"Limitation of" was: [FOM] New Axioms

W.Taylor@math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Wed Jul 2 02:33:17 EDT 2003

Harvey Friedman wrote:

> My own opinion is that the axioms of ZFC have a special kind of
> coherent simplicity that cannot be extended. I.e., ZFC is complete.
> And also the axioms of ZF have a related special kind of coherent simplicity
> that also cannot be extended. I.e., ZF is complete.

I would certainly like to hear more explication of these intuitive ideas,
which seem on the face of it to contradict each other, for one thing!

> 5. There are very simple fundamental sounding philosophies that suggest it.
> With 5, there is the simple fundamental sounding philosophy that
> suggests the axiom of choice:

I believe that there is a very simple sounding philosophy that most
blatantly *fails* to suggest it, (though neither does it suggest
the contrary).    This is Dana Scott's largely-ignored attempt
(Proc Symp Pure Math Vol 13 1974), to found ZF(C) on the more
fundamental idea of "stages" of constructing things.  He produces
some very basic and intuitive axioms for "stages", and shows how the ZF
axioms arise from these.  But he mournfully finishes up by noting that
nothing in this approach tends to suggest AC, and plaintively wishing
for a similarly fundamental and "obvious" principle to found it!

My heart bled sincerely for him, while my mind rejoiced in his failure.

Harvey then goes on (regarding 5):

> *anything imaginable whatsoever,
> subject to the well known size limitation, forms a set.*

This is troublesome on two counts.  The "imaginable"; and the "size".

> In particular, a set which meets every one of a set of pairwise
> disjoint nonempty sets, is at least IMAGINABLE, and therefore, by
> this principle, exists.

The trouble is, there are *many* imaginable things that contradict
each other.  In the current context, I find it equally imaginable,
indeed more so (being game-theoretically rather than set-theoretically
oriented), that every sequential game of complete information be
determinable.  (This involves "imagining" that certain choice-based
sets exist).  In other words, that AD be true.   This contradicts AC.

So this "imaginability" criterion is rather unhelpful.

But the other matter, "limitation of size", though, is what I chiefly
want to talk about.  I disliked this principle when I first saw it
written down long ago, and still do.

It seems to me to stress entirely the wrong thing; to be a mere hack
to avoid the paradoxes, which could be avoided by doing the same thing
in effect, but the (IMHO) "proper" way.

It seemed obvious to me in my naive undergraduate days, just what
the cause of all the paradoxes was, both set-theoretical and logical;
and in my old age, though no longer naive, yet perhaps still untutored,
it still seems the same.

And it is NOT due to things being "too big" - this does not (initially)
seem to be a problem with e.g. the Burali-Forti paradox.  Though technically
the B-F can be put down to it, it is not intuitively so, and the answer
to it AND all the others still seems plain to me.

And it is *impredicativity*; not in general terms, where it may be
quite harmless, but impredicativity BY CONSTRUCTION, as it were.

Basically, that we cannot "build" any set, until we already have building
blocks to put in it.  That the elements of a set MUST BE logically prior
to the set itself.     This fits in perfectly, as I see it, with Scott's
above-mentioned "stages"; and makes the axiom of foundation a sine qua non
of true set theory, and the ignoring of which positively *invites* problems!

So when we look at a set   S = { x : P(x) }   given by comprehension,
though it *may* be OK for P to refer to S itself, (maybe so, maybe not),
it can NEVER be OK for S to be *one of the possible x's being ranged over*.

This "Impredicativity of Candidacy", as I like to call it, is (IMHO)
plainly wrong, and is the simple cause of all the set-theoretic paradoxes,
and can be seen as so in the logical ones also, such as the Liar.

It has always baffled me that this is not universally seen to be so.

So there is no need for any "limitation of size" principle;
the "limitation of candidacy" does it all, and much more naturally.

         Bill Taylor               W.Taylor at math.canterbury.ac.nz
  I know EVERYTHING there is to be known about the members of the empty set!

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