Toms Technical Introduction To CGT/ Discussion

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ilan: Hi, I have a couple of comments, which are based on the structure of this page, so are not necessarily criticisms, but more additions to what you have written already. The basic comment is that CGT, from the mathematical point of view, should be taken as the combination of set theory *and* Dedekind cuts.

So, the first explicit comment is that your section on set theory should add to "lots of sets", the examples {} = 0, {{}} = 1, etc. and note that this construction only allows you to construct the natural numbers (and zero), with finite operations anyway. This is important in order to show that set theory can define some numbers and also some games, in case I, or someone else, ever gets around to explaining CGT by starting with Nim, since those game values correspond exactly to this construction.

Then, the mathematical introduction should recall how real numbers can be defined by sequences of rationals on the left and right: (3, 3.1, 3.14, 3.141,... | 4, 3.2, 3.15,...) will define pi as the "real number" which is greater than anything on the left and smaller than anything on the right.

Then you can say that the generalisation will be {A | B} where A will be "negative" and B "positive" and {A | B} will be in between, to be vague yet accurate, the "simplest" thing in between.

Tom: Thanks Ilan, I think what you are suggesting would make the page more complete, but at the expense of making it longer. I think it is possible to get across what I wanted to without explaining these details, though they are certainly illuminating. Maybe we could make another page for them, but that might be too off-topic. This discussion page might be a good compromise.

The usual method of defining natural numbers from sets is a little more complicated than the first two terms suggest, though n+1={n} would work too. The sequence I am used to starts





Defined by 0={} and n+1={0,1,...,n}. It has the advantage that the number is the size of the set, and so it ties up nicely with the definitions of cardinal numbers and ordinal numbers as equivalence classes of sets and equivalence classes of totally ordered (inductive?) sets respectively.

Of course, it is possible to define games using set theory only, so I am not sure what the precise statement about the relation between natural numbers and sets is. However, the relation between Conway numbers and Dedekind cuts is nice.

A page on Nim, and impartial games generally would be good. The relevance is to understanding CGT rather than go directly.

ilan: Notice that the definition n+1 = {0,1,2,...,n} is exactly the options you can move to in a Nim heap of size n+1. It is in this sense that Nim corresponds to set theory. The relation between Conway numbers and Dedekind cuts is not just "nice", it is fundamental. In general, I do not believe that you can present CGT without starting with Nim. This is the historical basis, and the foundation of the Conway theory is simply this: Generalise Nim theory to Go. Moreover, since mathematics is about complete understanding of simple things, it is better, pedagogically speaking, to understand Nim completely, which is easy to do, before starting on Go.

Tirian Surreal numbers are an extension of both Dedekind's concepts and Cantor's theory of ordinals, with some infinitesimal "games" thrown in for good measure. I hope that my grandchildren learn about surreals as the foundation of number theory, since the rules have a "Go-like" elegance to them.

Starting with Nim seems like an unfortunate idea for two reasons. First, Nim is a nonpartisan game (i.e. both players have the same moves available to them), while Go is partisan (i.e. Black cannot play a White stone and vice-versa). Also, there is an isomorphism between a class of elementary partisan games and the real numbers and ordinals that we're all familiar with, as opposed to "nimbers" which are fascinating but perhaps more esoteric. If I were to teach CGT, I would start where Conway does in "Winning Ways" and "The Book of Numbers", which is with Hackenbush.

victory conditions for CGT games

The CGT page as it stands is very well written! My congratulations.

One improvement I would recommend is to define the common denominator of CGT games i.e. that winning a game is defined as not losing and losing is defined as having no legal move when it is your turn.

This definition of winning and losing is the basis for the sum of games being intuitive. It also makes the later definition of a game with all that talk of sets and empty sets more to the point.

It should briefly be mentioned that modern Go rules can be mapped onto CGT rules . Later a link to how this mapping is achieved for territory rules and area rules can be supplied. To start with this mapping can be left as an exercise for the student.

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Toms Technical Introduction To CGT/ Discussion last edited by PJTraill on January 21, 2019 - 20:50
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