The Infinite. Part 9. Abraham, Omega, and Doing Things In Order.

[Here is part 8. All parts may be found here.]

What does it mean for one thing to be less than another? We are natural “orderers” aren’t we? We have ordering intuitions about size, strength, speed, beauty, importance, riches, standard of living, loudness, height, and other stuff. Some of this is reasonably quantifiable, some not so much. As a schoolboy, I learned the hard truth from a sixth grade seat mate, Susan Ortiz: the girls ranked guys on a handsome scale (that wasn’t the only scale, but it was one). Are some people (or animals) more “intelligent” that others? As Mormons we are familiar with this sort of ordering by Deity:

6 And the Lord said unto me: Now, Abraham, these two facts exist, behold thine eyes see it; it is given unto thee to know the times of reckoning, and the set time, yea, the set time of the earth upon which thou standest, and the set time of the greater light which is set to rule the day, and the set time of the lesser light which is set to rule the night.

The Children of Abraham and Sarah

7 Now the set time of the lesser light is a longer time as to its reckoning than the reckoning of the time of the earth upon which thou standest.

8 And where these two facts exist, there shall be another fact above them, that is, there shall be another planet whose reckoning of time shall be longer still . . .

17 Now, if there be two things, one above the other, and the moon be above the earth, then it may be that a planet or a star may exist above it; and there is nothing that the Lord thy God shall take in his heart to do but what he will do it.

18 Howbeit that he made the greater star; as, also, if there be two spirits, and one shall be more intelligent than the other, yet these two spirits, notwithstanding one is more intelligent than the other, have no beginning; they existed before, they shall have no end, they shall exist after, for they are gnolaum, or eternal.

19 And the Lord said unto me: These two facts do exist, that there are two spirits, one being more intelligent than the other; there shall be another more intelligent than they; I am the Lord thy God, I am more intelligent than they all.

Here, celestial ordering gives a kind of earth-centric reference to “less than” and spirits get some kind of ordering via their “intelligence.” Do the more intelligent folk get the perks? Is it really the case that this is a service opportunity? (smiley face here) Our fearless blogleader once labeled this abrahamic induction the Comparative Principle, Sam Brown styles it as overlaying his interpretation of a Mormon chain of being structure in his In Heaven as it is on Earth, see here for example (p.14). I want to come back to this but to do it justice we need a little machinery to make sense of it all.

So I now drag you back into Cantor’s paradise. A most useful and elementary abstraction is the idea of a collection of objects, grouped by whatever rule we wish. Last time we saw that there may be pitfalls lurking here, so we will be careful and gradually put some restrictions in place. But for now, let’s jump into the concept of “ordinal numbers.”

You are familiar with the idea. You use it every day with language like “first” “second” “5th” “ducks in a row” and so on. But now we must be a little formal about it. Don’t run an hide, it’s not that bad. I’m taking an approach partly suggested by John von Neumann (1903-1957), an amazing genius.

I’ll start by saying what it means for a collection to be “well-ordered.” We’re assuming that we can discern some order among the members of our collection in the first place and being “well-ordered” merely means that if we extract any subgrouping of items from our collection, it will have a “smallest” or least member in our perceived ordering. Not all orderings have this property but it is usually assumed that any collection, no matter what it consists of, can be supplied with an ordering that imposes the property of being well-ordered. N with its natural ordering is the paradigm of well-ordered sets[1].

Now a bit of notation. This will make life considerably easier. First, we use ∅ to stand for the collection with nothing in it. A bit of a place holder, like zero if you will. ∅ = { }. The empty warehouse, nobody home. Next, we use ∈ to stand for “is in”. Like 4 ∈ {5,6,4,10}. And ∉ to mean “is not in.” 70 ∈ N, but “raisin” ∉ N. One more abbreviation, the “union” of two collections of things gives the collection of things which include all stuff in either collection. {7,8} ∪ {8, 0, 45, 3} is {0, 3, 7, 8, 45} (8 is not listed twice because we’re just interested in content, not twins or triplets or whatever – you’re either in, or not).

∈ can be used to supply an ordering sometimes. We can arbitrarily say that A < B if it happens that A ∈ B. A collection X of sets is said to be transitive if whenever Y ∈ X and Z ∈ Y then Z ∈ X.

Here’s an example of a transitive set:

{∅ , {∅}, {∅, {∅}}, {∅ , {∅}, {∅, {∅}}}}. In fact we will call a set an ordinal when it is both well-ordered and transitive. It’s possible to show in a few pages of work that an ordinal is the set of all ordinals that precede it. This is clearly true for our example, {∅ , {∅}, {∅, {∅}}, {∅ , {∅}, {∅, {∅}}}}.

My first observation is this: if we make the identification

0 = ∅

1 = {∅}

2 = {∅, {∅}}

3 = {∅ , {∅}, {∅, {∅}}}

and so on,

we can say that the counting numbers are ordinals, a happy circumstance since they are the models we use for ordering things in everyday life.

Now a fact that we can easily prove, but won’t, is this: If C is an ordinal, then C ∪ {C} is the least ordinal greater than C. In this case it is natural to write C + 1 for C ∪ {C}. And again, if S is any collection of ordinals, then there is a least ordinal in S. Hence, the ordinals are well-ordered.

Finally, this very interesting definition. α is a successor if there is an ordinal β so that α = β + 1. γ is called a limit ordinal if it is not 0 and not a successor.

And, limit ordinals exist. The *smallest* limit ordinal is just N with the identification given above. With the identification, we can identify N as ω. That’s God. Maybe (he does claim the title after all). The Thomists were both pleased and horrified. They know what Cantor has up his sleeve. And that was, ω + 1. Sitting on the top of a topless throne as some antebellum preachers liked to have it (for one of a million, see Evangelical Magazine and Gospel Advocate, 5 (1834):396) is no guarantee that someone is not up there above you. And then there is ω₁. We are not done with either theology or ordinals.

(Part ten is here.)
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[1] {79, 40, 91, 10²⁰, 30, 10,000} is a subset of N and clearly has a least member (30). There are orderings which are not well-ordered and they can be quite useful. The set of all fractions with their natural order are not well-ordered, but we can reorder them so that they become well-ordered (not very interesting in practice). The ability to supply an ordering to any collection in such a way that it well-ordered, is a statement equivalent to the Axiom of Choice.