Beyond Reality (for math geeks)

Factual Questions
1

There is a long-standing thread of a mathematical nature – I am sure you know the one. It has raised some questions in my mind that deserve their own discussion – questions related to extensions of the real number set.

I understand the possibility of extending the Reals to include infinitesimals and infinities and grok how it is logically possible. I lack any kind of experience in working within these systems, am not familiar with their properties and do not know how they work.

So for the questions:
[ol]
[li]The thread has mentioned the hyperreals (I think at least two systems exist here), the surreals, R((X)) and others. What are the salient features of these systems and how do they differ from each other? What notations are used to describe numbers in these sets?[/li][li]In extending the reals, what axioms are relaxed or added? (I realise that there is more than one answer to this question depending on the system being used.)[/li][li]What operations are possible in these systems that are not possible with the reals?[/li][li]What is lost by extending the set of reals? (By analogy, extending N to Q gives us division but we lose prime numbers and unique factorisation.)[/li][li]Are these systems useful for modelling real world applications? Or is there a good reason why I never encountered them during my engineering degree?[/li][li]Are these sets well-orderable? If so, how does that work?[/li][li]What is the cardinality of these sets? Is it c?[/li][li]What does analysis look like for these sets? Do they give us additional tools for differentiation for example? If we take something like the blancmange function that in the reals is everywhere continuous but nowhere differentiable, does it become more well-behaved by extending the domain to one of the alternatives?[/li][li]What is the density of these number sets compared with the reals?[/li][li]Consider a function on the reals such that f(x)=0 when x∈Q, f(x)=1 otherwise. I know that integral(f(x)) between 0 and 1 equals 1. What happens when I consider an analogous function on the reals and hyperreals?[/li][/ol]
I think that is the short list of questions. I am sure there are some that I have omitted, but I think that will be enough to get the conversation started.
J.

2

There are many possible answers to your question, because there are many possible extensions to the reals. One possible extension, that’s fairly easy to describe using familiar mathematics, is to go from real numbers to functions. Comparisons in this case are considered to be in the limit of large argument. For instance, we can have the function f(x) = 0, or f(x) = 1, or f(x) = pi, and these represent the familiar real numbers… but we can also have the function f(x) = x, and this function is greater than every real number, since there’s always some x such that f(x) is greater. Likewise, the function f(x) = 1/x is greater than 0, but less than every real number.

I believe this also answers your question #5: These numbers are extremely useful, and you did in fact use them in engineering, it just never occurred to you to call them “numbers”.

The cardinality of this set depends on just what functions are allowed. If all functions are allowed, then it has cardinality 2^c. But there are a wide variety of possible restrictions you might put on (only polynomials, or only rational functions, or only analytic functions, or…), and most of the restrictions you’d think of would leave the cardinality at c.

This set probably is not fully ordered. Some elements are definitely greater than others, and when that happens, you’ll get things you’d expect like transitivity of order… but some pairs of numbers are incomparable. For instance, sin(x) is neither asymptotically greater than zero nor asymptotically less than zero.

3

IANAMG (I am not a math geek), but why would the cardinality be the speed of light? :slight_smile:

4

Everybody knows that you cantor go faster than light.

5

Let’s see if I have this.
Under such a scheme, the concept of limit would be replaced by BigO. Is that right?

6

If this is a serious question, in the context “c” is the cardinality of the continuum.

7

This doesn’t follow, I don’t think. If the “comparisons are … in the limit of large argument,” then the number f(x) when f(x) = 1/x is not greater than zero as its limit is zero

8

Set theory isn’t really my field, but I’ll take a stab at some of these:

  1. In brief, the point (well, a point) of the hyperreals *R is that first-order logic statements for R (i.e., quantifying over individual variables in R insteaed of subsets of R) also hold for *R. For an explicit construction, recall that the R were constructed from Q by taking Cauchy sequences modulo sequences that converge to 0. Killing off every sequence that converges to 0 gets rid of infinitesimals; doing the same construction with R but killing off a finer set of sequences gives *R. (I’m skipping over the requirements you need to enforce to get an interesting thing here; if you’re interested, the keyword to look up is ‘ultrafilter’ or ‘ultrapower’). The surreal numbers are trickier, not really my field of expertise, and I don’t have anything more to say than what’s on the wikipedia page, so I’ll pass on them for now. R((X)) is the fraction field of the space of formal power series R[]. Since R is a field, an element f = a_0 + a_1 X + … in R[] is invertible iff a_0 is invertible; thus R((X)) = R[][X^{-1}] consists simply of series f = \sum a_n X^n (with the sum over all integers n) for which a_n = 0for n sufficiently small. It’s ordered (use lex ordering with the order coming from R and the degree map) and complete, but it contains infinitesimals: take X^{-1}, for example, which satisfies n X^{-1} < 1 for all positive integers n.

I’ll also mention in passing that “things that look like numbers” can include cardinals, ordinals, quaternions, octonions, and so forth.

  1. This is a vague answer, but: At that point, you’re really narrowly working in set theory or related areas, rather than something like analysis or topology. (Not that nonstandard analysis is completely useless in those areas, but I’m not aware of anything like, say, a theory of manifolds modeled on *R^n, as opposed to the incredibly productive theory of real or complex manifolds, or even algebraic geometry.) There are more interesting extensions of Q for those purposes: algebraic number fields, p-adics, etc. Nothing in math is completely isolated from anything else, of course, but you’re going to have to give up on quite a bit of analysis, geometry, topology, etc. when you switch into the category of surreal numbers.

  2. Not really, though nonstandard analysis can be useful in ordinary real analysis. In terms of applications, they’re mostly of interest to set theorists, game theory, or (through forcing) number theorists and arithmetic geometers. Complex numbers are ubiquitous, of course, and quaternions have tons of applications as well. For the hyperreals, surreal, etc., probably not. Certainly they aren’t things I’d expect to be covered in a standard engineering degree.

  3. Do you mean totally ordered? Well-ordering means that every nonempty subset has a least element, which isn’t even the case for Z or Q. The hyperreals are totally ordered more or less from the definition of an ultrafilter (which is part of the machinery above I omitted). The idea is to take (a_1, a_2, …) <= (b_1, b_2, …) iff the set {i: a_i <= b_i} lies in the ultrafilter.

  4. The hyperreals *R have the same cardinality as R, which follows from unravelling the construction above. The surreals are huge; they contain all the infinite ordinals, for example.

  5. What exactly do you mean by ‘density’ here? The examples above have a natural embedding of R into them, but I’m not sure what kind of measure of density you would want on the result.

9

In addition to hyperreals, surreals, etc., here are two other constructions that you might find interesting:

  • The quaternions H are a four-dimesional non-commutative real algebra. Over the usual basis {1, e[SUB]1[/SUB], e[SUB]2[/SUB], e[SUB]3[/SUB]}, we have e[SUB]i[/SUB]e[SUB]j[/SUB] = - e[SUB]j[/SUB]e[SUB]i[/SUB] for i != j and e[SUB]i[/SUB][SUP]2[/SUP] = -1. They’re closely related to SO(3) and SU(2), and so pop up constantly in quanutm mechanics, computer graphics, etc. If you think of complex numbers as C = R + iR with i^2 = -1, then the quaternions are H = C + jC with j^2 = -1 and ij = -ji.

  • The octonions are an extension of the quaternion that have dimension 8 as a real “algebra.” I use the quotes there because even though the quaternions aren’t commutative, octonions aren’t even associative, which limits their usefulness. Still, they pop up from time to time: some of the exceptional simple Lie algebras can be constructed as groups of their isomorphisms, and there’s a brilliant construction that uses them to construct a space that is homeomorphic to but not diffeomorphic to S^7.

The sequence doesn’t quite continue; there isn’t another normed division algebra over R beyond the octonions. (It is possible to get something, but you’re really scraping the bottom of the barrel in terms of algebraic structure.) This fact is very closely related to the fact that the tangent bundle on S^n is trivial only for n = 0, 1, 3, 7. (It’s easy to prove that n must be of the form n = 2^k; it’s harder and requires more sophisticated machinery to prove that only 2^0, 2^1, 2^2, 2^3 work.)

10

Thanks Itself. I will have a close look at these. I might not get the serious brain space for a couple of days, just looking at my schedule.
I have come across quaternions and octonians before. They struck me as useful in some situations, but little different from a certain class of matrices. Losing the commutativity and the associativity as you say are large costs.

By density (q9) I guess I was thinking alongs the lines I asked in q10. Although between any two rationals there are an infinite number of irrationals, and between any two irrationals there are an infinite number of rationals, the cardinality of the irrationals is greater than the rationals leading to certain consequences such as the one I noted in q10. I was merely wondering if the same kind of thing occurred with some of the extensions to the reals.

If I understand Chronos’ contribution properly, some of the extensions to the reals are not numbers in the way that I have normally thought of them and so it is not really a comparison of like objects. Apples and oranges kind of thing.

11

Yeah, I was going to mention the quaternions in the other thread - because there are an infinite number of quaternions that satisfy the equation x*x=-1.

12

But 1/x is always greater than zero for large x. To more precisely state the comparison rule: “f is said to be greater than g (or alternately, g is less than f) iff there exists a number X0 such that, for all x > X0, f(x) > g(x)”. In the case of comparing f(x) = 1/x with g(x) = 0, it suffices to use X0 = 0.

Incidentally, I weasel-worded about the orderability because it depends on the sorts of functions allowed. If, for instance, we allow only polynomials, then we can get transfinite numbers, but no infinitesimals, the cardinality is the same as the reals, and we have total ordering. If instead we extend our polynomials to allow terms of negative order (e.g., x^2 + x + 1 + x^(-1) ), or to rational functions (i.e., ratios of two polynomials), then we can get transfinites and infinitesimals both, while (I think) retaining the total ordering.

One can view this as a statement about the mathematical objects themselves, or as a statement about how you have normally thought about them. But they’re really not as dissimilar as you’re thinking. You can accept complex numbers as numbers, apples-to-apples with real numbers, right? A complex number is just an ordered pair of real numbers, with a few extra rules for working with them. But these polynomial-numbers can likewise be cast as ordered n-tuples of real numbers, with an arbitrary value of n. If pairs are OK, why not other tuples?

13

Since the OP considers infinitesimals and infinite numbers, I will ignore things like the complex numbers and quaternions. You can adjoin i or i,j,k to any model of the reals and get non-standard models of C and H.

Now, there are many ways of doing so. Here is an especially simple one. Consider all “convergent” sequences of ordinary reals (allowing convergence to infinity, whence the scare quotes). Among the convergent sequences, the ones that converge to 0 are called infinitesimal. The inverses of infinitesimal are called infinite; the rest are finite. You still cannot divide by 0, but you can divide by non-zero infinitesimals. Every finite number is an ordinary (its limit) plus an infinitesimal. The real problem is taking a function, say f: R --> R and extending it to a function f*: R* --> R* (R* being these extended reals). Essentially, you can do this in a sensible way if and only if f is continuous (which means takes convergent sequences to convergent sequences). Assuming f is continuous, you can define Df(x) as the ordinary part of (f(x+h) - f(x))/h for infinitesimal h, provided this ordinary part doesn’t depend on h.

Is this set ordered? No, the sequence 1,-1/2,1/3,-1/4,… is neither greater than nor less than 0. While this model is enough to show that infintesimals do not give a logical contradiction, it lacks a number of properties you would like, most important the ability to extend a function. Notice that even if f is such a function, Df need not be, even if defined for all x.

The most important non-standard model is something called an ultrapower of R. It is based on dividing all subsets of the positive integers into two classes, which I will call small and large. This division must satisfy:

  1. Every finite set is small.
  2. The union of two (or any finite number) of small sets is small.
  3. The complement of a small set is large and vice versa.

The existence of an such a division (technically called a non-principal ultrafilter) requires the axiom of choice, so don’t ask me to produce one. Now you define R* to be the set of all sequences of reals but declare a_1,a_2,a_3,… = b_1,b_2,b_3,… provided that {n | a_n = b_n} is large. No convergence is needed. It gets complicated. Check out wiki for more information.

14

How is that any different from just saying “finite” and “infinite” instead of “small” and “large”? Every finite set is finite, the union of any finite number of finite sets is finite, and the complement of a finite set is infinite.

15

Given an infinite set with infinite complement, you have to choose of the two for one to be “small” and the other to be “large”.

There’s no problem extending discontinuous functions; they just happen to have outputs whose ordinary parts do not solely depend on the ordinary parts of their inputs.

16

Oh, I’m sorry, I forgot the model Hari Seldon was using in that sentence, having set up the restriction to only “convergent” sequences.

17

I would use the word “hyperreals” for any elementary extension of the reals; that is, any extension of the reals such that any sentence formed from quantifiers (“for all”, “there exists”), Boolean combinations, real constants, and basic operations and relations is true in the extension just in case it was already true in the reals.

Exactly what this amounts to depends on what our “basic operations and relations” are; if we take these to be addition, multiplication, and ordering relations, then it turns out to be equivalent to being an extension of the reals which is still a “real closed field”; that is, an ordered field satisfying the intermediate value theorem for polynomials.

The surreals are a particular example of a real-closed field extending the reals (and thus, in this sense of the term, a particular system of hyperreals); they’re what you get if you want a “class”-sized ordered field which contains every “set”-sized ordered field.

R((X)) (i.e., the formal Laurent series with real coefficients) is a different extension of the reals. Specifically, it is what you get if you add to the reals a particular positive infinitesimal ε, along with a sum for every series of the form “the sum, over all integers n from m on up, of r[sub]n[/sub] ε[sup]n[/sup]”, where the coefficients r_n are real. This is an ordered field, but not a real-closed one (as, for example, ε is positive with no square root); we can make it real-closed by also allowing for fractional exponents (as formalized minimally by Puiseux series, or more liberally by Hahn series).

We can consider more basic operations and relations than just those in the language of ordered fields. The ultrapower constructions described by others produce elementary extensions supporting every operation and relation already available for standard reals.

In fact, we can break the ultrapower construction down into two parts: the first part is the observation that, for any D and X, functions from D to Xes, manipulated componentwise, act an awful lot like Xes themselves. So, for example, if we have a standard notion of the mathematical universe U, then we can also make a mathematical universe U’, where elements of U’ are sequences of elements of U.

In this new universe, the type of Booleans (“true” or “false”) will amount to sequences of “true” or “false”. Thus, we might say that the claim that the real <0, 1, 0, 1, 0, 1, …> is positive is neither straight-up “true” or straight-up “false”, but rather, <false, true, false, true, false, true, …>.

In the second part of the ultrapower construction, we can seek to turn nonstandard Booleans into standard Booleans. A coherent way of doing this is an ultrafilter, as described by others above: it assigns to every nonstandard Boolean either “true” or “false” in such a way as respects all the Boolean operations.

We don’t have to go all the way to ultrafilters, though. There’s also a notion called a filter, which just amounts to a way of saying some nonstandard Booleans should be considered equal, without necessarily going so far as to say every nonstandard Boolean is equal to one of <true, true, true, …> or <false, false, false, …>. Chronos’s first post in this thread is along those lines: we could take our mathematical universe to be given by functions from reals to the standard universe, and we could say that two nonstandard Booleans should be considered equal so long as they are equal from some point on. (Note that in this model, every hyperreal is the result of applying some standard real-to-real operator to the particular infinite hyperreal given by the identity function; similarly, the models based on sequences have the property that every hyperreal is the result of applying some standard natural-to-real operator to the particular infinite hypernatural given by the sequence <0, 1, 2, 3, …>).

18

It’s also worth noting that even when people claim to use ultrafilters, they are in practice only using filters: they’ll say, for example, “I want ANY ultrafilter with the property that every eventually-true sequence counts as true”, and then proceed to talk about statements which they can prove true without worrying about which such ultrafilter was selected. But the statements which are guaranteed to come out true for ANY such ultrafilter are precisely those which would already come out true just using the (non-ultra) filter of eventually-true sequences.

19

Thanks for all of the answers so far. Much appreciated.

I guess this thread is going to suffer some of the same drawbacks as numerous other mathematical threads in that the one asking the questions does not have the necessary language to frame the question adequately. So thanks for bearing with me here.

There is more information given here than I can probably handle at this time, so let’s just limit discussion to the extended reals as described by Hari Seldonat this stage. Is it right to simply refer to these as hyperreals? Or is that too vague a term?

[Aside]
I am familiar with complex numbers of course. And know about quaternions and octonions. One of the appeals of the real numbers is the ability to visualise and estimate in my head – this gives a sense of familiarity. Complex numbers I can do something with, but If I ever again needed to be calculating z[sub]1[/sub]^ z[sub]2[/sub], I would begin by deriving the algorithm for myself from first principles and playing with some Argand diagrams to gain that sense of familiarity.

So, my questions did not really relate to extending the Real numbers to polynomials, functions, quaternions or any other “multi-dimensional” kinds of numbers. What was playing in my head while reading the 0.999=1 thread is that it was messing with my notions of continuity. I mean, if the Reals are already continuous, how is it possible to have something that is even more finely divided? (For want of better terminology.)

I vividly recall, a long time ago now discovering for myself the difference between Q and R. I hadn’t been taught any set theory formally and pretty much any context I was using mathematics was with the real numbers. (At the time I also had a bit of a fetish for fractals.)
I was playing with a nice little relation, y[sup]x[/sup]=x[sup]y[/sup]. Obviously, trivially, y=x is part of this relation. The interesting part of the curve is something looking superficially like a hyperbola with asymptotes at y=1 and x=1. I found a way of calculating rational solutions to the equation. For each solution I found, there was a corresponding solution in the third quadrant. However, when I first got hold of some (rather primitive) graphing software it gave solutions only for positive numbers. Actually better than that – the software worked by shading out regions. Therefore it initially drew a line in quadrant three which then dissolved to nothing. There were negative rational solutions to the equation, but in general, a negative real raised to a real power is undefined in the real numbers.
It then dawned on me that the real numbers were infinitely more numerous than the rationals. In my own words at the time, the rational numbers were perforated and had kind of a fractal structure. It was an independent discovery, and looking back, something of a glorious moment.
[/aside]

So, Itself and others, knowing that Q is embedded in R but the cardinality of R is greater, and reading about number sets that included infinitesimals – where two numbers that were equal in R may have a difference, this prompted my questions about how that all works. This is what I was driving at when I asked about density.
[ul]Do the hyperreals have a greater cardinality? Apparently not.
[li]Do they have some kind of hyper-continuity that allows for greater flexibility in analysis - including the ability to differentiate otherwise undifferentiabe functions? Again, apparently not.[/li][li]Do the hyperreals have uses in the physical world - the continuity of the real numbers seems to be sufficient in any applications I have encountered. Again, apparently not.[/li][li]Comparing the number sets that I am familiar with N, Z, Q, R, C – there are usefulnesses that are gained and lost as one progresses from one to the other. I was wondering what the gains and losses were for sets that allowed infinitesimals.[/ul][/li]
Hopefully this gives a bit of background to the questions I have asked and inserts a bit of meaning behind my bumbling attempts to describe what it is I am asking.

20

The reals are characterized as the only complete Archimedean totally ordered field. Field just means the usual arithmetic operations of addition, subtraction, multiplication and non-zero division. Totally ordered means that any two elements are comparable. My construction using convergent sequences was not, as I mentioned, totally ordered. Archimedean means that for any element x, there is a natural number n such that nx > 1. The non-standard models will not have that property. Any x for which that fails is an infinitesimal.

As for utility, it can be shown that anything that can be done with infinitesimals can be done without them (“infinitesimal calculus is a conservative extension”). But it might be more complicated. Anyone who has pored over an epsilon/24 argument will know what I mean. Also taking standard part is somehow a more natural concept than taking limits. All those epsilons and deltas are implicit in the construction of the non-standard models.

Very importantly I should point out that it is not necessary to actually understand the ultrapower to use infinitesimals. We teach calculus without talking about Dedekind cuts and construction of the reals. All you have to know is that there are extended reals and they include infinitesimals and otherwise have all the usual properties. Also all functions will be defined on these extended reals. The ultrafilter construction is needed only to verify the consistency of the model.