FOM: Conway's foundational ideas

[Stephen G Simpson]

Joe Shipman 19 May 1999 14:25:44 

 > I now realize I should have asked you to read not only pp.64-67 of
 > ONAG, but the whole book....

Actually, I did glance through it.  I didn't find anything in it that
would not make just as much sense and be just as meaningful when
restricted to hereditarily countable numbers and hereditarily
countable games.

 > When he talks about "axioms of infinity", I am certain he means no
 > more than inaccessibles

I am *not* certain of this.  But on the other hand, he gives no sign
of being aware of other axioms of infinity (Mahlo cardinals,
measurable cardinals, etc etc), so you could be right.

 > Conway's system is basically an alternative kind of set theory ...

What system?  He doesn't present any system.  He only presents a call
for a Mathematicians' Liberation Movement.

Maybe you could extract a system from his informal remarks.  But then,
wouldn't the system be included in a trivially obvious definitional
extension of ZF?  Or, at the outside, ZF plus an inaccessible
cardinal?  (After all, left and right membership could be introduced
into ZF by means of Kuratowski pairs, etc.)  If so, then this system
should not be called an alternative to set theory.  It is set theory
plain and simple.  Definitional extensions are part of the normal
mechanism of how one formalizes mathematics in foundational theories
such as ZF.

Simpson:

 > > I think part of the force of Fenner's question is: Does Conway's
 > > book contain any serious result or definition that doesn't work
 > > just as well or isn't just as meaningful for this restricted
 > > class of numbers, the hereditarily countable numbers?  And ditto
 > > for hereditarily countable games?

Shipman:

 > You might as well ask the same question for a book on ZFC.

For a book on ZFC, the answer to this question would be that there are
many interesting definitions and results that are meaningful only if
you go beyond the hereditarily countable realm.  Examples would be:
the hierarchy of alephs, singular cardinals, strong limit cardinals,
regular uncountable cardinals, Fodor's lemma on stationary sets, the
Erdos-Rado partition calculus, etc etc etc etc.

In Conway's book, I really don't see anything that would not be just
as interesting and meaningful if restricted to hereditarily countable
numbers and hereditarily countable games.  The rest seems like empty
generality for the sake of generality.

 > His "numbers" and "games" are as general as "sets".

I think this is empty or excessive generality.

 > Conway wants to be "liberated" form Kuratowski ordered pairs ...

This reminds me of hippies who want to be liberated from the need to
earn a living.

-- Steve