FOM: Conway's foundational ideas -- reply to Simpson

Joe Shipman shipman at
Wed May 19 14:25:44 EDT 1999

I now realize I should have asked you to read not only pp.64-67 of ONAG,

but the whole book....

Conway needs to use proper Classes freely or his exposition would be
greatly encumbered without any corresponding benefit.  Formalizing his
system in ZF suffers some of the same difficulties formalizing category
theory does, and these difficulties are curable with some fussing, but
it is perfectly obvious that everything in his book could be
straightforwardly formalized with an inaccessible or two.  To do without

inaccessibles requires dealing somewhat more freely with proper Classes
than can be done easily in VNBG (the system Conway asserts he can use if

forced to).  When he talks about "axioms of infinity", I am certain he
means no more than inaccessibles, which are to be used in order to treat

classes as freely as sets by modeling them as "large" sets.  Conway's
complaint is that "we regard it as almost self-evident that our theory
is as consistent as ZF, and that formalization in ZF destroys a lot of
its symmetry", and I agree with this statement.

Conway's system is basically an alternative kind of set theory, and
transfinite induction, Replacement, Foundation, Choice, and Powerset are

as essential to it as they are to the more standard set theories.  There

is a cumulative hierarchy which resembles V in many ways (and is
isomorphic to V if you restrict to "impartial games"), and restricting
to hereditarily countable objects misses the point.

> 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?

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

> I conjecture there isn't, so the next question to ask is: Why does
> Conway make such a big fuss over inessential issues: sets versus
> proper classes, inaccessible cardinals, etc?  These issues seem to be
> irrelevant to his main topic: numbers and games.

They are not inessential in his development!  His "numbers" and "games"
are as general as "sets".

> > Conway proposes that proper classes, or Universe-size sets, be
> > referred to as "Universities" by analogy with "infinities", ...
> > Conway calls collections which are larger than Universe-size
>  > "Improprieties", since they are not proper Classes, but does not
>  > attempt to do anything with the one such collection he defines ...
> I tentatively diagnose this as an advanced case of f.o.m. amateurism
> on Conway's part.  The clearest symptom is the excessive generality.

It is not excessive because Conway is not restricting his attention to
"small" objects.

>  But here on page 66
> is where he seems to show his stripes:
>   It seems to us ... that mathematics has now reached the stage where
>   formalisation within some particular axiomatic set theory is
>   irrelevant, even for foundational studies.
> This is completely crazy.  Formalization within specified axiomatic
> theories is of the essence in foundational studies, and will remain so

> for the forseeable future.

I agree that he shouldn't have said "even for foundational studies".

>   It should be possible to specify conditions on a mathematical theory

>   which would suffice for embeddability within ZF (supplemented by
>   additional axioms of infinity if necessary) but which do not
>   otherwise restrict the possible constructions in that theory.
> Note that ``axioms of infinity'' means large cardinal axioms.  Thus
> Conway is saying that there are interesting sufficient conditions for
> a theory to be interpretable in ZF plus large cardinal axioms.  But
> this implies a solution to a major open meta-problem in f.o.m.!

No, interesting *necessary* conditions would be a solution to a major
open meta-problem.  **Conway just wants a metatheorem justifying the
intuition that his system, and others which deal freely with classes and

superclasses in a "constructive" way (I interpret this to mean "built
up" rather than "cut out" from something preexistent), are interpretable

in ZF plus enough inaccessibles.**

>   Of course the conditions would apply to ZF itself, and to other
>   possible theories that have been proposed as suitable foundations
>   for mathematics (certain theories of categories, etc.),
> O Ye Gods.  ``Categorical foundations'' again.  By the way, Conway in
> his preface acknowledges help from, of all people, Johnstone.

There is nothing controversial here -- Conway is simply noting that you
can found category theory in set theory if you have an inaccessible
(or maybe a proper class of inaccessibles a la Grothendieck).
**He calls for a better understanding of what it is about category
theory or his system that make them easily interpretable in ZF plus
inaccessibles but only interpretable with painful reformulation in ZF.**

>   but would not restrict us to any particular theory.  This appendix
>   is in fact a cry for a Mathematicians' Liberation Movement!
> I think I could more or less agree with Conway, if he dropped the bits

> about axioms of infinity and ``categorical foundations'', and if he
> stopped trying to grandstand as a liberator of mathematicians.

Conway wants to be "liberated" form Kuratowski ordered pairs and other
devices which are METAmathematically essential because they allow
mathematics to be founded in the elegant and minimal theory ZFC which
has a nice model theory, but not relevant to ordinary mathematics.

I have said something like this before.  You could do something like
starting with the integers as urelements, adding a primitive for
ordered pair and appropriate axioms, adding the ZFC axioms without
Foundation, and then putting Zermelo set theory (without Replacement
or Choice) "on top" of that so that classes can be treated like sets,
and you get something obviously consistent and sufficient to do ordinary

mathematics, category theory, and so on WITHOUT the tedious but
METAmathematically worthwhile work of showing it can all be done in
"pure" ZFC.  (The system I have outlined is stronger than Morse-Kelley
but weaker than an inaccessible; if you made it "ZFC on top of ZFC"
instead of "Z on top of ZFC" it would be equiconsistent with one
inaccessible.)  Conway's system wouldn't require urelements but would
require two ("Left" and "Right") notions of membership.

Conway's system is obviously consistent because it is "built up" in
essentially the same way the set-theoretic universe V is; for category
theory this is not so obvious because the universe is unspecified and
Russell-style paradoxes can be constructed, but with a small/large
distinction it is also obviously consistent.

The two basic issues here seem to be "definitional extensions" and
"treating classes like sets".

Definitional extensions: "Pure ZFC" is unrealistically
parsimonious and we want to be able to extend it freely by defining
notations for operations on sets satisfying given properties.
Particular examples which are known to be OK:
ordered pair (no metamathematical problems after Kuratowski),
cardinal number (no metamathematical problems after Von Neumann),
global choice function (slightly stronger than ZFC but consistent with

Treating Classes like sets: we want to form classes, superclasses,
super-duper-classes, and so on, with appropriate comprehension
principles, while avoiding Russellish paradoxes by not mixing levels
or allowing unlimited comprehension.   We know we can do this with an
inaccessible and a small-large distinction, but we'd like to avoid
having to make this distinction.  VNBG is not flexible enough for
this; "Z on top of ZFC" probably is.

-- Joe Shipman

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