FOM: In defense of Conway

Joe Shipman shipman at
Thu May 20 11:22:52 EDT 1999

I have nothing to add to Julio Gonzales Cabillon's excellent summary of the
foundational issues involved in Conway's 'On Numbers and Games", but I would like
to clarify a couple of things I said for Steve Simpson:

>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.

First of all, such a restriction would be an unnecessary complication.  Secondly,
there is an essential use of the full power of set theory in Conway's
construction of the Surreal Numbers; he wants to have infinitesimals in order to
simplify real analysis, and to ensure both infinitesimals AND completeness his
inductive construction needs to "go all the way".  Thus proper classes are
indispensable.  Of course one COULD stop at an earlier stage and obtain a
substructure of the surreal numbers with some of the desired properties, but it's
silly to criticize Conway for not doing this.  The whole point of the
construction of the surreals is Conway's observation that Cantor's construction
of the ordinals and Dedekind's construction of the real numbers could be seen as
specializations of a more general construction with some amazing properties.

>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.)

I was referring to the system of Numbers developed in Part Zero of Conway's book,
from pages 1-63, and not simply to Conway's foundational remarks in his "Appendix
to Part Zero" on pp. 64-67.  This system does indeed represent an alternative
foundation for Number Theory and Analysis, if not for all of Mathematics.  It is
essentially set-theoretical, and of course Conway is aware that the system is
easy to formalize as a definitional extension of ZFC+Inaccessible, but you seem
to be missing the point that Conway, Gonzales Cabillon, and I keep making, which
is that Conway is not required to do anything more than REMARK that his system
can be so rendered, and that students of number theory and analysis who seek a
rigorous foundation of the elementary part of these subjects may start with
Conway's book rather than a text which follows the usual Cantor/Dedekind

I didn't mean that the Surreal Numbers were an alternative to Set Theory as a
foundation for all of mathematics, just an alternative for the parts of
mathematics Conway develops.  That Conway doesn't need Fodor's lemma or the
Erdos-Rado partition calculus shouldn't disqualify him from using the level of
generality he does, especially when it makes the development so elegant and
powerful.  I didn't learn about Fodor's lemma and the partition calculus until I
was in graduate school, but it was perfectly appropriate that I learned
transfinite induction and axiomatic set theory as an undergraduate, the better to
rigorously develop topology, analysis, algebra, etc.  It was not "empty
generality" that led my professors to ignore the fact that it could all be done
in second-order arithmetic with appropriate coding, it was good pedagogy.

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

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

The only reason for Kuratowski ordered pairs is to justify eliminating the axiom
"There exists an operation OP on sets with the property that OP(a,b) = OP(c,d)
iff a=c and b=d".  Kuratowski accomplished something important when he noticed
that {{a},{a,b}} had this property, but if someone wishes to ignore this and take
the ordered pair operation as a primitive, it should not offend you.

-- Joe Shipman

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