877 research outputs found
Generalized Indiscernibles as Model-complete Theories
We give an almost entirely model-theoretic account of both Ramsey classes of
finite structures and of generalized indiscernibles as studied in special cases
in (for example) [7], [9]. We understand "theories of indiscernibles" to be
special kinds of companionable theories of finite structures, and much of the
work in our arguments is carried in the context of the model-companion. Among
other things, this approach allows us to prove that the companion of a theory
of indiscernibles whose "base" consists of the quantifier-free formulas is
necessarily the theory of the Fraisse limit of a Fraisse class of linearly
ordered finite structures (where the linear order will be at least
quantifier-free definable). We also provide streamlined arguments for the
result of [6] identifying extremely amenable groups with the automorphism
groups of limits of Ramsey classes.Comment: 21 page
On Decidability Properties of Local Sentences
Local (first order) sentences, introduced by Ressayre, enjoy very nice
decidability properties, following from some stretching theorems stating some
remarkable links between the finite and the infinite model theory of these
sentences. We prove here several additional results on local sentences. The
first one is a new decidability result in the case of local sentences whose
function symbols are at most unary: one can decide, for every regular cardinal
k whether a local sentence phi has a model of order type k. Secondly we show
that this result can not be extended to the general case. Assuming the
consistency of an inaccessible cardinal we prove that the set of local
sentences having a model of order type omega_2 is not determined by the
axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesi
Local Sentences and Mahlo Cardinals
Local sentences were introduced by J.-P. Ressayre who proved certain
remarkable stretching theorems establishing the equivalence between the
existence of finite models for these sentences and the existence of some
infinite well ordered models. Two of these stretching theorems were only proved
under certain large cardinal axioms but the question of their exact
(consistency) strength was left open in [O. Finkel and J.-P. Ressayre,
Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ].
Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In
fact, we show that the stretching principles are equivalent to the existence of
n-Mahlo cardinals for appropriate integers n. This is done by proving first
that for each integer n, there is a local sentence phi_n which has well ordered
models of order type alpha, for every infinite ordinal alpha > omega which is
not an n-Mahlo cardinal
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