Compactness Theorem for Some Generalized Second-Order Language

  •  Zakharov V. K.    
  •  Yashin A. D.    


For the first-order language the {\em compactness theorem} was proved by K. G\"odel and A. I. Mal'cev in 1936. In 1955, it was proved by J.~\L o\'s (1955) by means of the {\em method of ultraproducts}. Unfortunately, for the usual second-order language  the compactness theorem does not hold. Moreover, the method of ultraproducts  is also inapplicable to second-order models. A possible way out of this situation is to refuse the most vulnerable place in the construction of ultraproducts connected with the  factorization relatively an ultrafilter, i.e., to stay working with the  ordinary non factorized product. It compels  us instead of the single usual set--theoretical equality $=$ to use several {\em generalized equalities} $\approx_{\mathrm{first}}$ and $\approx_{\mathrm{ second}}$ for first and second orders, and instead of the single usual set-theoretical belonging $\in$ to use several {\em generalized belongings $\inn_{\mathrm{ second}}$}. Following that it is necessary to refuse  the usual set-theoretical interpretation $(\gamma(x_0),\ldots,\gamma(x_k))\in\gamma(u)$ of the second basic (after equality) atomic formula $(x_0,\ldots,x_k)u$ and to replace it by the generalized interpretation $(\gamma(x_0),\ldots,\gamma(x_k))\inn_\tau\gamma(u),$ where $x_i^{\tau_i}$ are variables of the first-order types $\tau_i$, $u^\tau$ is a  variable of the second-order type $\tau=[\tau_0,\ldots,\tau_k]$ (i.e. predicate), and $\gamma$ is some  evaluation of variables on some mathematical system $U$.

This paper is devoted to rigorous development of the expressed general idea. For the generalized in such a manner second-order language the compactness theorem is proved by means of the {\em method of infraproducts} consisting in rejection of the \L o\'s factorization. In the end of the paper the method of infraproducts is applied for the construction of some uncountable models of the second-order generalized Peano--Landau arithmetic.

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