Since we talked about the surreals, I thought it would be interesting to take a *very* brief look at an alternative system that also provides a way of looking at infinites and infinitessimals: the *hyperreal* numbers.
The hyperreal numbers are not a construction like the surreals; instead they're defined by axiom. The basic idea is that for the normal real numbers, there are a set of basic statements that we can make - statements of first order logic; and there is a basic structure of the set: it's an *ordered field*.
Hyperreals add the "number" ω, the size of the set of natural numbers, so that you can construct numbers using ω, like ω+1, 1/ω, etc; but it constrains it by axiom so that the set of hyperreals is *still* an ordered field; and all statements that are true in first-order predicate logic over the reals are true in first-order predicate logic over the hyperreals.
For notation, we write the real field ℜ, and the hyperreal field ℜ\*.
The Structure of Reals and Hyperreals: What is an ordered field?
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If you've been a long-time reader of GM/BM, you'll remember the discussion of group theory. If not, you might want to take a look at it; there's a link in the sidebar.
A field is a commutative ring, where the multiplicative identity and the additive identity are not equal; where all numbers have an additive inverse, and all numbers except 0 have a multiplicative inverse.
Of course, for most people, that statement is completely worthless.
In abstract algebra, we study things about the basic structure of the sets where algebra works. The most basic structure is a *group*. A group is basically a set of values with a single operation, "×", called the *group operator*. The "×" operation is *closed* over the set, meaning that for any values x and y in the set, x × y produces a value that is also in the set. The group operator also must be associative - that is, for any values x, y, and z, x×(y×z) = (x×y)×z. The set contains an *identity element* for the group, generally written "1", which has the property that for every value x in the group, "x×1=x". And finally, for any value x in the set, there must be a value x-1 such that x×x-1=1. We often write a group as (G,×) where G is the set of values, and × is the group operator.
So, for example, the integers with the "+" operation form a group, (Z,+). The real numbers *almost* form a group with multiplication, except that "0" has no inverse. If you take the real numbers without 0, then you get a group.
If the group operator is also commutative (x=y if/f y=x), then it's called an *abelian group*. Addition with "+" is an abelian group.
A *ring* (R,+,×) is a set with two operations. (R,+) must be an abelian group; (R-{0},×) needs to be a group. If × is commutative (meaning (R-{0},×) is abelian), then the group is called a *commutative* group.
A *field* (F,+,×) is a commutative ring with two operators "+" and "×"; where the identity value for "+" is written 0, and the identity for "×" is written 1; all values have additive inverses, all values except 0 have multiplicative inverses; and 0 ≠ 1. A *subfield* (S,+,×) of a field (F,+,×) is a field with the same operations as F, and where its set of values is a subset of the values of F.
*Finally*, an *ordered* field is a field with a total order "≤": for any two values x and y, either "x ≤ y" or "y ≤ x", and if x ≤ y ∧ y ≤ x then x=y. The total order must also respect the two operations: if a ≤ b, then a + x ≤ b + x; and if 0 ≤ a and 0 ≤ b then 0 ≤ a×b.
The real numbers are the canonical example of an ordered field.
*(The definitions above were corrected to remove several errors pointed out in the comments by readers "Dave Glasser" and "billb". As usual, thanks for the corrections!)*
One of the things we need to ensure for the hyperreal numbers to work is that they form an ordered field; and that the real numbers are an ordered subfield of the hyperreals.
The Transfer Principle
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To do the axiomatic definition of the hyperreal numbers, we need something called *the transfer principle*. I'm not going to go into the full details of the transfer principle, because it's not a simple thing to fully define it, and prove that it works. But the intuitive idea of it isn't hard.
What the transfer principle says is: *For any **first order** statement L that's true for the ordered field of real numbers, L is also true for the ordered field of hyperreal numbers*.
So for example: ∀ x ∈ ℜ, &exists; y ∈ ℜ : x ≤ y. Therefore, for any hyperreal number x ∈ ℜ\*, &exists y ∈ ℜ\* : x ≤ y.
Defining the Hyperreal Numbers
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To define the hyperreal numbers so that they do form an ordered field with the right property, we need to two things:
1. Define at least one hyperreal number that is not a real number.
2. Show that the *transfer principle* applies.
So we define ω as a hyperreal number such that ∀ r ∈ ℜ, r < ω.
What we *should* do next is prove that the transfer principle applies. But that's well beyond the scope of this post.
What we end up with is very similar to what we have with the surreal numbers. We have infinitely large numbers. And because of the transfer principle, since there's a successor for any real number, that means that there's a successor for ω, so there is an ω+1. Since multiplication works (by transfer), there is a number 2×ω. Since the hyperreals are a field, ω has a multiplicative inverse, the infinitessimal 1/ω, and an additive inverse, -ω.
There is, of course, a catch. Not quite everything can transfer from ℜ to ℜ\*. We are constrained to *first order* statements. What that means is that we are limited to simple direct statements; we can't make statements that are quantified over other statements.
So for example, we can say that for any real number N, the series 1,1+1,1+1+1,1+1+1,1+1+1+1,... will eventually reach a point where every element after that point will be larger than N.
But that's not a first order statement. The *series* 1, 1+1, 1+1+1, ... is a *second order* statement: it isn't talking about a simple single statement. It's talking about a *series* of statements. So the transfer principle fails.
That does end up being a fairly serious limit. There are a lot of things that you can't say using first-order statements. But in exchange for that limitation, you get the ability to talk about infinitely large and infinitely small values, which can make some problems *much* easier to understand.
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Clearly you mean that rings and fields have two operators, one of which forms a group, and the other of which almost forms a group except for the inverse.
billb:
Yes, you're right. I always have a bit of trouble remembering where the 0 restriction hits; I frequently get scrambled and think that the issue that the additive identity doesn't have a multiplicative inverse doesn't hit until fields.
Not only that, but also, can you define rings with non-commutative addition? I presume in a way you can, but will the structure you get make any sense?
I actually studied some Calculus using Hyperreals instead (or rather along side) of the old epsilon-delta. It did actually make some things easier to understand. I think it was Jerome Keisler who did this. Abraham ... somebody ... did the original work here as I recall (from 25 years ago).
You left something out of the definition of ordered field: not only does it need an order (like you said), but the order actually has to be related to the field operations. For example, if a ⤠b then a + x ⤠b + x.
... oops, hit Post too early. Also, 0 ⤠a and 0 ⤠b needs to imply 0 ⤠ab (ie, the product of nonnegatives is nonnegative).
Sorry to be the annoying guy bringing no real value, but there's an "&omega:" in your text, with a colon instead of semi-colon... Hope you can fix it and avoid more people like me taking valuable comment space. ;-)
Alon, as I learnt it a ring implicitly has commutative addition (i.e., that (R,+,0) is an abelian group) and that the label 'commutative' or 'non-commutative' is applied depending on the commutativity of the multiplication operation; with the default assumption being the more general case of non-commutative (for instance, the quarternions can safely be called 'a ring').
Of course, there are addition operations which are non-commutative, such as concatenation of strings- I don't know if you can then establish a multiplication to get a ring-like structure.
This note demonstrates that with a multiplicative unit and distribution of multiplication over addition, the addition will necessarily be commutative. So to break this you'd need a ring without a '1' (mult. id), which by most definitions is no ring at all. But I've heard speak of "a ring with 1", which implies that at least sometimes people think of rings without 1...
Alon:
The commutativity of the addition operation comes from the abelian constraint. You start with an abelian addition group, and then add the multiplication operation.
Dave:
Yes, you're right that I forgot to include the details of the ordering constraint on the ordered field.
I was trying to get through the algebraic definitions quickly (the definitions of abstract algebra weren't the point of the post, and I didn't want to discourage readers by spending too much time dwelling on it); I ended up going a bit *too* quickly.
@Markk:
∃ an online book on the subject, Elementary Calculus: An Approach Using Infinitesimals by H. Jerome Keisler.
From what I recall, rings don't need to have multiplicative inverses. For example, the even integers under normal addition and multiplication form a ring. As long as the multiplication is closed, associative, and distributes over addition, it forms a ring over the abelian group. The multiplication does not need to commute, either. For example, the set of 2 x 2 matrices form a ring.
Also a point from the main post: If you have an additive identity (0) and a multiplicative identity (1), and they are the same, there is only one element in the ring. IOWs, in any ring with more than one element and 1, 1 != 0. The proof is simple. Given any a in F:
0 = 0
0 = 0 + 0
0 * a = (0 + 0) * a
0 * a = 0 * a + 0 * a
a = a + a
0 = a
One Brow, the proof Graeme links to is based on additive inverses, which exist by the assumption that the structure forms a group under addition.
Graeme, I learned that rings were necessarily additively abelian, but didn't have to have 1. It helps think of ideals as special kinds of subrings. On the other hand, every ring without 1 is an ideal of a ring with 1, so it's a big tradeoff.
Markk:
Blakes reference on Keisler should contain this. I have heard somewhere that one simplifying point is that cheats of physicists becomes correct.
For example, the area of the circle. An infinitesimal added circle ring have area 2*pi*delta_r after invoking "welldefined Hyperreals on First order expressions". Presto, integrating 0...R over delta_r gives pi*R^2.
Going epsilon-delta needs changing integration and limit order, it seems. (Long time since last time...; and I can't get HTML symbol font to work, sigh!)
Should be 2*pi*r*delta_r going round the circle with radius r.
A small point; Abraham Robinson originally defined the Hyperreals axiomatically, but they can also be defined contructivly.
I have never seen the axiomatic construction before. I picked hyperreals up from Goldblatt's "Lectures on the hyperreals" (which I highly recommend). The constructive definition basically is similar to the Cantor approach to defining the real numbers as equivalence classes of rational sequences. Two rational sequences are called the same real number if the difference between their terms eventually gets arbitrarily small (in other words they converge to the same real number).
For hyperreals, two real sequences are considered the same if a 'large' number of terms of the sequences are equal. Large is defined using an ultrafilter on the natural numbers. An ultrafilter on a set X is a family of subsets of X which is closed under finite intersections, unions and has the property that for a subset Y of X, either Y or the complement of Y is a member of the ultrafilter.
A sequence is a function from the set of natural numbers to the set of real numbers. If two sequences are equal at a set of natural numbers, Y, and if this set is "large" (belongs to the ultrafilter F) then the two sequences are said to be equivalent. The equivalence classes under this equivalence relation are the hyperreals. Infintesimals are basically sequences that approach zero in the limit e.g. (1/2, 1/4, 1/8, ... ,1/(2^n)). Infinitely large hyperreal correspond to sequences of reals that approach infinity in the limit. The hyperreals also have sequences representing real numbers like for instance (1, 1, 1, 1, ...) which represents the real number '1'. There are also hypernaturals, hyperrationals etc which are infinity large hyperreals in which the sequences are natural numbers (1,4, 8, ....) or rational numbers (1/2, 5/2, 9/2,..). These hypernaturals, hyperrationals have similar properties to natural numbers and rationals.
The construction described above is called an ultrapower. There are a few problems with this construction. Firstly the ultrafilter F is arbitrary and is not explicitly constructed. The existence of an ultrafilter on N is inferred using the axiom of choice. However if the axiom of continuum is assumed it can be shown that the hyperreals as an ordered field are unique up to isomorphism.