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

A look into the Infinite: How it works Conceptually and In Reality

Well I’m both a little surprised and impressed with the reception my last post received and am glad to continue with a topic with many similarities with the last one. This post will look at the concept of the infinite by looking at a couple of paradox’s and in the end it will show how and idea can work in theory, and in reality. Now as a forewarning I am firstly a philosopher and only further down the line a mathematician/physicist, but I’m going to play those roles for this post. If there are any mistakes or clarifications that are needed, feel free to post the information in the comments and I’ll make the corrections as soon as time permits.

First, I’ll just introduce the subject matter through an illuminating argument I had with a former professor a few years ago. He brought up the topic of the infinite and how it worked as a mathematical concept, and in doing so he pointed out that in math you could have a number that was infinite ( Z for example) and also a number that was double that infinite (2Z). Fair enough it does make sense within the framework of mathematics, but a problem arose when I was asked to define what the infinite meant.

Now when asked I said that the infinite is really a concept that is irrational when applied to the world around us (Look back that may be going a bit to far, but it points out how much the meaning changes when applied to real world issues). The infinite really breaks down and can’t be used in real applications, except as a concept that doesn’t exist in reality. Take for instance the simple equation above, and try to put that into an example that makes sense.

There is an infinite number of apples, but there are twice as many oranges as apples. Try to prove that statement is true. It makes sense as an idea when the sentence is read, but to really prove it all the apples and oranges would have to be counted, which cannot be done. Yet there is also a problem with being double the infinite of another infinite, because as soon as you hit infinity you’ve reached the limit, there can no longer be anything more. It is the equivalent of saying that something is more unique than another unique item. Unique means one of a kind, and once you’ve hit the status of one of a kind there can no longer be anything more unique. If there was something more unique it wouldn’t exist.

Now it could be argued that a circle or something caught in a loop is doing something perpetually, and that is an infinite. For example someone running in a circle on a track never really comes to an end point, but instead has to pick on arbitrarily.

So, what if someone is running on a circled track and there is another person running twice as fast as the other person. Couldn’t it be said that the one person is traveling twice the distance of the other, while they are both in an infinite loop? Well I think that would be a fair but inaccurate description of that is going on. The relative speed is the important part, while noting the infinite distance, just implies that it isn’t known when or if they will stop. It doesn’t add any information.

Now this seems to be an example of the infinite, and in a way it is. The problem comes with trying to find a home for the infinite within out observable universe, without there being a loop. The earth, due to it having a curved surface, has no endpoint. A person could fly around the earth and end up in the exact same spot they were in, but it wouldn’t be fair to characterize the earth as having a surface of infinite size.

This brings up how the infinite is a supernatural answer to some questions. How far can the universe expand? How long will space/time exist? How dense is a black hole? If the answer to those questions was given as infinite, it just means that the answer really isn’t known, but there is no reason to think that there is an upward limit.

The infinite is a placeholder when used to talk about most real things, and an effective tool when used conceptually.

Sorry I didn’t get to some of the interesting paradoxes, but I will in my next post.

Thanks for reading
-the moral skeptic