FOM: Conway's foundational ideas

[Stephen G Simpson]

Stephen Fenner 3 May 1999 16:32:23 

 > I think a reasonable question to ask in this case is: Is there any
 > interesting property of H that doesn't reflect down to H_kappa for
 > some hereditarily accessible kappa?  If not, then the bigness of H
 > doesn't really amount to anything interesting.

Joe Shipman 4 May 1999 00:36:48

 > Read Conway's book "On Numbers And Games" -- surprisingly large
 > ordinals crop up very quickly.  ...

Joe, you haven't really answered Steve Fenner's question.  For
example, let's look at HC, the set of hereditarily countable sets.
Clearly Conway's basic definitions regarding numbers and games can be
trivially restricted to HC by inserting the word ``countable'' at
appropriate places.  For instance, Conway's inductive definition of
*number* on page 4 says:

  If L,R are any two sets of numbers, and no member of L is greater
  than or equal to any member of R, then there is a number {L|R}.  All
  numbers are constructed in this way.

and we can insert ``countable'' in front of ``sets'' and in this way
we get the *hereditarily countable numbers*, i.e. the numbers which
belong to HC.

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?

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
essentially irrelevant to his real subject: numbers and games.

 > 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.

Joe Shipman 05 May 1999 12:32:09 

 > See "Appendix to Part Zero", ONAG pp.64-67, which is foundationally
 > well-informed ....

OK Joe, I am looking at it, and I don't necessarily agree that it is
foundationally well-informed.  Yes, Conway understands how flexible ZF
is with respect to transfinite inductive constructions, or at least he
understands this better than most mathematicians.  But here on page 66
is where he seems to show his f.o.m. amateur 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.

  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.!  Many
famous and difficult set-theoretic consistency results have, as their
outcome, specific interpretability results of this kind.  (Example:
Shelah's proof of the consistency of the Proper Forcing Axiom relative
to a supercompact cardinal or whatever.)  Conway seems unaware of this
kind of thing.  Clearly Conway is underestimating the difficulties
inherent in his vague meta-conjecture.

  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.

  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.

-- Steve