August 3, 2010

The Infinite Continues: A Look at Paradox's and Problems From The Previous Post

Well, I am fairly sure that the topic of the infinite isn't one of my strongest subjects, but this has still been and continues to be an interesting learning experience. I've learned a fair bit since my last post by reading up on Supertasks, and taking the advice of a commenter to read The Infinite Book: A Short Guide to the Boundless, Timeless, and Endless by John Barrow. I look forward to reviewing the book and blogging about it in the future, but today I'm just going to go over a few things that have been pointed out and talk about a few interesting paradox's involving the infinite.

I'll start with just clearing up a couple things. 

1. In the last post, before I edited it, I made false causal relationship between density and mass that really isn't there and I had to adjust it, as it was a mistake. Density is somethings mass divided by how much volume it has, while mass can be a bit more complicated, but is commonly looked at as how much something weights. The mistake was obviously pointed out by the example that a car could crushed and it would have the same mass, but the density would have changed.

2.This is just a matter of clarification.My thinking is so far still in line with Keith Mayes who is of the opinion that, "Infinity exists only as a means of description, such as found in mathematics for example, or any other thing that exists only in the abstract. I do not believe that it has any real existence in the universe such as infinite mass or infinite size. The word 'infinity' is a descriptive term and not a measure of size, and I therefore do not see how it can be applied to anything 'real', as real things can be measured." This is, like any view I hold, is still open for further evidence to be weighted, so if you think there is something I should look at let me know.

3. The idea of distinction without a difference really helps to understand larger and larger sets of infinite's. There isn't a difference because when you hit the infinite you have hit the upwards limit, there is no going any further. Yet there can be infinite series that involves larger numbers, and therefore larger a larger infinite (A better way to think about it is that it would approach the infinite more quickly). This is where it becomes hard to understand because two sets can be at the highest possible limit, but one can still greater than the other. There seems to be a logical inconsistency that's contained within that idea, but that's how it works.

It's that type of inconsistency created by the idea of the infinite, that makes many a paradox. Take for instance something infinite in size and infinitely full, yet still has room for you. That is the summation of what makes up the first paradox created by the infinite that I will talk about today.

The Grand Hotel Paradox -  This problem comes from David Hilbert, and is a neat and counter-intuitive problem created by the use of infinite. The set up is described exactly as it was above, there is a hotel with an infinite number of rooms, but each of those rooms is full. So if the hotel had a number of new guests arrive looking for a room it would at first seem that, there would be no way for the hotel to accommodate the new guests due to it being full. This is where it gets slightly mathematical and interesting. To make room for the new guests all the desk clerk has to do it make an equation for room changes, like every new guest is put in room two. But then what happens to the guest that was in room two before? The guest originally in room two would be put in room four, the guest originally in room four would move to room six which would cause the guest from six to be moved to room eight and that series would continue infinitely, thus making room for any new guest that arrived. This process could then be repeated with any new guest that arrived, so the hotel would be infinitely full, yet still have room for everyone. William Lane Craig, gets a lot further into this idea and uses it to say that infinite's are impossible and therefore the universe had a first cause, which I don't agree with but find interesting.

Zeno, not to be confused with Xenu
Zeno's Achilles and the Tortoise - This is a classic paradox that is brought up from time to time and really never goes away. Like the previous paradox this one involves only a very simple idea that when applied creates a counter-intuitive situation. Zeno's idea was that motion can be broken down into sections seemingly without end. For instance a Tortoise starts 100 meters ahead of Achilles and for Achilles to catch the Tortoise he first has to go half the distance to the Tortoise then he has to go two thirds the total distance then he has to go three fourths the total distance....This continues so that Achilles gets fractionally and fractionally closer to the Tortoise, but he never reaches it. Simply there are an infinite number of points for Achilles to travel in a finite amount of time (This is also referred to as a Supertask).

Zeno relies on either time or space being divided up into infinite pieces. A question arises from this, is it possible in a finite amount of time to accomplish an infinite amount of tasks? That question could be answered or the problem can be dealt with by there being a bottom limit of divisibility which breaks the cycle. A few people have posited a Planck Length as that bottom limit, but I have yet to see that really flushed out in detail.

Zeno's Arrow - Like the race above this paradox involves breaking up motion into a series of points and looking at it from that perspective. Instead of breaking up and worrying about the distance that the arrow has to travel Zeno instead breaks up motion into a series of time. He goes so far in his break up that he looks at the arrow flying through the air in instants, the way a camera would take a picture of a moment in time. As it can  be seen in a picture the arrow isn't moving, at any precise instant in time the arrow is at rest. Since the arrow is at rest at any instant you look at it, it never moves and makes motion impossible.

This last paradox has been solved, and if you'd like to learn more about it or other paradoxes look for Michael Clark's Paradoxes from A to Z.

Well, I hope I have instilled a greater appreciation of the depth of infinity and the problems/disagreements it creates.

Thanks for reading,
-the moral skeptic


  1. Have you read the book GEB (Gödel, Escher, Bach - An Eternal Golden Braid)? Chances are that you have judging from your blog, but if you haven't, check it out! It's the greatest book ever written.ödel,_Escher,_Bach

  2. Skeptic,

    Have you considered publishing the contents of this blog? I know it is easier for a talented writer / thinker to become a rock start than it is to find a publisher or agent. However, if you are not in it for the money, you could try a print on demand shop or even a pure self-publishing shop. If you put no marketing into it, then maybe only I would buy a copy, and you may make 8.95 or something, which you could frame. I must tell you, if you publish your words, I will buy a copy.

    I did not have time to read this full thread. I bookmarked it. I did see that your discussion of infinity dovetails into Boolean set theory. As I recall, aleph null is the label for an infinite set and C is the label for infinite set of aleph nulls. If we label everything, it is much easier to comprehend. You may have, I will let myself know when I finish reading.