# FOM: Conway games; question about CCing to Conway

Stephen G Simpson simpson at math.psu.edu
Wed May 26 19:52:18 EDT 1999

```Joe Shipman 26 May 1999 14:15:15

> if you move on to Part One of the book (the "Games" part, ...

As I said before, I have glanced through the whole book.  But I don't
claim to understand the games part very well.  In particular, the
definition of the ``normal play convention'' on page 71 doesn't make
sense to me.  When is a player ``called upon'' to move?  Do they
alternate?  Do they move simultaneously?

In any case, it looks to me as if Conway games are pretty much the
same thing as clopen Gale-Stewart games.  (A clopen Gale-Stewart game
can be viewed as a rooted tree with no infinite path.  The game starts
at the root node.  A move is a step from a node to an immediate
successor node.  Two players take turns moving.  The loser is the
first player who cannot move.)

I am wondering how Conway games fit into the context of general
Gale-Stewart games, as studied by set theorists like Tony Martin, Hugh
Woodin, etc.

> There is an amazingly interesting function from sets to ordinals
> ... the value of the function on a set is the least ordinal not the
> value of the function on any element of the set

That does sound pretty interesting.  It makes sense also for rooted
trees with no infinite path, i.e. the value of the function on a node
would be the least ordinal not the value of the function on any

CCing to Conway:

A while back somebody on FOM got publicly upset because I wasn't CCing
my Conway-related postings to Conway.  So I started doing so, and
others have done so also.  But apparently there has been no response
from Conway, and for all I know Conway is uninterested in receiving
this stuff.  So I'm thinking I may stop CCing him.  Joe, what do you
think?

As always, Conway would be more than welcome to subscribe to FOM at
<http://www.math.psu.edu/simpson/fom/> and join the discussion.

-- Steve

```