Friday, February 8, 2013

Down the rabbit hole--infinity, paradoxes, and weirdness, Part 1

I have been thinking about infinity a lot more than usual for the last month, and my trained opinion on the matter is that the deeper into this rabbit hole I go, the weirder the place gets. Lewis Carroll wrote Alice's Adventures in Wonderland, a satire of abstract mathematics, a few decades too early! Abstract mathematics had not gotten properly weird until Georg Cantor made it so by blowing open the topic of infinity about a decade later and sparking one of the most intense debates in the philosophy of mathematics, requiring the sure hand of David Hilbert (and a complete reformulation of set theory, inter alia, plus Kurt Gödel indicating that the problem cannot be resolved from within--some half a century after Cantor opened Pandora's Box here) to bring it down to the steady simmer it has enjoyed ever since. This really is a deep rabbit hole, and it is littered throughout with highly perplexing and uncomfortable paradoxes, one of which I want to address here.

Before getting into the meat of this post, I should note that I've written about infinity a few times before (and even made an infinity tag for posts related to the topic). I think it would be particularly useful, especially since I expect this post to become a series of posts to manage some of the length and difficulty, to at least take a look at two of my previous posts about infinity:
  • "Coming Clean about an Error that Is but Isn't," where I remark on a place in my very own God Doesn't; We Do where I fell prey to the siren song of "intuitiveness" while dealing with infinity--noting, however, that for my point in my book, the error I committed doesn't really matter (indeed, were I to rewrite the relevant part of that chapter for a revised edition, I think I could make the argument far stronger now in the vein of the theme of the book, even if the conclusion superficially sounds weaker).
  • "On Reality and Logic," where I maintain that reality is not subject to logic but rather that logic has been built around our conception of reality. This creates a powerful illusion that our logic must be the way that it is because it is very, very difficult to conceive of a different way to do logic. That turns out not to be the case, though, which forms a deep, dark rabbit hole of its own (possibly a topic for a future post). (Teaser: logic is an abstract system by which we decide if statements are valid, but as an abstract system, logic is built out of the very concepts--axioms--that logic ultimately acts upon in other logical-axiomatic systems. Should I get the red pill or blue one now, or would you prefer to drink from the little bottle that reads "Drink Me"?)
The point I want to make in this post is going to be relatively light, given the topic, and it deals with the concept underlying the second of those two blog posts.

In the first paragraph of this post, I indicated that it was eventually revealed to be undecidable from within whether or not Cantor was right about what he did. This raises the question of "from within what?" That's what I plan to discuss briefly here.

Up until that time, number theory had been developing rather nicely, and a set of axioms that still underlie number theory, the Peano Axioms, had been clarified and refined. There was an implicit problem with these axioms, though, and without Gödel's Incompleteness Theorems, these problems led to unresolved paradoxes and an awful lot of controversy.

Without getting into the nitty-gritty about the Peano Axioms, I can say that they lead to something very familiar: the idea that each number has a successor, literally exactly in the way that 1 follows 0, 2 follows 1, and n+1 follows n for any (whole) number n. This implies that there cannot be finitely many (natural) numbers because if there were, there would be a largest one, but any choice of largest has a successor that is larger, which indicates a logical contradiction. Thus, the Peano Axioms predict the concept of infinity, but they also indicate that we never get there. Adding one always gives us another number, not infinity.

It's sort of worse than that, though. Successorship is a very slow way to get anywhere with numbers. Exponentiation is much, much, much faster. If we choose some natural number greater than 1, say 2, raising that number to successive powers (i.e. recursively doubling numbers) gets big fast. Just ten iterations of this process, and we're over 1000; only twenty and we're over a million. But then, this raises the question of what "getting big" means.

To wit, if we look at this process, there is no end to the successive doublings that we can do. Suppose at some arbitrary point, we say that we're at a "big" number. Then the next step doubles it, and our "big" number is only half as big as where we are now. Repeat ten times, and our "big" number is like 1 to 1000. Repeat ten more times, and our "big" number is like 1 to a million. Feeling small yet? If we choose to select this new "bigger" number as "big," we just do it all again because the choice of "big" was arbitrary in the first place. To say that some number is the largest power of 2 that we can use is identical to saying that there is a largest number, which the Peano Axioms preclude. Thus, successive multiplication by any number (here: using two as a model) gets us to another number, not infinity, and every one of those numbers is arbitrarily small against the whole.

This concept, that we can "never get to infinity" from the Peano Axioms has been refined quite a bit, and we now say that the infinity that counts the natural numbers (countable infinity) represents a "strong limit cardinal," meaning that it is a cardinal number (a number that enumerates sets) that cannot be reached by succession or exponentiation.

Here's the paradox, then: when we go to build set theory from number theory, the Peano Axioms predict infinite sets, but the set theory that comes out of the Peano Axioms requires the negation of the axiom of infinity (essentially because "getting to"  infinity is impossible from below). This issue is part of the Pandorish Box that Cantor opened, leading to David Hilbert dedicating some of his famous questions (23 unsolved questions listed in 1900 as the most important ones for 20th century mathematics) to its resolution, with Gödel's incompleteness theorem telling us that from within the axiomatic system of number theory created by the Peano Axioms, we cannot decide if infinity exists or not. The result was a reformulation of set theory according to the Zermelo-Fraenkel Axioms of set theory, which are essentially the accepted foundation today--with some modifications.

The Zermelo-Fraenkel Axioms include the axiom of infinity: "There exists at least one infinite set." This raises other questions, like whether or not we can choose from infinite sets (oversimplified statement), that themselves cannot be answered from within the Zermelo-Fraenkel axiomatic system (this referencing the Axiom of Choice, which most mathematicians--but not all--accept now). As you might expect, if you've been following along or know what Gödel's Incompleteness Theorems say, adding more axioms leads to more unresolvable questions that lead to choices about more axioms, ad infinitum, if we accept that axiom.

This tells us two very important things:
  1. Claiming that "infinity exists" is equivalent to accepting the axiom of infinity. 
  2. All of this takes on a very peculiar human-made feel when examined closely (because it is).
The first point just above means that infinity is an abstraction defined by an axiom. The way we "get to" infinity (remember, it is a strong limit cardinal) is by defining it. This leads to a whole fringe group of the philosophy of mathematics called "ultrafinitism" that rejects the axiom of infinity--possibly for good reasons. On the other hand, it leads to other fringe groups of mathematicians--also possibly for good reasons--that argue for expansions of our number systems to very bizarre concepts like "surreal numbers" (click, read, feel dizzy, agree that "surreal" is a good name for them) and other branches of non-standard analysis along with other groups that suggest that there are strong limit cardinals beyond the infinitely many infinities (follows from Cantor if we accept the Zermelo-Fraenkel axioms) themselves, possibly an infinite (?!) progression of such hyper-inaccessible sizes of "things." Frankly, this shit gets weird.

This all brings me to the second point: for now, infinity is an abstract concept. Indeed, this is the mainstream interpretation of infinity among mathematicians: it's an abstraction--a useful one--that represents the idea embodied by the Peano Axioms of "can't say only finitely many" and it "exists" only as such--as an axiomatically defined abstraction. This is the camp I find myself in, intrigued by the fringe groups that explore infinity and that reject it but not swayed to any of their causes. For what it is worth, this stuff gets even weirder (and vastly harder) when we try to define probability theory--itself a field with raging and bitter (read: unsettled) debates in the philosophy of mathematics--in terms of it.

This brings me back to the argument that it is we who judge the worth of our axioms, usually by comparing them to the real world. This is the question of "actual infinities": can/do infinite things exist in reality (incidentally, Christian apologists like William Lane Craig argue that they cannot, possibly with the usual bullshit loophole that God isn't subject to such limitations). This question is not settled and is rather hotly debated. Accepting that they can presents some very uncomfortable paradoxes (not unlike "Can God create a sandwich so big that He cannot eat it?"). Rejecting it creates logical paradoxes against the Peano axioms (what is the largest meaningful number? what about that+1 or times 2?). Getting out of these problems may require reformulating logic without the rule of the excluded middle, or it may not be possible at all.

Incidentally, the naive approach to claiming that actual infinities exist is fought with the same problem presented by the strong limit cardinal property that creates this paradox in the first place. It has been suggested, for instance, that unless something were to destroy it, the universe must necessarily have an infinite timeline (at least in one direction) because at moment T presumed to be the last moment, there is the moment T+1 later than it (in whatever units). On the other hand, though, at any given time T, no matter how long it has been, the universe is still only finitely old--it never becomes infinitely old. There's that strong limit cardinal thing again--there is no way to get to infinity without jumping to it, which is the same as accepting a particular axiom, which is not proving "existence" in actuality.

For my purposes, this is very powerful. If theists want to claim infinite properties for their God, which they must to maintain statuses like "Most High" and "Almighty," then it necessarily follows that their God is defined as an abstraction at least until actual infinities can be proven to exist. Abstractions don't do anything except serve as the basis for (potentially useful) ideas.

If they want to claim their God is finite, then that's a major concession for them to make. It puts them in a rather serious bind--not least that they're stuck back into the memetic arms race that made them play the infinity card in the first place. (Actually, they're still in this race once they understand the implications of Zermelo-Fraenkel, only now they're having to argue it on ever more abstruse concepts--the real point I wanted to make in God Doesn't; We Do. This has the result of making their argument less and less coherent and more and more obviously "making stuff up" and isn't good for their cause.) They become very vulnerable in this position for the same reason the Peano Axioms are unresolvable with set theory: here, infinite Gods cannot exist but neither can there be a limit on finite ones if they are truly to be omni-grade Gods.

I look forward to exploring and developing more ideas on this theme. Stay tuned!

Edit (9 Feb. 2013): Part 2, on uniform distributions on infinite spaces, has been added.
(11 Feb. 2013): Part 3, on infinity from the other side, has been added.
(15 Feb. 2013): Part 4, on forcing a uniform PDF, has been added.
(21 Feb. 2013): Part 5, on choosing axioms, has been added.


If you enjoy my writing, you can read more of it in my first book, God Doesn't; We Do: Only Humans Can Solve Human Challenges. If you choose to pick it up, I thank you for your support of myself, my family, and indie authors in general.

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