Monday, February 11, 2013

Down the rabbit hole--infinity, paradoxes, and weirdness, Part 3

This post is the third in a series discussing the topic of the weirdness of infinity. It is recommended to read Part 1 (Link to Part 1) for some foundation for this part, and Part 2 (Link to Part 2) is of related interest, developing the idea at the center of this post from one angle and suggesting at the psychology of why infinity is so counterintuitive.

In Part 1 of the weirdness of infinity, I introduced a concept known as a "strong limit cardinal." The point of this post today is to explore that concept from the other direction as compared to Part 2. In the previous posts in this series, I introduced that the meaning of infinity (technically countable infinity, which is the smallest infinity--yes, they come in different sizes under current set theory) is a strong limit cardinal by pointing out that that means we cannot get to it from below. Specifically, if we use the successor function, add 1, repetitively, although we cannot claim that the total number of numbers is finite, we never reach infinity by repeatedly applying this technique. Additionally (and making it a strong limit cardinal instead of just a limit cardinal), if we take successive powers of any finite number, we always get a finite number as a result and thus cannot get to infinity this way either. The way we get there is to jump to it by defining and accepting the Axiom of Infinity.

In short, then, that countable infinity is a strong limit cardinal implies that we cannot get to countable infinity from below, and how that plays with our intuitions was the point of the first two posts in this series. This post starts at infinity and looks at the matter from the other perspective.

Imagine if we have an infinite collection, the natural numbers {1, 2, 3,...} being a very handy and intuitive set to work with. Now suppose I want to remove half of them (skipping the case where I just remove one or some at a time). The first weird thing we notice is that we cannot simply go up to some value and say that's the halfway point (the surreal number axioms allow us to do this, but they also reveal that the halfway point is not actually a number but is instead infinite). Anywhere we choose to pick has finitely many numbers below and infinitely many above. One way to put this is that "every number is smaller than most." At any rate, this certainly isn't cutting the set in half.

There is an easy way to cut the natural numbers in half, though, which is to consider divisibility by 2. If a number is divisible by 2 (i.e. even), we'll put it over here, and if it is not (i.e. odd), we'll put it over there. According to the Zermelo-Fraenkel axioms of set theory, the ones currently accepted, both of these sets are equally numerous, and so we've cut the natural numbers in half... or have we? Those same axioms provide that we have in each of these two sets, the Evens and the Odds, exactly the same number as we had in all of the natural numbers. Okay, weird.

Still, for the argument I want to present, I will call this "removing half of the natural numbers," because that really does make sense in a meaningful way (described by the natural density approach to understanding some subsets of the natural numbers). More importantly, since I'm exploring where our intuition gets weird with infinity, this notion is intuitive enough to be getting on with.

Now, what do I have after removing half of the natural numbers? Well, I already said: I have the same number. If I remove half of those, I still have the same number (one in four of the originals now). If I remove half of those, I still have the same number (one in eight of the originals). If I remove half of those, I still have the same number (one in sixteen of the originals). No matter how many times I do this process of removing half of them, I still have the same number that I started with.

This means that the concept of a strong limit cardinal is really a two-sided beast. On the one hand, if I start with something finite, I cannot get to infinity by repeated successorship or exponentiation, and on the other hand, if I start with infinity, I cannot get to finite by repeated finite removals or (negative) exponentiation. (Sure, I could get to a finite set by removing everything larger than five, for example, but this is the inverse of adding infinitely many and is the exact kind of leap I'm saying is required to traverse to infinity from below.)

In overgeneralized summary, then: because (countable) infinity is a strong limit cardinal, one can neither reach it from below nor get out of it from above without having to jump between the two paradigms.

That's pretty weird.

What kinds of consequences does it have?

For an example, consider what it would mean if there were a star an infinite distance away, shining its light. Photons from that star would never reach us. Since the speed of light is finite, this is the successor process. Its light would never get here; its gravity would have no impact on us (or anything in our universe). In every meaningful sense, a star imagined infinitely far away is completely identical to a star that does not exist.

Weirder still, if I follow the ideas suggested by the surreal numbers, if another star were half as far away (n.b.: This doesn't mean anything without a structure like the surreals to define it, although our intuition kind of likes it), the same would be true. If there were a third star half again as far away, the same would be true yet again. If a fourth star were half again as far away, same thing. No matter how many stars I define like this, be that tens, hundreds, billions, or numbers that are still every bit as embarrassingly small even if they don't feel like it (like a googolplex to the googolplex power) all are still infinitely far away (and infinitely far away from each other).

As this starts to sink in, the idea of an infinite universe starts to feel pretty uncomfortable. This leads me to the same suggestion I made throughout Part 2 of this series that our intuition is pretty poorly equipped to handle these questions, doing us the serious disservice of graying out the part where we jump between finite and infinite without really facing that such a chasm cannot be jumped without merely defining it to have been done.

Does this have anything to do with God?

Actually, although the psychology of intuition around infinity is one of the main things I'm exploring in this series (while getting ideas out there), I think in this case there is something pretty important at play.

In God Doesn't; We Do, I make the argument that the omni-grade (meaning infinite in scope) properties of God are haphazardly applied as the result of a memetic arms race. The quality of this arms race of memes can be summarized by the following exchange familiar to almost everyone who has ever been a child ever (at least in the United States):
  • "My God is bigger than yours!"
  • "Nuh uh, my God is twice as big as yours!"
  • "Nuh uh, my God is a hundred times as big as that!" (No child or theist--do I repeat myself?--I am aware of has ever noticed the recursion here that breaks this game here. They also seem not to notice that making the numbers bigger doesn't really mean all that much in terms of their goals.)
  • "Yeah, well my God is thousand times as big as that!"
  • "Well, my God is a million times as big as that!" (Emphasis makes a million no bigger, but it doesn't preclude the behavior.)
  • [This continues until the names for bigger numbers cannot be thought of, which is an important point to make.]
  • "My God is infinitely bigger than your God!"
  • [This often continues with "infinity plus one" or "infinity times two," but we know clearly now that this reveals a lack of understanding of infinity--and thus of the underlying God--as opposed to clarity of any sort, unless we want to get surreal with this in which case those ideas technically mean something and yet what requires an awful lot of formalism that might not be interpretable.]
Here, there are a few points to make (besides that this same sort of memetic arms race also started to characterize the entire Dragonball Z Japanese cartoon series). First, it is rhetorically very easy to make the jump to infinity, but because infinity is a strong limit cardinal, it is impossible to make this jump without just appealing to rhetoric (which includes, necessarily, accepting the axiom of infinity).

Second, as I argue in God Doesn't; We Do, and mentioned just above, this reveals a fundamental lack of understanding of what is going on. It is expected out of children, but it is not acceptable out of supposedly sophisticated adults who happen to be theists (who certainly wouldn't want to commit heresy by mischaracterizing their God, right?).

Third, the jump happens precisely when we run out of names for numbers, which happens (infinitely) long before we run out of numbers. This strongly underscores my point that we gray out the jumps between finite and infinite specifically because eventually we just aren't good at conceiving of larger numbers (probably we actually aren't able to do so at all) than some fuzzy and arbitrary cutoff point.

Fourth, as I argue in God Doesn't; We Do, because of the exact same phenomenon mentioned in the first part of this series (link again), I think this memetic arms race is a necessary consequence of the definitions of God people use. Each culture, religion, denomination, sect, or church merely has to argue convincingly that their God-concept is bigger and more effectively inclusive than the others, and they can win superstitious converts who don't want to be on the wrong side of a stronger deity.

One conclusion I draw from point four above is that the very definition of God that we use, not least because the evidence for His existence has always been numinous instead of material, is actually simply just an abstraction (to be distinguished from "physical" or the diabolical "spiritual," which depends upon substance dualism, a philosophical construction that isn't doing so well against examination in the marketplace of ideas--however well it seems to sell). My conclusion from that is that as an abstraction, God doesn't do anything and cannot be said to "exist" without equivocating somewhat on what is meant by that term.

A second conclusion I draw from point four above is that just as the Peano axioms that underlie number theory essentially suggest that infinity needs to exist (as an abstraction) and yet cannot show that it actually must, a finite God never provides security against the problem of simply arguing for a bigger God while an infinite one is just a rhetorical trick resting upon the acceptance of an abstraction.


I hope you're enjoying this series as much as I am. Be sure to check out Part 1 and Part 2 if you haven't yet, and keep your eyes open for more parts to come. I've got at least two or three more with sketched outlines.

Edit: (15 Feb. 2013): Part 4 is now available as well, discussing the question of uniform PDFs on infinite spaces in more rigorous detail.
(21 Feb. 2013): Part 5 is now available also, discussing how the question of infinity forces us to choose among various axioms that underlie everything logical.




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