CGT Annotation

[Bill Spight]

Thx Bill. I know about Peano and about Conway. But your answer does not explain to me why say complex numbers are completely different mathematical objects from Conways `numbers`. As Flover claims. Both are numbers aren´t they? Wait! Could it be that there is no multiplication of Conway´s numbers?

You can multiply Conway numbers. (I would have to look up the definition, though. )

Peano axioms define the Natural numbers. On top of them, you can of course constructively define more types of "numbers", like rational numbers, real numbers, complex numbers and so on.

Conway's "numbers" are entirely different mathematical objects, even though they share the same name and some of their properties (essentially, forming an ordered field).

flOvermind does not mean that rational numbers, real numbers, complex numbers, etc., are different from Conway numbers. Conway numbers include the rationals and reals. To get complex numbers, you add i to Conway numbers, just as you add it to numbers based on natural numbers. (Or you can define a complex number as an ordered pair of real numbers.)

[Redundant]

To get complex numbers, you add i to Conway numbers, just as you add it to numbers based on natural numbers. (Or you can define a complex number as an ordered pair of real numbers.)

Or there's my favorite way of taking the ring of real polynomials in one variable and quotienting by the maximal ideal generated by x^2+1 ... I'll show myself to the door.

[RobertJasiek]

Conway numbers, but they are not counting numbers.

Why?

[Bill Spight]

Conway numbers, but they are not counting numbers.

Why?

By counting number I mean natural number, one that conceptually rests upon the sensory-motor operation of counting. Conway numbers start with the integers.

[RobertJasiek]

It is an advantage that Conway does not restrict himself to a special case set of numbers:)

[flOvermind]

In a way, you could say that Conway's numbers are the most "general" set of numbers, because many other kinds of numbers are embedded in them. The natural numbers, rational numbers, real numbers, hyperreal numbers and more are all embedded in Conway's numbers (by the way, complex numbers are not, since they form no ordered field, but of course we can get "complex Conway's numbers", "Conway's quarternions" and so on using the same constructions as with the reals). That's of course aesthetically nice, but it doesn't offer as big an advantage as you might think, because most interesting results of real analysis don't generalize to bigger number sets.

Conway's numbers also come with a problem: Strictly speaking, they don't form a set, they are a proper class. In a way, Conway's numbers are the biggest proper class that forms an ordered field. For example, they are more or less a superclass of the ordinal numbers (only "kind of", because there is a Conway number for each ordinal number, but you can only find a homomorphism if you use a non-standard addition operation on the ordinals). That's a big disadvantage, working with proper classes is not easy

That's the reason why we usually work with real numbers as the "standard" number set, they are mathematically speaking more "well-behaved" (at least sort of, there are enough headache-inducing results in real analysis as it is ). Of course, it would be possible to define the real numbers by first constructing Conway's numbers up to S_omega, and then throwing away the infinites and infinitesimals. But this top-down construction doesn't really have much practical advantages over the bottom-up construction from Peano's axioms. And in practice, most of the time the reals are defined axiomatically...

[cyclops]

I am afraid I resign here. Searching on Wikipedia about extended reals, hyperreals, superreals and surreal I get only more confused.
If I understand Flover correctedly Conways numbers do not add in the same way as reals do. And an individual Conway number is not a sets but a classe.
One final question though. I understood the Game * is not comparable to the Game 0. How then are the Conway Numbers totally ordered?

[Bill Spight]

I am afraid I resign here. Searching on Wikipedia about extended reals, hyperreals, superreals and surreal I get only more confused.
If I understand Flover correctedly Conways numbers do not add in the same way as reals do.

What do you mean, in the same way? The addition of games is defined such that adding two numbers gives the usual result. You can prove that the usual result is correct by playing a game. But why bother?

And an individual Conway number is not a sets but a classe.
One final question though. I understood the Game * is not comparable to the Game 0. How then are the Conway Numbers totally ordered?

Games are partially ordered. (In go, the biggest play is not always the best play, for instance.) But numbers are ordered. Here is another example of where something that is ordered can be part of something that is partially ordered. The points on the (x,y) plane are partially ordered. However, the points on the x-axis are ordered.

[cyclops]

I am afraid I resign here. Searching on Wikipedia about extended reals, hyperreals, superreals and surreal I get only more confused.
If I understand Flover correctedly Conways numbers do not add in the same way as reals do.

What do you mean, in the same way? The addition of games is defined such that adding two numbers gives the usual result. You can prove that the usual result is correct by playing a game. But why bother?

mm, difficult to resign. I only try to interpret Flover's "because there is a Conway number for each ordinal number, but you can only find a homomorphism if you use a non-standard addition operation on the ordinals". So I gather Conway adding does not behave. Something like half + half unequals one. Seems quite weird to me, tough. But that is how I read him. Why bother? To learn something, to have a good time, curiosity, to trigger big Bill telling exciting math, why not.

Games are partially ordered. (In go, the biggest play is not always the best play, for instance.) But numbers are ordered. Here is another example of where something that is ordered can be part of something that is partially ordered. The points on the (x,y) plane are partially ordered. However, the points on the x-axis are ordered.

Baffled again! ( biggest play not always best play ). But here I infer the set/class of Games and the set/class of Conway Numbers are not the same or isomorf. I am tempted to guess a Conway Number is a class of mutually incomparable Games. But then what about resigning, Bill will come after me.

[hyperpape]

Not that I know about Conway numbers, but you did switch from ordinals to reals, going from Flover's comment to yours.

[flOvermind]

If I understand Flover correctedly Conways numbers do not add in the same way as reals do.

Conways numbers add in the same way as reals do (at least the subset that corresponds to the reals). But they add differently than ordinal numbers.

For example, in ordinal arithmetic, omega < (omega+1), but (1+omega) = omega (where omega is the smallest infinite number). Conway's numbers behave better in that regard, there (omega+1) = (1+omega), and both are stricly larger than omega.

As long as you stay in the finite numbers, there are no differences.

And an individual Conway number is not a sets but a classe.

As long as you only talk about individual Conway numbers, you're ok. But the "set of all Conway numbers" does not exist, in the same way that the "set of all ordinal numbers", or the "set of all sets" do not exist. In mathematics, these "non-sets" are called "proper class" (that is, a "class" is either a "set" or a "proper class"). Often you can get away with just pretending they form a set (but see http://en.wikipedia.org/wiki/Russell%27s_paradox for an example of what can happen if you're not careful ).

But here I infer the set/class of Games and the set/class of Conway Numbers are not the same or isomorf.

Exactly. Each number is a game (or, more accurately, an equivalence class of games), but not each game is a number.

I am tempted to guess a Conway Number is a class of mutually incomparable Games.

Not quite. Numbers are always comparable to each other, and a number is an equivalence class of comparable and "equal" games, where equal is defined as being both <= and >=.

If two games are not comparable, one of them is not a number. For example, * is a game but not a number


EDIT: Of course omega < omega+1, not the other way round