Up numbers
So sometimes I invent some maths, though never anything too smart or complex and I very strongly doubt ever anything original. Lacking a full maths degree is handy here because there is a greater* space of maths to be discovered the less you know in advance. And now I've got a place to share such I occasionally will, starting with the subpar invention of "Up numbers".
Classically there is no good answer to the question "What's the smallest number that's bigger than 2?", with the most elegant and useful answer being "The smallest number that's bigger than 2". This is unsatisfactory** and easy to fix by defining some more numbers that extend the reals. Let the smallest number greater than 2 be 2↑ and the largest number less than 2 be 2↓. Simple.
So what's the smallest number greater than 2↑? Well it's 2↑↑ but lets use the notation 2↑2. If we want to root this in set theory, let a↑n be the ordered pair a,n where a∈R and n∈Z. Let a↓n = a↑-n for convenience and a↑0 = a↓0 = a
So far so good, but we've called these "numbers". We have a base point and a means of incrementing but do they hold up to sense in respect of behaving like numbers and being just bigger or smaller than real numbers. To preserve the definition we need the following to be true:
1) anZ and ∀a, b∈R
Still good, arithmetic is next. Hopefully the following definitions of addition and subtraction are intuitive:
2) a↑n+b = (a+b)↑n
3) a↑n+b↑m = (a+b)↑(n+m)
4) a↑n-b = (a-b)↑n
5) a-b↑n = (a-b)↑-n = (a-b)↓n
6) a↑n-b↑m = (a-b)↑(n-m)
I introduced Up numbers as subpar and that's because we now need to define multiplication and division, and it's not going to be sound.
Lets start with:
7) a↑n*b = (a*b)↑b*n where b∈Z
This works with our definition of addition, any other definition doesn't seem to to me. To extend this we get to
8a) a↑n*b = (a*b)↑b*n where b∈R
except that (n*b) may not be ∈Z. Hmm... second attempt:
8b) a↑n*b = (a*b)↑floor(b*n) where b∈R
Why floor? Well it's basically arbitrary but seems more right than any other option to me somehow. Finally for multiplication:
9) a↑n*bm = (a*b)↑floor(b*n)+floor(a*m)
I feel by now I'm stood on a house of cards, the base definitions seemed good but following them down it all gets shakier. The reasoning that a↑n*bm = (a*b)↑x for some x is based around 1↑z. Any real multiplied by 1↑z shouldn't be some larger real or 1) above looks in trouble. But it may legally be a larger Up number. The rest of 9) is harder to justify, better definitions are welcome.
Now the killer, division. It should be the inverse of multiplication so looking at 7) lets go for:
10a) a↑n/b = (a/b)↑n/b where b∈Z
Except n/b may not ∈Z. Lets add another arbitrary floor although you can replace it with ceiling or round to nearest if you wish, it won't change the next issue.
10b) a↑n/b = (a/b)↑floor(n/b) where b∈Z
But we can now see:
( 6↑5/2 ) * 2 = ( 3↑2 ) * 2 = 6↑4
which isn't nice. Poking and playing around with all the definitions we always either run into this or an issue whereby a↑n + a↑n ≠ a↑n * 2 which is equally broken.
So I came up with Up numbers (and no doubt re-created well known work done elsewhere) and this is about as far as I've progressed them. Comments, holes poked, better definitions, expansions all welcomed.
*using an intuitive English language definition of greater as clearly the space of discoverable maths is infinite (is it a corollary of Gödel's theorems that it is a countable space?)
**to me, your milage may vary.
Classically there is no good answer to the question "What's the smallest number that's bigger than 2?", with the most elegant and useful answer being "The smallest number that's bigger than 2". This is unsatisfactory** and easy to fix by defining some more numbers that extend the reals. Let the smallest number greater than 2 be 2↑ and the largest number less than 2 be 2↓. Simple.
So what's the smallest number greater than 2↑? Well it's 2↑↑ but lets use the notation 2↑2. If we want to root this in set theory, let a↑n be the ordered pair a,n where a∈R and n∈Z. Let a↓n = a↑-n for convenience and a↑0 = a↓0 = a
So far so good, but we've called these "numbers". We have a base point and a means of incrementing but do they hold up to sense in respect of behaving like numbers and being just bigger or smaller than real numbers. To preserve the definition we need the following to be true:
1) anZ and ∀a, b∈R
Still good, arithmetic is next. Hopefully the following definitions of addition and subtraction are intuitive:
2) a↑n+b = (a+b)↑n
3) a↑n+b↑m = (a+b)↑(n+m)
4) a↑n-b = (a-b)↑n
5) a-b↑n = (a-b)↑-n = (a-b)↓n
6) a↑n-b↑m = (a-b)↑(n-m)
I introduced Up numbers as subpar and that's because we now need to define multiplication and division, and it's not going to be sound.
Lets start with:
7) a↑n*b = (a*b)↑b*n where b∈Z
This works with our definition of addition, any other definition doesn't seem to to me. To extend this we get to
8a) a↑n*b = (a*b)↑b*n where b∈R
except that (n*b) may not be ∈Z. Hmm... second attempt:
8b) a↑n*b = (a*b)↑floor(b*n) where b∈R
Why floor? Well it's basically arbitrary but seems more right than any other option to me somehow. Finally for multiplication:
9) a↑n*bm = (a*b)↑floor(b*n)+floor(a*m)
I feel by now I'm stood on a house of cards, the base definitions seemed good but following them down it all gets shakier. The reasoning that a↑n*bm = (a*b)↑x for some x is based around 1↑z. Any real multiplied by 1↑z shouldn't be some larger real or 1) above looks in trouble. But it may legally be a larger Up number. The rest of 9) is harder to justify, better definitions are welcome.
Now the killer, division. It should be the inverse of multiplication so looking at 7) lets go for:
10a) a↑n/b = (a/b)↑n/b where b∈Z
Except n/b may not ∈Z. Lets add another arbitrary floor although you can replace it with ceiling or round to nearest if you wish, it won't change the next issue.
10b) a↑n/b = (a/b)↑floor(n/b) where b∈Z
But we can now see:
( 6↑5/2 ) * 2 = ( 3↑2 ) * 2 = 6↑4
which isn't nice. Poking and playing around with all the definitions we always either run into this or an issue whereby a↑n + a↑n ≠ a↑n * 2 which is equally broken.
So I came up with Up numbers (and no doubt re-created well known work done elsewhere) and this is about as far as I've progressed them. Comments, holes poked, better definitions, expansions all welcomed.
*using an intuitive English language definition of greater as clearly the space of discoverable maths is infinite (is it a corollary of Gödel's theorems that it is a countable space?)
**to me, your milage may vary.