A first-order theory $T$ has the {\em finite model property} if every first-order sentence that is implied by $T$ has a finite model. We give a characterization of when a complete $\omega$-categorical theory $T$ has the finite model property. The characterization makes it possible to transfer stability theoretic results in [29] to certain classes of finite structures. This approach also gives information concerning properties that a sequence of finite ``base structures''of ever larger size must have in order for its random expansions to a larger vocabulary to have a chance to satisfy a convergence law for first-order logic.
We prove that if a minimal ordered structure $(M,<, \ldots)$ with infinite chains in an arbitrary language extending the language of strict orders interprets (in some power $M^n$) the linear order $(\omega,<)$, then this fact can be witnessed for $n=1$ via the incomparability relation of an $<$-definable strict order $...
We construct a sentence of $L_{\omega_1,\omega}$ having exactly $\aleph_1$ many countable models up to isomorphism, but no perfect set of non-isomorphic models. This disproves Vaught's conjecture for infinitary logic. The construction follows a framework devised by Robin Knight, and the proof has been formally verified...
N. Ackerman, Cameron E. Freer, R. Knight· 0 citations
We study the finite spectra of the syllogistic logic with cardinality comparisons $S^\dagger(card)$. Since the language does not contain conjunction at the sentence level, we consider the spectrum of a theory $\Gamma$, namely the set of positive integers $n$ for which $\Gamma$ is satisfiable in an $n$-element model. We...
We present a typed two-dimensional $\lambda$-calculus whose equality evidence is \emph{computational}: a path between two terms is an explicit finite sequence of one-step conversions (the $\beta$- and $\eta$-contractions, the congruences, and the structural rules), and every property of paths is proved \emph{by recursi...
Daniel O. Martínez-Rivillas, A. F. Ramos, R. D. de Queiroz· 0 citations
Let X and Y be finite ordered sets and let $X^Y$ denote the ordered set of order-preserving maps from $Y$ to $X$. Let $t$ be a~term formed from a~single variable by exponentiation. We prove that, for every term $t$ with at most twelve variable occurrences and arbitrary finite ordered sets $A,B$, $t(A)\cong t(B)$ implie...
We formulate a proof of Part~I of Martin's conjecture for order-preserving maps directly on the quotient of the reals by Turing equivalence. The assumptions are countable choice for reals, the perfect-set property for sets of reals, and continuous uniformization on a comeager domain. The argument applies under $\mathsf...
Liang Yu· 0 citations
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