Now…
Let me write down my thoughts on hyperreals.
I don’t think the concept requires any introduction. Anybody who’s been reading along so far probably gets the gist that they are numbers bigger than infinity or smaller than infinitesimals (and by “bigger than infinity” we mean greater in absolute magnitude so that negative infinity is included).
I only grant the validity of hyperreal numbers given an incoherent assumption: that you can have quantities greater than infinity. I won’t go into the reasons why I think this is incoherent (anyone can read my numerous arguments throughout this thread), and given that I am granting that assumption in this post (and any discussion on hyperreals), it would be unnecessary anyway.
But if we grant that assumption, a whole world of possibilities opens up with hyperreals, a whole lota math and, yes, even arithmetic. Mathematicians wouldn’t have been able to establish a whole branch of mathematics if this were not so. It should be noted that hyperreals aren’t universally accepted even among mathematicians, but there are a sizable number of them who do accept them for them to warrant serious consideration.
Here are a few assumptions I hold which will likely come up for debate:
- Hyperreals don’t represent specific quantities.
The vsauce video which I’ve been trying to urge Magnus to watch talks about a similar concept. Michael Stevens of vsauce talks about counting past infinity with ordinal numbers. Ordinals are numbers used to describe the order of things. Contrast them with cardinals which are used to describe the quantity of things. Examples of cardinals are 1 apple, 2 oranges, 3 shoes, etc. Examples of ordinals are 1st place, 2nd best, 3rd contestant, etc. Stevens makes the point that you can’t count past infinity with cardinals, that there is no quantity greater than infinity that can be represented by a cardinal. But you can represent numbers greater than infinity with ordinals. If your purpose is to count items in the order they are arranged or added or appear, then it is sometimes impossible to do so when you start with an already infinite set unless you have omega, the first ordinal number. The idea is that if you have an infinite number of things, and then one additional thing shows up, you can’t use cardinal numbers to count that additional thing since all the cardinals would have been used up in counting the initial infinity of things. So you need to bring in omega. If order didn’t matter, you could count that additional thing as item #1, and then begin counting the infinite set starting at 2, 3, 4,… which would never give you a result greater than (\infty). But if order does matter, you have to count all the items in the original set first, then move onto the additional item.
Stevens stresses that once you get into omega, you are no longer talking about quantities. You are simply talking about order. The omegath item does not make the sum of items (\infty) + 1, it is simply the item that comes last. Likewise if you have a (omega+1)th item, that is simply the item that comes after the omegath item, and so on for omega+2, omega+3, etc.
I don’t know if Stevens would agree that ordinals greater than infinity are synonymous with hyperreals (even if we focus strictly on the hyperreals greater than (\infty)), but I think there’s more to hyperreals than Stevens’ treatment of ordinals greater than infinity. For one thing, hyperreals are on the number line and form a continuum with the reals. This makes it awefully difficult to dismiss their function in representing quantities. It seems we’d have to at least say they represent quantities greater than infinity, and that if R is a hyperreal number, R+1 is greater than R by 1. But I would agree with Stevens if he said that hyperreals don’t represent a specific quantity. They can’t. Exactly how much greater than infinity is some arbitrary hyperreal number R? If the hyperreal numbers extend infinitely in both directions, then the only answer to this questions seems to be “infinity”. R is infinitely larger than infinity. So whereas Stevens might say hyperreals don’t represent quantities at all (assuming his treatment of ordinals greater than infinity can be carried over to hyperreals), I’m willing to be a bit more leneant and say they do represent quantities, just not specific ones (other than being greater than infinity).
- Infinitesimals don’t represent the smallest numbers possible.
Just as the infinitely large hyperreals extend infinitely in both directions, infinitesimals also extend infinitely in both directions, but in terms of scale rather than position on the number line (though they extend infinitely in that sense too). By scale, what I mean is the amount of division one must do to get to infinitesimals. Take any real number and divide it infinitely many times, and you get to an infinitesimal. But that doesn’t get you to “the end of the line” so to speak. Infinitesimals can still be divided further to get infinitesimals that are even infinitesimal in relation to the infinitesimal you started with. And it goes without saying that you can multiply any infinitesimal upward an infinite number of times to get back to the reals.
It also goes without saying that infinitesimals don’t stand for specific quantities for similar reasons that infinitely large hyperreals don’t stand for specific quantities. It is completely arbitrary where you land after dividing a real number an infinite number of times. If we call the infinitesimal where you land (\epsilon), is this where you land after dividing (\infty) times? (\infty) + 1 times? (\infty) + (\infty) times? Since we can’t say specifically where you land, we can’t say how much smaller (\epsilon) is from 1 (or any real number), not specifically. We can, of course, say how much (\epsilon) is relative to other multiples of (\epsilon). For example (\epsilon)2 is twice as large as (\epsilon). We can even say how much larger (\epsilon) is from 0–it’s excatly (\epsilon) larger than 0. But beyond that, we can’t say specifically how much a given infinitesimal is.
- You can do arithmetic with hyperreals but you might get unexpected results.
For example, multiplying an infinitely large hyperreal R by 2 gives you a hyper-hyperreal number. That is, a hyperreal number that is hyperreal even relative to R–i.e. R2 is infinitely larger than R. You can see why just by giving it a moment’s thought. If R is already infinite in size, then multiplying it by 2 should give you a result that is twice as infinite.
Unlike the infinitely large hyperreals, infinitesimals play by the opposite principle. Multiplying them by even astronomically large real numbers still gives you an infinitesimal on the same order. But doing something as simple as adding 1 to an infinitesimal causes it to take an infinite leap to another spot on the real number line. And it can be seen why: if you have (\epsilon) (the first infinitesimal after 0 at a given scale), and you add 1 to it, you get 1 + (\epsilon), which, relative to (\epsilon), is an infinite distance away.
If X = 1, then you can count up by multiplying X by the natural numbers or by adding the natural numbers:
X1, X2, X3, X4… = 1, 2, 3, 4…
X+1, X+2, X+3, X+4… = 1, 2, 3, 4…
If R is an infinitely large hyperreal, you cannot count up by multiplying it by the natural numbers (as each multiplication after 1 would give you a hyper-hypereal number), but you can count up by adding the natural numbers:
R, R+1, R+2, R+3…
If (\epsilon) is an infinitesimal, you cannot count up by adding the natural numbers (as each addition after 0 would give you a number an infinite distance away from (\epsilon) relative to (\epsilon)), but you can count up by multiplying by the natural numbers:
(\epsilon)1, (\epsilon)2, (\epsilon)3, (\epsilon)4…
- The standard treatment of infinity we usually give before bringing hyperreals onto the table carries over to hyperreals after bringing them to the table.
For example, we say there is no largest real number, that the reals just go on infinitely. And while we don’t stop saying this once we bring hyperreals to the table, we must say the same thing of hyperreals. There is no more reason to say there is a largest hyperreal number than there is to say there is a largest real number. This reason applies repeated to any higher order of hyperreals (so hyper-hyperreals, hyper-hyper-hyperreals, etc.). Similarly for infinitesimals. Just as we say there is no smallest number (down in scale, though also down through the negatives), we must also say there is no smallest infinitesimal.
What this means is that a lot of the issues that makes the concept of infinity so problematic won’t necessarily go away just because we bring hyperreals to the table.
For example, in an earlier discussion I had with Magnus, I argued that removing every odd point from an infinite line and shifting the remaining points to fill the gaps would give you a perfectly identical line. Magnus’s response was to argument that beyond the infinitieth point, you would see a difference. The line would go from this:
(\bullet \bullet \bullet … + \bullet \bullet \bullet …)
… to this:
(\bullet \bullet \bullet … + \circ \circ \circ …)
He suggested that all the points beyond infinity would have to be used up to fill the gaps on this side of infinity.
But if the hyperreals have their own infinity (beyond which there are the hyper-hyperreals), and indeed there is no end to the orders of hyperreals, then a more extended picture of the line might look like this:
(\bullet \bullet \bullet … + \bullet \bullet \bullet … + \bullet \bullet \bullet …)
And if you remove every odd point from the line (at least on this side of infinity) and shift the remaining points into the gaps, then maybe the result would look like this:
(\bullet \bullet \bullet … + \bullet \bullet \bullet … + \circ \circ \circ …)
…but of course, this line of reasoning carries on indefinitely, meaning that you could have an infinite string of infinities, none of which are ever depleted of points because the points from the next infinity over get shifted down.
One eventually wonders: why not just say the entire series of infinities is one great infinity? And you’re back at square one. This is just one example of how introducing hyperreals won’t necessarily solve all the problems plagued by infinity.
I’ll stop there. I could go one with several more points, but I believe this is enough to get a fruitful discussion going, if not between Magnus and I (because he’s scared of hyperreals even though that’s what he’s talking about without knowing it) then between other members of ILP. Should be fun.