
aleph_0, smallest infinite cardinal
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Today's Podcast (For a nonbible post? That's new!)
Let's start by considering the set of all positive integers {1,2,3,4,...}. Notice that there are infinitely many numbers in this set, and yet the difference between any pair of numbers is always finite. Even though we have an infinite set, there is no member of that set that is infinitely big, there is no integer of size infinity. Of course if we include negative numbers (and zero) to get all integers {...,3,2,1,0,1,2,3,...} the same holds true. We have a set which limits to negative infinity in one direction and positive infinity in another, and yet every number in the set is some finite distance away from zero. This type of thing is very common when we deal with infinities and it demonstrates that precision and care is very important. Your first instinct when considering this infinite set is that there must be some point in that set which is infinitely far away from zero. However, by looking at it from this perspective, I hope it is easy to see that it's not quite right. (Although what you can say is that for any number N, there is some integer which is more than distance N away from 0)
While arguing with apologists, I will commonly see it very casually mentioned that an infinite regress is impossible. Although the explanation is typically quite sparse, I've luckily had a few good conversations with apologists lately where they have tried to explain themselves more fully on this topic. The first such conversation took place on my blog in a
comment from
The Rational Zealot
Let’s say the past is represented by negative numbers, zero is the present, and positive numbers are the future. Let’s say you never start counting but have been counting from an infinite past. An infinite amount of time later, you are still counting negative numbers. An infinite amount of time after that, you are still counting negative numbers. An infinite amount of time after that? Still negative numbers. To say otherwise means you haven’t really been counting from negative infinity, but have changed infinity into a number.
So the situation we find ourselves in is an infinite regress. Every moment has a moment before it. If we take the entire timeline at once we have an infinite number of moments. I would argue that there is nothing wrong with this, there is an infinite past, so what? Every moment has an infinite number of moments preceding it, and yet the distance between any two moments is finite. Strange for sure, but there is no contradiction here. Let's look at The Rational Zealot's argument one step at a time.
Let’s say you never start counting but have been counting from an infinite past.
I'm with you so far, we have been counting forever into the past, there was no start.
An infinite amount of time later, you are still counting negative numbers.
Here is where he loses me. An infinite amount of time later
from when? It seems that we have assumed a starting point at negative infinity (we'll explore what this means in a moment). He continues
An infinite amount of time later, you are still counting negative numbers. An infinite amount of time after that, you are still counting negative numbers.
I believe this is the key problem to this conversation every time I have it. Every point on the timeline is a finite distance from the current moment. There is no point on the timeline from when I can count for an infinite amount of time and still land on the timeline. Again, every pair of points on the timeline are a finite distance apart. So what does it even mean to count for an infinite amount of time?
Sometimes for convenience we will
compactify the space and put a point at infinity. In the case of a line we will probably put a point at negative infinity and another point at positive infinity. These two points are not standard, they are instead a mathematical abstraction. They are special points very different from the mundane numbers on the rest of the timeline. One way to look at this is that starting at negative infinity in the compactified space is the same as saying that you have always been counting without a start in the noncompactified space. Similarly, saying you will end on positive infinity in the compactified space is the same as saying that you will never stop counting in the noncompactified space. It is sometimes nice to translate "I will count forever" into a more manageable form.
If we do allow these points at infinity and we allow ourselves the ability to actually count an infinite number of moments, there are three possibilities as far as I can tell
 You start on the timeline and after an infinite number of moments you are at positive infinity
 You start at negative infinity and after an infinite number of moments are at positive infinity
 You start at negative infinity and after an infinite number of moments are anywhere on the timeline
To say that we count for multiple infinities worth of counting and still are at negative numbers makes no sense.
To say otherwise means you haven’t really been counting from negative infinity, but have changed infinity into a number.
Ultimately, I believe that the mistake that has been made is to assume that this special point at infinity exists, but then treat is as a standard point in some ways. The easiest way to solve this problem is to not allow this point at infinity. Only consider normal points, even though there are an infinite number of points, there is no first point and any pair of points is a finite distance from one another. The complaint evaporates because there is no start, there is no point from which we can count an infinite amount of time and still be in negative numbers.
Humblesmith provided a very similar complaint
over on his blog where he discusses an example of an infinite string of dominoes that is falling over. At first I thought the addition of physical objects would make things more complicated (where did the dominoes come from, etc), but if we ignore those problems, having dominoes set up does wind up being instructive. He was talking about various problems with such a setup and we get to his third point
Third, as we observe this string of dominoes falling over, if it were infinite, we must ask ourselves “how did the falling get to me?” If the line of dominoes were infinitely long, it seems the falling would always be an infinite distance away from me. The atheist might reply, “Well, the ones currently falling over have to be somewhere. It just so happens that it is next to you.” But this misunderstands infinites. If the line were truly infinite, then the falling would always, at all instances, be an infinite distance away from any one point on the line. Pick any domino, and the falling would have been an infinite distance away. Since the falling is happening in sequence, it is impossible to select a domino where the falling is not an infinite distance away. The dominoes are always falling, but never arriving anywhere, which is an absurdity.
This is essentially the same objection, we have an infinite string of dominoes, which have been falling forever, and therefore the falling cannot get to the current position. But we have a similar problem, it seems to me that what is happening is they are trying to put a domino at that "point" at negative infinity, flick it, and say it can never hit any of the other dominoes. But there is no domino at negative infinity, the fact that we have an infinite number of dominoes just means that every domino has one before it. To say that this string has been falling forever simply means that every domino has been hit by the one before it. Let's again, highlight the real misunderstanding here
If the line were truly infinite, then the falling would always, at all instances, be an infinite distance away from any one point on the line.
No, this is incorrect. Every domino is on the line, so if any dominoes are falling then the falling is somewhere on that line. If the falling is an infinite distance away from the domino that I am standing next to, then which domino is falling? Remember, every domino is a finite distance from the one I'm standing next to, if falling is happening it has to be happening
to some domino. Either the falling is some finite distance from my domino, or no dominoes are falling and it doesn't really make sense to say that falling is happening at all. Falling can't be happening an infinite distance away because there is no domino at distance infinity from mine.
In both cases here, the key is to ask what is meant when things are happening "at infinity". If we count for an infinite amount of time and are still stuck at negative numbers, then when did we start? If a domino is falling an infinite distance away, where is that domino? The problem here seems to stem from the same counterintuitive notion that we can have an infinite number of points such that every pair of points is actually a finite distance from one another.