Where mathematics starts out

Miscellaneous and Personal Stuff I Must Share
1

It’s time to fight some ignorance in my field of specialty - mathematics. A lot of people have some mathematical problems that aren’t so much mathematical as philosophical in nature but look like math, and a lot of people just “hate math” for whatever reason. So I’m starting up this thread, which I’ll get back to at least once a day as long as people are interested, and people with questions about mathematics can ask them here and I’ll try to answer.

The first thing I’m going to do here is to explain where to start in mathematics. Some people get ideas in their heads, and they think it’s math, but a mathematician says, “That’s not math, that’s philosophy,” which really tells the poor person nothing. Well, It’s time to define what mathematics is. If you want to know the fundamentals of mathematics, this is the thread for you.

Mathematics is a logical system based on certain axioms that uses the rules of predicate calculus to explore the nature of sets and classes.

But what the heck do I mean by that? Isn’t math supposed to be about numbers? What is a set, or a class? And what’s predicate calculus?

Well, when it comes down to brass tacks, a number is a certain type of set. A class is, well, this is where mathematicians sort of wave their hands and say, “Use your intuition”. In mathematical terms, a class is something that has elements, and a set is a class that satisfies the basic axioms. (An axiom is a statement that is assumed to be true. Different flavours of mathematics are arrived at by adding or subtracting axioms to or from the basic set.) This is the best definition of a set we can possibly get. Don’t worry about “predicate calculus”; it’s just a fancy name for the basic rules of logic, and if someone wants to start a parallel thread about it, be my guest. If I try to get too formal, I’m going to cloud the beauty of the mathematics.

So what are the basic axioms of mathematics? To the best of my recollection:

Axiom of Extensionality. If A is a set and B is a set, then A = B if and only if for every x, x is an element of A exactly when x is an element of B. (In short, two sets are the same if they have the same elements. So, for instance, the sets {a}, {a,a,a}, and {x | x = a} (read this as “all x such that x is equal to a”) are all equal; in a mathematical sense, they’re the same set, just written in three different ways. From a purely mathematical point of view, there is no way to distinguish them.)

Axiom of Foundation. There exists a set that has no elements. (This one is sometimes known as the Empty Set Axiom, for obvious reasons. So {} is a set. Furthermore, by the Axiom of Extensionality, it’s the only set with no elements, because if there’s some other set that has no elements, it’s automatically equal to the empty set by the previous Axiom.)

Axiom of Specification. If A is a set and P(x) is a statement that is either true or false for every x, then there is a set B whose elements are exactly the elements x of A for which P(x) is true. (This means, if you start with a set A, and you want to consider all the elements of A that have a certain property, you’re guaranteed to be looking at a set.)

Because of the Axiom of Specification, we can in fact replace the Axiom of Foundation with the less specific statement: “A set exists”. Since a set A exists, then by the Axiom of Specification there is a set B that consists of all x in A for which the statement “x does not equal x” is true. But everything is equal to itself by the Axiom of Extensionality, so B has no elements.

The Axiom of Specification also gives you set intersections. If you have two sets A and B, and you want to get their intersection, you take the elements x of A satisfying the statement, “x is an element of B”.

Pair Set Axiom. If A and B are sets, then there is a set C whose elements are exactly A and B. (In other words, {A,B} is a set whenever A and B are sets. Exercise: Starting with just one set A, how can you use the Pair Set Axiom to get the set {A}?)

Union Set Axiom. If A is a set, then there is a set UA satisfying: x is an element of UA if and only if, for some element a of A, x is an element of a. (So you get the normal union operation A U B, the set consisting of all elements of A and B, as follows. Start by using the Pair Set Axiom to give you the set {A,B}, then use the Union Set Axiom on {A,B} to get A U B. How do you get A U B U C?)

The above are the axioms that almost every mathematician in the world will agree to. The following are the standard axioms that invite controversy from time to time, in increasing order of controversy.

Power Set Axiom. If A is a set, then there is a set P(A) satisfying: x is an element of P(A) if and only if x is a subset of A, i.e., every element of x is an element of A.

Axiom of Infinity. There exists a set A satisfying the following conditions:[ul][li]The empty set is an element of A.[*]If x is an element of A, then there is an element y of A that is a set and whose only element is x.[/ul]Axiom of Replacement. Suppose A is a set and P(x,y) is a statement satisfying: for each x in A and for all y,z, P(x,y) and P(x,z) are both true only if y = z. Then there is a set B satisfying: y is an element of B if and only if P(x,y) is true for some element x of A.[/li]
The last axiom is the Axiom of Choice, and every statement of this axiom that I’ve seen involves ordered pairs, so I’m going to leave that for tomorrow’s ramble.

What’s so tricky about ordered pairs, you might ask? Well, go back to the axioms and have a look. By the Axiom of Extensionality, the sets {a,b} and {b,a} are the same set, and we can’t tell just by looking at it which element is supposed to be the first element of the ordered pair and which is supposed to be the second. We have to be a bit smarter, and I have to date seen three ways to do up ordered pairs, each of which has their advantages. I’ll go over the easiest-to-explain one tomorrow, but from a logical point of view it usually doesn’t matter which one you use.

So tomorrow: The Axiom of Choice, and we’ll get started on numbers (finally!):slight_smile: Until then, start the questions coming!

2

I can’t offer the same level of expertise as MrDeath, but I do know some math, and have a taste for philosophy. I’ll try to help him out, answer what I can. I can check back more than once per day, and that’s the main thing I bring to the table.

In the event that he and I disagree on something, go with what he says. :slight_smile:

3

Oh goody. My father’s a mathematician (and a quite articulate one), but I never seem to remember the right questions when he and I are together. So here’s one, provoked by that other thread (and forgive me for my bad phrasing):

Tell me about infinity and infinitesimals (is that a word?). Our discussion was of 0.999…=1, and its predicate, .000…1=0. Let’s take a look at the second. I have an intuitive sense (the most pernicious of senses in math, it seems) that an infinitesimal shouldn’t equal zero - that it should, well, exist. Yet, if infinity is to be meaningful, and we stick infinite zeros before that 1 (or 2 or 13,762), by definition we never get to that 1 or 2 or 13,762.

It seems too pat. I understand it, and I accept it for mathemitcs, but it curdles my tummy and I start wondering about the difference between insanity and a minority of one.

What happens if we assume that an infinitesimal must be greater than zero?

4

Good 'ole limits!

The limit of (1/10[sup]x[/sup]) as x->infinity

equals zero!

Simplified, 0.00…1 = 0

Problem solved.

5

Is this going to be on the test?

6

Yes, Mr. Smartypants, there will be a test. Thing is, you get to write it yourself.
Peace,
mangeorge

7

Woo hoo, I passed! I am vindicated! I got an F on my Calc 201 final exam, I had an exam conflict and had to take it at an alternate time. Unfortunately, there was a fire in the building. As I frantically tried to complete the test amid the clamoring klaxons, a security guard came in to check the room and told me to leave. I refused because the instructor was coming back and he said if I left the room I’d get an F. He pulled out a nightstick and MADE me leave. Turned out that the fire was pretty serious, I could have been killed.
I argued with the professor, told him the story, and said I’d risked my life to complete his damn test. He relented and gave me a D+ for the course. I passed.

8

So, this differs from philosophy how?

9

Part of the difference lies in the origin of mathematics; math used to be much more of a “hard science”.

Math wasn’t always just a matter of manipulating formal symbols; originally math was much more like physics. By that I mean that the rules of mathematics were modelled on the behaviour of systems in the real world. Set theory, for example, wasn’t always defined by the axioms MrDeath listed; originally a set was just another word for a collection of things.

But starting roughly at the turn of the 20th century, a number of mathematicians started noticing real problems with the “common sense” definitions; in set theory and logic those problems took the form of paradoxes. (For example, does the “set of all sets which do not contain themselves” contain itself?) In order to put mathematics on a more stable foundation, things like Axiomatic Set Theory (what MrDeath is describing) were developed. It must have been a heck of a time to be a mathematician, as people like Russell, Cantor, and Godel were yanking the rugs out from everybody’s feet.

Carry on…

10

But seriously, getting back to the OP, I have to wonder about your whole premise that “Mathematics is a logical system based on certain axioms that uses the rules of predicate calculus to explore the nature of sets and classes.” Seems to me you’re describing set theory and not mathematics as a whole. The mathematical realm got pretty far before it ever encountered set theory, and mathematicians like Goedel punched big holes in any attempts to write a Principia with a set of logically consistent axioms based on set theory.
So maybe you can explain this to me, and remember, I passed Calc 201 (albeit 25 years ago).

11

Thanks for your insightful questions. Let’s take them in roughly the order in which they shuffle into my mind.

And the last shall be first. Chas.E: Much like (in an extremely general sense) all of science is physics, all of mathematics is set theory. Yes, there was mathematics, a lot of really good, profound mathematics, before set theory was discovered; but it wasn’t mathematics as we know it today. They had lines, and planes, and derivatives, and polynomials, and numbers, and sequences, and whenever someone asked, “Well, what is a point?”, or “…polynomial?”, or “…derivative?”, all the mathematician could say is, “Well, er, um.” Now, we have turned all these into various kinds of sets, and the only unanswerable question that we have left to answer is, “What is a class?” Most people see this as an improvement. Set theory gave mathematics a foundation, and made its results philosophically unassailable, if you’re willing to accept the validity of predicate calculus and the consistency of the axioms I described in the OP (which is much less than any other belief system will ever ask you to swallow). As to how we define these things in terms of sets, I’m going to start on that tomorrow.

Since you made it through Calc. 201, I’ll give you another example. You may have heard of group theory, a particularly rich, vibrant area of mathematics. At some point, someone realised that every group is (isomorphic to) a permutation group, which is a very specific way of describing a group. Therefore, group theory is the study of permutation groups. However, most group theorists don’t look at a permutation group from one year to the next, and we don’t call it permutation group theory, we call it group theory. But it’s still a nice way to approach it sometimes. Likewise, most mathematicians don’t work with the nuts and bolts of how sets interact; they’re much more interested in Hilbert manifolds and wavelet analysis and Henselian valuations and tensor algebras, but if you pry it all apart into itty bitty pieces, it’s all sets and classes at the bottom. And that’s why I answered the question “What is mathematics?” in the peculiar way that I did.

dropzone: “Mathematics is an inexpensive discipline; the only materials required are paper, a pencil and a wastebasket. And when mathematicians turn philosopher, they don’t even need wastebaskets.”

Mathematics differs from philosophy as quails differ from birds. Mathematics restricts itself to statements it can prove or disprove about sets and classes, starting from the axioms of your choice (this usually includes the axioms I described above and sometimes a few others as needed). Others, usually non-mathematicians, have other definitions of what mathematics is, but certainly everything that is usually considered mathematics can be turned into statements about sets and classes.

Philosophy examines, in addition to sets and classes, everything else: God, strawberries, war, joy, sentence fragments, and what colour next Thursday is. The axiom system used by one philosopher will often conflict with that used by another, and most of the axioms are of a somewhat more complex nature than, “A set exists”. Mathematics is often used to describe certain aspects of some aspects of science, which is another branch of philosophy, but it is important to keep the mathematical concept of a probability cloud distinct from the philosophical/quantum mechanical concept of a proton shell, for instance. Mathematics (as we know it today) is deliberately kept simple and uncluttered, the axioms as self-evident and general as possible, to minimise the holy wars fought over its foundations. Besides, even with this minimalistic set-up, there are literally uncountably many things to explore (and a surprisingly large number of arguments).

Soup_du_jour, OxyMoron: Except for the fact that I have a wife, child and sleep cycle to keep happy, I’d be writing out answers to your questions right now. I expect that after I’ve gone a ways on numbers, I’ll be able to tackle limits. You might want to come back in two days, because I don’t expect I’ll get to limits before then.

12

Sorry about this, folks. Real life intervenes again. I’ve started to write up some stuff, but I can’t finish it, and I don’t want to post it until it’s complete. It’ll be tomorrow then.

13

That’s okay. I was back pondering how, until a couple months ago, I had always thought of Russell solely as a philosopher. Then I was reading some books on AI and the history of computing and up pops Bertie as the King of the Set Theory and I say to myself, “I didn’t know that! I thought that he was just a simplistic explainer of Einstein. Why the hell were they teaching us set theory in the fourth grade when I didn’t find a use for it until the class in database design thirty years later?”

Yeah, one problem with me and math is staying on topic. I’ll leave the floor to thems what knows what they’re talking about and will print this thread out so I can stare at it some.

14

For those who prefer a hardcopy of this type of stuff, I’d recommend Paul Halmos’s Naive Set Theory. This book, which will probably cost about $40 US, is considered to be a classic in the field. It’s written for non-set theorists and can be followed by an interested layperson (although some of the later sections might be a bit much).

Anyway, carry on.

15

Huff huff

Wait, you mean I’m only good enough for…Naive Set Theory?

Huff!

(OK, it’s a term of art, I’m sure, but still…)

And Mr. D., take yer sweet time, it’s not like infinity’s going anywhere or anything :).

16

It’s naive cause it’s in English. :slight_smile:

17

So is is the set of non-polymonial equations a proper subset of the set of polynomial equations.

18

Can’t be. No non-polynomial equation is a member of the set of polynomial equations.

19

ultrafilter’s recommendation isn’t a bad one, but I personally found Halmos’ approach (and the whole concept of “naive set theory”) a bit too mushy. I do like the idea of starting from stuff people have an intuitive feel for, but I also prefer to start building my castles on the ground. I’m trying to do both here.

dropzone: If you’re really a big Russell fan, you might want to hunt down a copy of his Principia Mathematica. I haven’t read it myself, but I’ve heard good things about it. Russell isn’t really the “king” of set theory, but he made the observation (Russell’s Antinomy, often called Russell’s Paradox) that really got the ball rolling.

The term “naive”, by the way, is intended to set the approach apart from “axiomatic set theory”, which is what I’m presenting. Naive set theory is what was done between 1873, when Cantor started seriously looking at sets, and 1902, when Russell’s Paradox brought it crumbling down. Basically, up until Russell, the basic idea was, “If you can specify it, it’s a set.” So people wrote about things such as the set of all sets and other things that we know today are “proper classes”, or classes that don’t satisfy the basic axioms. In 1902, Russell wrote a letter to Gottlob Frege, another mathematician/philosopher, and I don’t have the exact quote in front of me, but he basically said, “What happens when you consider the set whose members are those sets that are not members of themselves?” Simple enough concept, right? Let’s call this “set” X. Now every set Y, the statement “Y is a member of X” must either be true or false, right?

Well, hang on a sec. What about X itself? Suppose X is a member of X. Then, by the definition of the set, X is not a member of itself. Likewise, if we suppose X is not a member of X, then it has to be a member of itself. This is a massive, serious contradiction, and getting around it is a bit tricky. One way is the way I’m presenting, in which we introduce axioms, and we say that things that satisfy the axioms and are consistent are sets, and things that don’t satisfy the axioms or are inconsistent are proper classes. X, being inconsistent, cannot be a set.

Exercise: Show that the class {X | X is a set} is not a set.

Moving right along. . . .

So we know what how to manipulate sets. How do we turn this into mathematics?

Well, the first thing that’s usually on the table is “functions”. But in order to describe a function, we need an ordered pair.

The concept of an how an ordered pair should work is simple: you want to have (a,b) = (c,d) if and only if a = c and b = d. This means that, for instance, if a != b, then (a,b) != (b,a). (I’m going to use the symbol “!=” to mean “is not equal to”.) So we can’t just use the set {a,b} to represent the ordered pair, since {a,b} = {b,a}. You can’t even get away with something fancier like {a,{b}}. To see that this doesn’t work, let’s say that a is itself the set {c}; so {a, {b}} = {{c},{b}}. Now is this the ordered pair ({c},b) or the ordered pair ({b}, c)? It’s still not strong enough.

Something that is strong enough is the “Kuratowski ordered pair”, which defines (a,b) to be the set {{a}, {a,b}}. I’m not going to prove that this works as advertised, but it’s not that difficult. This is not the only way to describe ordered pairs; another one that was used before Kuratowski’s idea was {{{a}}, {{},{b}}}, which also works but is uglier. A third one that is used by some mathematicians is trickier to describe but has the nice property that when you go to ordered n-tuples, the “level” of the set (the number of nested brackets needed to write out the set) does not increase over the ordered pair.

Enough about that. We want to define a function. Intuitively, a function is a “map” f that matches elements in one set (the “domain”, D) with elements in another (not necessarily distinct) set (the range, written either as R or as f(D)). This is exactly where ordered pairs shine. We say a set f is a function if:[ul][li]Every element of the set is an ordered pair; andIf (a,b) and (a,c) are elements of f, then b = c.[/ul]The domain of f is the set {a | for some b, (a,b) is an element of f}, and for the range just replace (a,b) with (b,a). If (a,b) is some element of a function f, then we write f(a) = b. This should make the “mapping” operation pretty clear, and this is why we require that any given element not occur more than once as the first element of an ordered pair; we don’t want one “point” mapped to two or more destinations.[/li]
Now we’re ready for the controversial but oh-so-useful Axiom of Choice. There are at least a half-dozen statements that are all equivalent, so I’ll give one that I hope looks pretty obvious:

If A is a nonempty set, then there is a function f whose domain is P(A), and whose range is a subset of A, such that the statement “f(a) is an element of a” is true for every element of P(A) except for the empty set.

I’m not going to get into why it’s controversial, but it sure is mighty useful, which I might get into later on. I will say that there are mathematicians who deny the validity of the axiom of choice, and from a logical standpoint it turns out they are just as correct as those who affirm it; however, there are many things one simply can’t do without using it. Just as one f’rinstance, ordinal numbers are way messier if you don’t use the axiom of choice.

I’m breaking this post in two, because it’s getting awfully long; however, I’ll be posting the next one soon, i.e., later today. Next post: Numbers! (Finally! :))

20

Oh, and I forgot to say how the axiomatisation resolves Russell’s Paradox, so I’ll do that quickly. Up in my previous post, I said that the statement “Y is a member of X” has to be true or false for each set Y. This is still the case, but since X is no longer considered a set, but a “mere” proper class, we no longer need to know whether “X is a member of itself” is true or false. It’s a bit ugly, but on a purely logical level, it works, like sweeping all the dust under the bed works to make your bedroom look clean.

Numbers. We all know what a number is, do we not? Let’s list some things that numbers satisfy.[ul][li]The “counting numbers”, 1, 2, 3 and so forth, are all numbers.[]You can add numbers together, to get another number.[]You can multiply numbers together, to get a number.[/ul]Let’s continue with some statements that at least one prominent mathematician over time has disagreed with:[ul][]Addition and multiplication of numbers are commutative, associative and distributive.[]0 is a number.[]Whenever you subtract two numbers, you get another number.[]Whenever you divide two numbers, you get another number, as long as the denominator isn’t zero.[]Every number has an additive inverse, i.e. for every number n there is some number -n satisfying n + -n = 0.[]Every number except 0 has a multiplicative inverse, i.e., if n != 0, then there is some number n[sup]-1[/sup] satisfying nn[sup]-1[/sup] = 1.[]The ordinals are numbers.[]Whenever you take a limit of numbers, you get another number.[]Whenever you take a Dedekind cut of numbers, you get another number.[]You can order all the numbers.The solutions of a polynomial equation are numbers.[/ul]I present the list first, although it has a bunch of things we haven’t defined yet. And, it turns out that the last two statements are contradictory - you can’t order the set of solutions of polynomial equations.[/li]
Specifically, most of the things above are more examples of axioms. Since every axiom is, in a sense, a restriction, it makes sense that, the more axioms you put on, the fewer things you have that satisfy them all. So, for instance, there are many sets (things satisfying the set axioms) that are not numbers (things satisfying the number axioms.)

There are many things that satisfy some of the above axioms. For instance, the “natural numbers” satisfy some of them, the rational numbers satisfy more yet, the complex numbers satisfy even more. If we leave off the very last axiom, there is a very nice class that satisfies everything in the list (because of this, mathematicians say it isn’t algebraically complete). But given that it satisfies everything else, and that it can be “completed” very easily, it’s a very interesting structure to study. These objects are the “surreal numbers”, which were discovered in 1970 by John Conway, but from here on in we’ll just call them all “numbers” because it makes things easier to write. Also, I have never seen something referred to as a “number” that you can’t get by some sort of completion of the surreals.

We shall define a number, then, to be an ordered pair (L,R) satisfying:[ul][li]L and R are sets of numbers; andevery number in L is less than every number in R.[/ul]That’s it. Well, that’s not it it, we still have to define what me mean by “less than”, not to mention addition and stuff, but that’s all you need to get things like real numbers and ordinals and much, much more.[/li]
First, we shall define what it takes for numbers to be “alike” (I stole the term from Donald Knuth, who introduced it in that little book I referred to in my first post). Most people would call this “equal”, even most mathematicians, but I prefer to use a distinct word, because numbers can be alike without being “equal” in the set-theoretic sense. Remember this! I shall use “==” to mean alike, and “=” to mean set-theoretic equal.

The definition of alike is:

(A,B) == (C,D) iff (A,B) <== (C,D) and (C,D) <== (A,B).

“<==” is “less than or alike”, which is defined as:

(A,B) <== (C,D) iff a < (C,D) for all a in A, and (A,B) < d for all d in D.

And finally, “<” is “less than”, which is defined as:

(A,B) < (C,D) iff (A,B) <== (C,D) and (C,D) !<== (A,B).

BUT WAIT! Aren’t these definitions circular? I mean, we define less-than-or-alike in terms of less-than, but then we define less-than in terms of less-than-or-alike! This would be a problem, except that we can define them recursively! Look closely. In the definition of less-than-or-alike, you’ll see that we define the relationship between (A,B) and (C,D) in terms of elements of A and elements of D. We can define the “level” of a number in a certain way, and we can see that the < comparison being used is comparing numbers of a lower “level”; so you’re going to go down through the levels until you “bottom out”. This is the hand-wavey argument, but it can be formalised using a well-known trick called mathematical induction that I’m not going to go into for the moment.

Without proving it, I’m going to use the fact that for every a in A and every b in B, if (A,B) is a number, then a < (A,B) < b, and also that (A,B) == (A,B). These make an intuitive sense, and if you want to see a proof, read Knuth’s book (or come up with one yourself!); I’m trying to keep this relatively short.)

So where is the bottom? Well, every number is an ordered pair of sets of numbers. What’s the simplest set of numbers? Well, trivially, it’s the empty set. So the ordered pair ({},{}) is a number. (From here on in I’m going to adopt Conway’s notation and write ( | ) instead of ({},{}).) For convenience, let’s call this number 0. Now we have another set of numbers, namely, the set {0}, so we get three new number candidates: ({0},{}), ({},{0}) and ({0},{0}), or, using our new notation, (0| ), ( |0) and (0|0). (0|0) can be ruled out, since it would give us 0 < (0|0) < 0, which implies that 0 !== 0, which is false, so (0|0) is not a number. (0| ) is a number; is it less than, greater than, or like 0? We have to test the two statements (0| ) <== ( | ), ( | ) <== (0| ). The second statement is vacuously true since the empty set has no elements; the first statement is immediately false since {0} has an element 0 and 0 !< 0. So we have 0 < (0| ); we shall call the new number 1. Likewise, we can show that ( |0) < 0, and we call that number -1.

I shall build up the next level explicitly and point the way for future levels. Using -1 < 0 < 1, we then have the numbers: (-1| ), (-1,0| ), (-1,0,1| ), ( |-1), ( |-1,0), ( |-1,0,1), (-1|0), (-1|0,1), (-1,0|1), (-1|1), (0,1| ), (1| ), (0|1), ( |0,1), ( |1). (Did I miss any?) I will leave it as an exercise to show that:

( |-1) == ( |-1,0) == ( |-1,0,1)
( |-1) < ( |0,1) == -1
-1 < (-1|0) == (-1|0,1)
(-1|0) < (-1| ) == (-1|1) == ( |1) == 0
0 < (0|1) == (-1,0|1)
(0|1) < (-1,0| ) == 1
1 < (1| ) == (0,1| ) == (-1,0,1| )

Using the order as our guide, we shall call ( |-1,0) -2, (-1|0) -1/2, (0|1) 1/2, and (0,1| ) 2. Why those particular choices? Well, it turns out that for the purposes we’re considering, all like numbers behave the same. This is another important theorem that I’m not going to prove here. I could have picked any other representatives, but it turns out the ones I picked are particularly good for other reasons.

It should be fairly obvious now how to build up at least the integers; the positive integer n + 1 is just the number (0,1,. . .,n| ), and the negative integer -(n + 1) is the number ( |-n,. . .,-1,0). If n is an integer, to get n + 1/2, we can take the number (n|n + 1).

What about the rest of the rational numbers? Before we go to that, we will need addition. Addition is an operation that takes two numbers and gives you another number. Formally, it is a function whose domain is the ordered pairs of numbers, and whose range is the numbers. We want our addition to satisfy the following addition axioms:[ul][li]For any number a, 0 + a == a.[]For any numbers a and b, a + b == b + a.[]For any numbers a, b, c: (a + b) + c == a + (b + c).If a < b, then for any number c, a + c < b + c.[/ul]Formally, we define addition as follows:[/li]
(A|B) + (C|D) = (a + (C|D), c + (A|B) | (A|B) + d, (C|D) + b),

where a, b, c and d run over all the elements of A, B, C and D respectively. Again, by induction, the level of each sum on the right is less than the level of the sum on the left, so the process eventually ends.

The multiplication axioms, except for the last one, look similar to the addition axioms:[ul][li]For any number a, 1a == a.[]For any numbers a and b, ab == ba.[]For any numbers a, b, c: a(bc) == (ab)c. []If a < b, then for any number c > 0, ac < bc.[]For any numbers a, b, c: (a + b)c == ac + bc.[/ul]The multiplication rule looks a bit more complicated, which is only fair since it has to satisfy more axioms:[/li]
(A|B)(C|D) = (a(C|D) + (A|B)c - ac, b(C|D) + (A|B)d - bd | a(C|D) + (A|B)d - ad, b(C|D) + (A|B)c - bc).

Don’t worry, you don’t have to remember all this mumbo-jumbo; even I had to look it up, and it’s one of my areas of specialisation. What’s important is that, when we define the surreal numbers in a certain way, this addition and multiplication turn out to be the same as addition and multiplication on real numbers.

Tomorrow: More numbers, and a totally different way of writing the same numbers. You’ll have to trust me, I am getting somewhere!