[FOM] Axiomatic Dimension (additional examples)
joeshipman@aol.com
joeshipman at aol.com
Tue Mar 7 13:28:26 EST 2006
I wrote:
>We could formalize this by saying an axiomatization A has dimension n
if it contains disjoint subsets F U B1 U B2 ... U Bn, where F >is
finite and the Bi are infinite, such that any subset of A which
includes F and includes infinitely many elements of each Bi implies >A.
A theory T has dimension n if it has an axiomatization of dimension n
but no axiomatization of dimension n+1, and dimension >infinity if it
has axiomatizations of all finite dimensions >0.
>This suggests that a finitely axiomatizable theory has dimension 0; it
seems a bit odd that a theory of dimension infinity can be >given
axiomatizations of all degrees >0 but not of degree 0, but maybe no
odder than the fact we can map omega^omega into >omega^1 but not into a
finite set.
>Is this is a well-known model-theoretic concept in a new guise?
This definition needs a bit of fiddling. We need to require the
condition to go both ways:
An axiomatization A has dimension n if it contains disjoint subsets F U
B1 U B2 ... U Bn, where F is finite and the Bi are infinite, such that
any subset of A which includes F and includes infinitely many elements
of each Bi implies A, **AND** any subset whiich includes only finitely
many elements of one of the B's fails to imply A.
Also, for clarity we restrict the discussion to axiomatizations of
complete theories which have an axiom of infinity (that is, some
sentence in the theory has no finite models).
A key construction here turns an axiomatization of dimension n+1 into
an axiomatization of dimension n, by replacing the subsets {A1, A2,
..., } and {B1, B2, ...} with {A1 & B1, A2 & B2, ...}.
(This does not work for n=0!)
We can also turn a "tight" axiomatization {A1, A2, ...}, where each
axiom is independent of all the others, into an axiomatization of
degree aleph_zero, by combining and partitioning as follows:
{A1, A1&A3, A1&A3&A5, A1&A3&A5&A7,...}, {A2, A2&A6, A2&A6&A10,...},
{A4, A4&A12, ...}, {A8,...}, ...
Clearly a subset the union of these sets of axioms is sufficient iff it
contains infinitely many members from each.
The interesting issue is how the classification of axiomatizations
translates into a classification of theories, and of models. *Do there
exist, for each n, theories with axiomatizations of degree n but no
higher degree?*
In the case of the theory of real closed fields, we know that a single
axiom involving the order relation < encapsulates all the information
in the infinite set of axioms of the form "-1 is not a sum of n
squares". So the dimension of the theory seems to depend on whether <
is in the language. If we add this relation to the language, and
include the axioms of order but NOT the statement "every positive
element has a square root", we don't get a complete theory in the
expanded language, so we really seem to decrease the "dimension" by
expanding the language (we have to add the square root axiom, and then
the infinite set of axioms about -1 not being a sum of squares becomes
superfluous).
But there may be some totally different axiomatization of degree 2 for
the theory of RCFO.
I would like to find a condition on an axiomatization of dimension 1,
which ensures that every equivalent axiomatization has dimension 1.
Note that a condition like "the length of the axioms is linear in the
number of quantifiers" rules out constructions like {A1, A1&A2,
A1&A2&A3,...} but does not rule out constructions like {A1&B1, A2&B2,
A3&B3,...}.
-- Joe Shipman
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