Re: The iterative conception of sets and the (unrestricted) axiom of replacement

On Fri, 26 Jun 2009 01:28:56 -0700 (PDT), Marc Alcobé García
<malcobe@xxxxxxxxx> wrote:

Say a function is a set of ordered pairs such that etc, and
we define cardinality in terms of functions. (We don't
need to get into the definition of card(S) as a certain
ordinal or equivalence class or whatever, we can just
take "card(A) <= card(B)" as an abbreviation for
SmallerThan(A,B), which is in turn defined in terms
of the existence of a certain function.)

I think this is standard. See, for example, Def. 10.1 p.27 in Kunen.

Then the notion
of size may well make no sense for non-sets, ie proper
classes.

Obviously.

But now say that an ffunction ("formal function") is
something defined by a formula of the language,
without being required to be a set of ordered pairs.
For example P is an ffunction if P(x) is the power
set of x, S is an ffunction if S(x) = x union {x}, etc.

If I have understood well, I am pretty sure this is standard too. See,
for example, Kunen p. 24.

It seems perfectly reasonable to extend the notion
of "card(A) <= card(B)" to proper classes A and B,
changing the definition to allow ffunctions instead
of just functions.

I am afraid this 'extension' would involve a jump onto the realms of
the metatheory, since we would need to talk about the existence of
classes (i. e. formulas) instead of sets...

?????????? What could your point possibly be here?

Let's review. Kunen said something about justifying
the axiom of replacement on the grounds that a
class that's no larger than some set should be a set.
That was an _informal_ comment on his part.
He was _referring_ to classes - when I try to
attempt to explain what he may have meant my
explanation is going to say something about classes!
_Presumably_ you were not just pointing out that
the standard definition of cardinality makes no
sense for proper classes; _presumably_ you were
asking for an explanation of his comment as
opposed to "pointing out" for us that his informal
comment makes no sense if "size" is taken to be
the standard definition of cardinality.

We're _talking_ about classes. I point out that
a notion that's _identical_ to the standard notion
of "A is no larger than B" except that it allows
proper classes as "functions" allows one to make
sense of his comment, and you complain that
I'm talking about proper classes?

This doesn't make much sense as far as I can see.

If we do that then I'm pretty sure
the following are equivalent in a set theory that
includes proper classes (where a proper class
is {x : R(x)} for any definable unary relation R;
in particular the elements of proper classes are
sets):

(i) A is a set
(ii) There exists a set B such that card(A) <= card(B).
(iii) There is an alpha such that every element of
A is in V_alpha.

The same remark applies here.

And to your comment "From the point of view of the iterative hierarchy
to be or not to be a
set is not a matter of size (the notion of size does not even make
sense for not-sets), but just of climbing up unboundedly through
stages."

What's your point?

It seems to me that we _should_ believe that that's possible.
But we can't define a formal system by saying that anything
we should believe is possible is allowed - we need an
axiom that _implies_ that it's allowed.

The problem with the axiom schema of replacement is that it seems to
forbid the existence of certain proper classes which seem reasonable
from the realist conception of sets based on the iterative hierarchy
(unless they are forbidden by the language itself).

The axiom doesn't forbid the existence of anything - presumably
you mean that it implies that some things which "should" be
proper classes are in fact sets. If that's not what you meant,
what did you mean? If that _is_ what you meant: What's an example
of this phenomenon?


David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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