[FOM] Does `reduction to set theory' reduce?

[John Baldwin]

I respond to Bill Tait's remarks on Tim  Chow's `formalization thesis' 
post but
rename the thread as Bill and I have both veered from Tim original 
question.

On Wed, 26 Dec 2007, William Tait wrote:

> One challenge to the Formalization Thesis, or maybe better, to a
> satisfactory formulation of it, arises from the 'reduction to sets' of
> various types of objects: ordered pairs, functions, ordinal numbers,
> cardinal numbers, real numbers, etc. In each case, the reduction adds
> structure to the type that is not part of its real meaning. In fact,
> this was Dedekind's challenge. (And, as Benacerraf pointed out, there
> is in general some arbitrariness about what structure is added.)


Below is a several paragraph excursion on how the `reduction' complicates 
the situation.  In a separate post I will note how the extra structure of 
the reduction sometimes gives valuable information.

Most people have an intuition for only a few infinite structures:
arithmetic on the natural numbers, the rationals, and perhaps on
the reals.  Most mathematicians extend this to  the complex
numbers and then to a deeper understanding of various structures
depending on their own specialization: $(SL_2(\Re)$, $\bP_1$,
initial segments of the ordinals Recall Paul Cohen's intelligence
test: for what ordinals can you visualize how the termination of
descending chains witnessing well-foundedness of the ordinal?
Cohen claimed $\epsilon_0$. But all these structures have
cardinality at most the continuum. There are few strong intuitions
of structures with cardinality greater than the continuum.
However, there is a crucial exception to this remark. It is rather
easy to visualize a model that consist of copies of a single
countable or finite object. Consider a vocabulary with a unary
function $f$. Assert that $f(x)$ never equals $x$ but $f^2(x)=x$.
Then any model is a collection of $2$-cycles.  On the one hand we
have the notion that there are models of arbitrarily large
cardinality but we  have no really different image distinguishing
among the models of different large cardinalities. This situation
generalizes when the number of disjoint copies of the same
structure is replaced by the dimension of a vector space or field.
Thus we might consider the class of structures $A_\kappa$, a
direct sum of $\kappa$ copies of $Z_2$. The isomorphism type of
the model depends solely on the number $\kappa$ of copies (and NOT
AT ALL ON THE INTERNAL STRUCTURE of the cardinal $\kappa$).  We
say a class that has, up to isomorphism, a unique model of
cardinality $\kappa$, is $\kappa$-categorical.

  Such structures
arise in a standard way; they are the class of models of a first
order theory that is categorical in all uncountable cardinalities.
Categoricity is not a necessary condition for such a clear
visualization: consider an equivalence relation with two infinite
classes and  fix a totally  categorical theory and make  each
class a model of the given theory; the model is determined by two
cardinals- the cardinality (or more precisely the dimension) of
each class. More sophisticated investigations that slightly relax
the notion of `visualize' show  categoricity does provide a
sufficient condition for such a visualization.  And then
interpreting `visualizing' as: admitting a structure theorem, we
can obtain exact conditions for being able to `see' all models of
a first order theory.  Let us specify the situation somewhat. For
simplicity in this proposal languages are by default countable. If
we fix a logic and a class $\bK$ of models defined in that logic,
we can sometimes `see' models  of arbitrarily cardinality in
$\bK$.

A crucial and non-trivial fact is Morley's theorem that if $\bK$
is defined by a first order theory then categoricity in one
uncountable cardinal implies categoricity in all uncountable
cardinals.  The Baldwin-Lachlan proof of this result yields some
further information.  There is a definable `strongly minimal' set
which admits a dimension. Moreover, each model is `prime' over a
generating set for this strongly minimal set (each elementary map
defined on the basis extends to the entire model). Zilber provided
a way to give a finite analysis of each model to explicate `prime
over'.  This analysis provides a mathematical explication of
`visualize'.