Comments: 
... so we have an equivalence class of systems that can be modeled with whole numbers? What sorts of things can or can't be handled like this? All sorts of things are encoded as streams of bytes  is this the right intuition to take to this?
(tell me all about this, please :) )
Okay, so, Gödel worked out a way of assigning unique integers to each symbol and wellformed string in PM's notation, such that string manipulations in PM could be modeled with numerical manipulations (you know, sophisticated stuff like, oh, addition and multiplication). It turned out that there were two kinds of integers in the world: those that coded for provable strings of PM, and those that didn't. (Hofstadter cutely calls the former the "prim numbers".) (Interesting aside: you can't build a machine that will tell you if any given number is prim or not, 'cuz halting problem.)
But "primness" turns out to be a purely numbertheoretical property and therefore representable in PM! That means that you can write a (really frickin' long) string of PM symbols that will assert, "A certain integer g is not a prim number." But! If you are Kurt Gödel in 1931, you will have rigged it so that said integer g is in fact the integer that codes for that very (really frickin' long) string of symbols! Aha! Paradox! I've caught you in your web of deceit, Bertrand Russell! And stuff.
Anyway, to quote Hofstadter, this is all possible because "numerical patterns have the flexibility to mirror any other kind of pattern". So if "mirror" means "model", and "pattern" means "formal system", and if you can model a nonnumerical formal system with numbers, then you can surely model a formal system that's already numerical with different numbers, right? And if you can create a mapping from nonnumerical system to numerical system, then you ought to be able to create mappings between numerical systems, right?
And speaking of Wolfram, does his principle of computational equivalence say the same thing? (Except he might be full of crap.)
Also, I think I'm'a start saying "scaleinvariant" instead of "fractal". It's more descriptive and less trendy.
Also, I want an umlaut in my name. Let's change our names. You know you do, too.
In other startling news, any two systems that can be modeled with whole numbers can also model each other.
But will we like what we get?
In parallel (major) thirds with oniugnip, a question: are there links you can share? I've run a few searches and haven't come up with anything.
 From: pmb 20080128 06:59 am (UTC)
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This is basically a restatement of Godel's theorem, where he proved that if you could model whole numbers, then you could model all known mathematics, including the statement "this statement cannot be shown correct using manipulations of whole numbers in a general mathematical system".
Oh. ... but I thought you couldn't do that.
 From: pmb 20080128 07:18 am (UTC)
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You can make the statement, but not prove its truth value. Or if you can, then the statement is false and your system just proved a false thing. Thus, every mathematical system sufficiently complicated to model the whole numbers will either contain unprovable truths or provably true falsehoods. The system that modern mathematicians use  ZermeloFrankel set theory + the Axiom of Choice (aka ZFC)  contains unprovable truths.
It also contains things whose truth value is independent of the axioms, which is a slightly different condition. The most famous of these independent things is the Continuum Hypothesis (which comes in both normal and Grand forms), aka "How many real numbers are there, really?" The current drive is to find some "natural" or "intuitive" axiom that will imply an answer to the CH. Nobody knows any right now, so the Continuum Hypothesis sits around being of indeterminate truth value. As for me, as I advance in mathematical maturity I find it hard to even accept the Axiom of Choice, which apparently allies me with the crankiest of old mathematical codgers.
I got here by thinking about Gödel's theorem, so, points for you, but I don't think it is quite a restatement of Gödel's theorem. I make no claims about being able to completely model whole numbers!
 From: pmb 20080128 07:57 am (UTC)
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The system I'm talking about models them to the point of addition and multiplication, and that the numbers can get very big and that's okay. If you have that, and zero and one, then Godel has a trap for you.
If you have integers but cannot multiply or add them, or only have a limited number somehow, then your system is strange, but also may not necessarily contain a trap.
Sure, yeah. But that's not really what I'm talking about. I'm not talking about modeling the whole numbers, I'm just claiming that if you have a system A and a system B and they can both be modeled with whole numbers, then you can also model system A in system B.
 From: pmb 20080128 08:11 am (UTC)
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Looking back over what I said, I totally look like a pedantic jerk.
'sokay. How'd your talk go  did they like the drawing?
 From: pmb 20080128 08:55 am (UTC)
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I actually am giving this lecture on Thursday. For some reason, the prof asked me to give the lecture a month before I would need to give it, and then asked for a picture, which made me think about how I was going to talk long before I needed to. http://www.cs.uoregon.edu/classes/08W/cis170/
But will we like what we get?
Oh, probably not. See, this is why I secretly want to be a mathematician. Mathematicians get to say, "I have found an completely dreadful way of doing X and thus proved that X is not impossible! Woohoo!" Then they hand it off to the computer scientists, whose job it is to find ways of doing X that don't make people want to claw their eyes out.
 From: pmb 20080128 07:58 am (UTC)
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Math is "What", CS is "How".
Yeah. I mean, maybe I could turn a celery bread recipe into 12tone music, but why would I want to?
 From: lindseykuper 20080128 07:37 pm (UTC)
how most twelvetone music probably gets written  (Link)

Because it's 2 a.m. and the assignment is due at 9 a.m.?
 From: keystricken 20080128 07:50 pm (UTC)
Re: how most twelvetone music probably gets written  (Link)

Hmm.
BizarroMonica is offline, but your theory has merit.  