Strong 0-1 laws in finite model theory (Q2710601)

Recall that a logic \({\mathcal L}\) is said to have the 0-1 law, if for every \({\mathcal L}\)-sentence \(\varphi\), \(\mu(\varphi)= 0\) or \(\mu(\varphi)= 1\), where \(\mu(\varphi):= \lim_{n\to\infty} \mu_n(\varphi)\) and where \(\mu_n\) is the uniform measure on the set of finite models of size \(n\). Among others, first-order logic FO and the infinitary logic \({\mathcal L}^\omega_{\infty\omega}\) have the 0-1 law. In general, the function \(\mu\) mentioned above is not \(\sigma\)-additive. The author introduces a new framework for asymptotic sentences with a \(\sigma\)-additive measure. Among others, he uses his framework to strengthen the 0-1 laws for FO and \({\mathcal L}^\omega_{\infty\omega}\). The sample space consists of all sequences \(({\mathcal A}_1,{\mathcal A}_2,\dots)\) of structures, where the universe of \({\mathcal A}_n\) is \(\{1,\dots, n\}\).