Why, briefly, ordinal hyper is stupid
Jun. 17th, 2014 01:39 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo, do you remember this old entry? Well, Jeff is having me turn it into an actual paper. We'll see if it's new; I think it is, but I should actually, y'know, ask a set theorist.
(At this point you may be asking: Wait, why is Jeff relevant? Didn't you, y'know, finish your degree? Yes, but he's continuing to help me out for the time being. Ordinarily he'd have me forge ahead with complexity-related stuff, but I said I could get this done super-quick, so he's OK with it.)
Anyway, in my previous exploration of the subject, I mentioned that continuing past exponentiation into tetration and general hyper is pretty stupid for ordinals, but I never explained why. I thought I'd do that here.
I could actually go into quite a bit of detail on this, because I spent quite a bit of time thinking about it a few days ago, but I expect people would find it mind-numbing so I'll keep this short.
(Note: I am not claiming anything in this entry is new, except the second-to-last parenthetical. And while that's new, it's also kind of stupid. :P )
So what's wrong with ordinal hyper?
Let's start with ordinal addition. This is built up from the successor operation. To compute a+b, you apply the successor operation b times to a. Note that the resulting operation is continuous in b, by definition.
OK, that was pretty simple. How about ordinal multiplication? To compute ab, you add a to itself b times. Now here we have a choice; when we say "add a to itself b times", what we really mean is "start with 0, then add a to it b times". But are we adding a on the left or on the right? It makes a difference!
The correct choice is to add a on the right. As long as b is finite, of course, this makes no difference. But addition, recall, is continuous in the right summand. Which would mean that if we were to take aω under this weird modified mutliplication, we would get a fixed point of left-addition by a. Multiplying by any higher ordinal would still get you aω. That isn't what we want at all.
Thus we have to add on the right. Similarly, when it comes to defining exponentiation, we have to multiply on the right. But what about tetration?
For natural numbers, we define tetration by doing our exponentiation on the left, and there's a good reason for this. (x^x)^x is the same as x^(x*x), which makes doing it on the right a bit silly. Addition and multiplication don't have this problem. They're associative, sure, but associativity of, say, multiplication, doesn't involve any operations simpler/smaller than multiplication. By contrast, this relation turns two exponeniations into an exponentiation and a multiplication, and in general (if you keep putting more exponents on the right) turns n exponentations into 1 exponentiation and n-1 multiplications. So this is not very good.
Thus when we try to generalize to ordinals, we have a conflict of which way things need to go in order to be nonstupid. If we continue to do things on the right, we run into the same problem we do for finite numbers. If, on the other hand, we break that convention and switch to the left, we run into the problem of continuity and stabilization. (We can't have "been on the left all along", or we'd have run into that problem even sooner!)
Now the reader may point out here that "left" and "right" are just directions, without any particular meaning, but in fact they have been used here with a consistent meaning: Left is what's getting iterated, right is the number of iterations. So this is indeed a real problem.
And so -- assuming we do switch to the left, because we want finite things to work -- we run into the continuity problem and things become stupid pretty quickly. Tetration is pretty stupid; hyper-n for 5≤n<ω is very stupid; and for n≥ω is maximally stupid.
(There is also the problem that H4(0,ω) is undefined, but oh well.)
(Note, by the way, that if you're defining hyper so that things are done on the right, you should define H0(a,b)=Sa, not Sb.)
(Also of note is the fact that while "ordinary tetration", "Jacobsthal tetration", and "semi-Jacobsthal tetration" all remain distinct, once you go to hyper-5, they all become the same.)
(Tangentially -- the other day I had the idea: while trying to define "natural exponentiation" using surreal exponentials doesn't work... what if you just rounded up to the next ordinal? Turns out, this is a pretty bad idea. No algebraic relations there that I can see.)
-Harry
(At this point you may be asking: Wait, why is Jeff relevant? Didn't you, y'know, finish your degree? Yes, but he's continuing to help me out for the time being. Ordinarily he'd have me forge ahead with complexity-related stuff, but I said I could get this done super-quick, so he's OK with it.)
Anyway, in my previous exploration of the subject, I mentioned that continuing past exponentiation into tetration and general hyper is pretty stupid for ordinals, but I never explained why. I thought I'd do that here.
I could actually go into quite a bit of detail on this, because I spent quite a bit of time thinking about it a few days ago, but I expect people would find it mind-numbing so I'll keep this short.
(Note: I am not claiming anything in this entry is new, except the second-to-last parenthetical. And while that's new, it's also kind of stupid. :P )
So what's wrong with ordinal hyper?
Let's start with ordinal addition. This is built up from the successor operation. To compute a+b, you apply the successor operation b times to a. Note that the resulting operation is continuous in b, by definition.
OK, that was pretty simple. How about ordinal multiplication? To compute ab, you add a to itself b times. Now here we have a choice; when we say "add a to itself b times", what we really mean is "start with 0, then add a to it b times". But are we adding a on the left or on the right? It makes a difference!
The correct choice is to add a on the right. As long as b is finite, of course, this makes no difference. But addition, recall, is continuous in the right summand. Which would mean that if we were to take aω under this weird modified mutliplication, we would get a fixed point of left-addition by a. Multiplying by any higher ordinal would still get you aω. That isn't what we want at all.
Thus we have to add on the right. Similarly, when it comes to defining exponentiation, we have to multiply on the right. But what about tetration?
For natural numbers, we define tetration by doing our exponentiation on the left, and there's a good reason for this. (x^x)^x is the same as x^(x*x), which makes doing it on the right a bit silly. Addition and multiplication don't have this problem. They're associative, sure, but associativity of, say, multiplication, doesn't involve any operations simpler/smaller than multiplication. By contrast, this relation turns two exponeniations into an exponentiation and a multiplication, and in general (if you keep putting more exponents on the right) turns n exponentations into 1 exponentiation and n-1 multiplications. So this is not very good.
Thus when we try to generalize to ordinals, we have a conflict of which way things need to go in order to be nonstupid. If we continue to do things on the right, we run into the same problem we do for finite numbers. If, on the other hand, we break that convention and switch to the left, we run into the problem of continuity and stabilization. (We can't have "been on the left all along", or we'd have run into that problem even sooner!)
Now the reader may point out here that "left" and "right" are just directions, without any particular meaning, but in fact they have been used here with a consistent meaning: Left is what's getting iterated, right is the number of iterations. So this is indeed a real problem.
And so -- assuming we do switch to the left, because we want finite things to work -- we run into the continuity problem and things become stupid pretty quickly. Tetration is pretty stupid; hyper-n for 5≤n<ω is very stupid; and for n≥ω is maximally stupid.
(There is also the problem that H4(0,ω) is undefined, but oh well.)
(Note, by the way, that if you're defining hyper so that things are done on the right, you should define H0(a,b)=Sa, not Sb.)
(Also of note is the fact that while "ordinary tetration", "Jacobsthal tetration", and "semi-Jacobsthal tetration" all remain distinct, once you go to hyper-5, they all become the same.)
(Tangentially -- the other day I had the idea: while trying to define "natural exponentiation" using surreal exponentials doesn't work... what if you just rounded up to the next ordinal? Turns out, this is a pretty bad idea. No algebraic relations there that I can see.)
-Harry
