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...

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?

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?



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.

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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.


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.

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

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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.


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 ).


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


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