power sets.

Every pertinent non-empty set generates a power set that is made up of all combinations of the elements in the pertinent set. A set of cardinality-3, for example, one containing three integers, {1,2,3} generates a power set of cardinality-8 consisting of sub-sets: one empty set, {0}, with cardinality 0; three sets {1},{2}, and {3}of cardinality 1; three sets {1,2}, {1,3}, {2,3} of cardinality-2; and one set {1,2,3} of cardinality-3. The power set is always greater than the set that generates it.

The relationship that Cantor derived for generating power sets is:
“2^(cardinality of the pertinent set) = cardinality of the power set”, and in the example we have “2^3 = 8”

The 2 in the relationship is a simplification of the relationship by working with base-2 instead of base-10 number system. Base-2 needs only two numerals 0 and 1. All ordinals can be written in this binary base.

He then proved that the power set of aleph(0) is aleph(1) using
2^aleph(0) = aleph(1) , which is called the continuum hypothesis.

Some of the theorems of transfinite math are:

aleph(0) + n = aleph(0)
aleph(0) * n = aleph(0)
aleph(0) * aleph(0) = aleph(0)^2 = aleph(0)

The last theorem can be generalized to: “aleph(0)^n = aleph(0)” and for aleph(0), at least, exponentiation does not change the cardinality

Cantor proved by a method called diagonalization that there are no more points on an a square area projected from a line interval than there are on the interval; and there are no more points in a volume of a cube projected from a square then there are points on a line interval from which the square and hence the cube are projected. Increasing the dimension of space does not increase the number of points (cardinality) of the greater dimensioned object. This is a geometrical analog of the math shown above,

You of course are free to make your point by successively raising infinity to an infinite power, but I don’t believe that you can make a claim that it is rigorous math. The point that God is beyond comprehension is obvious and I make no contention that you haven’t made that point.

The interesting point about your post #18 is that, in a way you are pictorially describing the first three levels of Cantor’s infinities. The points, which must be rational since they can be counted (or establish a “state” that can be counted), describe aleph(0). Since each rational number on the real number line is immersed in an infinity of irrational numbers, your infinity of circles represents aleph(1); and your colors, shapes, and rotations are analagous to aleph(2), according to Cantor, the set of functions on a real line.

Your message is that numbers get unimaginably huge but are dwarfed by God. Cantor imagined something surpassing his infinite string of infinities; he called it the Absolute that he believed to be God.

Yppop

 

[James_S_Saint]

James,
I am seriously lagging behind in this discussion; you have me at a distinct disadvantage because: (1) you know what you mean when you explain it; I have to expend time and energy to extract the meaning, sometimes incorrectly, from your explanation; (2) you are a far more prolific writer (averaging 14 posts per day on this forum) than me;

Not to mention the other 6-10 forums that I might be on at any one time.

And I apologize for choosing difficult wording. It is probably as difficult for me to find the “right words” as it is for you to read them.

(3) in keeping with Friday 13 I inadvertently deleted a long post I spent considerable time to create (I do that often),

Oh man. I can relate all too well with that one. You try to get a clear picture in your mind of what is being presented. You see a point to make concerning it. You then concentrate in an effort to find the right words and order them into reasonable sentences and those sentences into coherent rationale toward a conclusive message. You check the spelling and reread the post to see if better wording might be revealed. And finally you see that you have accomplished your task.

Then terror strikes as you suddenly realize that you just did something that irrevocably deleted everything you had just done. Hours of concentrated effort gone in an instant.

Of course if you look at your watch, you find that it wasn’t really hours, but certainly felt like it. Oh but such strain in finding the right words. The thought of having to dredge them up again. How unlikely to be able to reconstruct the entire thing as well as you had done the first time. Such strain of mere concentration “shouldn’t” exist, but does anyway. So, “Maybe tomorrow”.

Been there, done that, got the gray hair.

If it helps any, try, as I do, to think of this exchange more as a couple of old guys wondering out onto the back porch for a while to perchance meet to discuss an old issue but always free to wonder back inside at any time. No pressure. Not that many people read these kind of threads anyway

And please don’t take my responses as defense in the combative or competitive sense, but rather in the explicative sense as in defending a thesis rather than an ego. I enjoy someone scrutinizing the details and helping me remove any potential stone unturned. I am not a politician, but a logician (although perhaps equally poor at both).

---

I think you might still be missing an important point, or perhaps I am. Cantor was defining a numbering system for the sake of mathematics. I had devised a similar one some time ago for a similar purpose. But there is a difference between “all potential numbers” and “all potential points of existence”.

As you pointed out, along an infinite line we can presume the entire set of rational and/or irrational numbers. That line can represent that set. And if we have another such line, we have not added any new numbers. The second line will have the same numbers as the first = aleph[0,1].

But by having 2 such lines, we have twice the number of items which does not leave us with “2 * inf[0] = inf[0]”. That expression reveals that we have added no new numbers but it falsely implies that we have no more *items *when in fact, we do have twice as many items.

The items are what I am counting, not the number of numbers involved.

This is a different kind of concern as I am talking about the number of sets of a given infinite set type. This can be describe by Cantor’s aleph(2) until I add more dimensions, which leads eventually to aleph(inf).

And what Cantor thought of as God, to me was merely one angel.

But as you say, this is somewhat of a trivial issue in that any of the alephs are WAY beyond imagination.

My concern is whether there is actually any error in logic or has anything been left out of consideration before the conclusions? To me “holiness” means that;

a) all things have been considered.
b) all that has been considered is coherent
c) the total is relevant

…taken a another break…

 

[MindOverMatter]

Oh man. I can relate all too well with that one. You try to get a clear picture in your mind of what is being presented. You see a point to make concerning it. You then concentrate in an effort to find the right words and order them into reasonable sentences and those sentences into coherent rationale toward a conclusive message. You check the spelling and reread the post to see if better wording might be revealed. And finally you see that you have accomplished your task.

Then terror strikes as you suddenly realize that you just did something that irrevocably deleted everything you had just done. Hours of concentrated effort gone in an instant.

You describe my pain very well. Some of what i consider to be the greatest posts i have ever written have been lost to the hellstorms of digital chaos.

 

[James_S_Saint]

Hey yppop,

It occurred to me, how would you measure a “degree of freedom”?

Normally we think in somewhat primitive terms of “either you are free or your aren’t”. But such is hardly the real case. In a sense, everything is free IF the impetus is strong enough to push everything out of its way. Thus was born the idea that as the freedom/opportunity rises, the need for impetus reduces. But I thought it would be better if I could find a reasonable “freedom gauge” to use in mathematics of the thesis.

Any thoughts?

 

[MindOverMatter]

Hey yppop,

It occurred to me, how would you measure a “degree of freedom”?

Normally we think in somewhat primitive terms of “either you are free or your aren’t”. But such is hardly the real case. In a sense, everything is free IF the impetus is strong enough to push everything out of its way. Thus was born the idea that as the freedom/opportunity rises, the need for impetus reduces. But I thought it would be better if I could find a reasonable “freedom gauge” to use in mathematics of the thesis.

Any thoughts?

Are not free choices like random events? How can they be determined within a mathematical frame work?

 

[James_S_Saint]

Are not free choices like random events? How can they be determined within a mathematical frame work?

Well that isn’t how I define “random events”, but yeah is a good question. How do you measure “free choice” other than merely counting specific known options?

Got any suggestions?