arXiv subjects

Michael C. Laskowski

Publications and source records attributed to Michael C. Laskowski.

39 records · Page 3Linked to original sources

arXiv2000

The uncountable spectra of countable theories

Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality κ. We denote by μ(respectively \hatμ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. We prove that I(T,κ), as a function of κ> \aleph_0, is the minimum of 2^κ and one of the following functions: 1. 2^κ; 2. the constant function 1; 3. |\hatμ^n/{\sim_G}|-|(\hatμ- 1)^n/{\sim_G}| if \hatμ<ωfor some 1 = ωsome group G <= Sym(n); 4. the constant function \beth_2; 5. \beth_{d+1}(μ) for some infinite, countable ordinal d; 6. \sum_{i=1}^d Γ(i) where d is an integer greater than 0 (the depth of T) and Γ(i) is either \beth_{d-i-1}(μ^{\hatμ}) or \beth_{d-i}(μ^{σ(i)} + α(i)), where σ(i) is either 1, \aleph_0 or \beth_1, and α(i) is 0 or \beth_2; the first possibility for Γ(i) can occur only when d-i > 0.

math.LO↗

arXiv1993

Forcing isomorphism

A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory T is classifiable then each model of cardinality lambda is described by a sentence of L_{infty, lambda}. In fact this sentence can be chosen in the L^*_{lambda}. (L^*_{lambda} is the result of enriching the language L_{infty, beth^+} by adding for each mu < lambda a quantifier saying the dimension of a dependence structure is greater than mu .) The truth of such sentences will be preserved by any forcing that does not collapse cardinals <= lambda and that adds no new countable subsets of lambda. Hence, if two models of a classifiable theory of power lambda are non-isomorphic, they are non-isomorphic after a lambda-complete forcing. Here we show that the hypothesis of the forcing adding no new countable subsets of lambda cannot be eliminated. In particular, we show that non-isomorphism of models of a classifiable theory need not be preserved by ccc forcings.

math.LO↗

arXiv1993

On the existence of atomic models

We give an example of a countable theory T such that for every cardinal lambda >= aleph_2 there is a fully indiscernible set A of power lambda such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size lambda where the principal types are dense, yet T(A) has no atomic model.

math.LO↗