On Skolem's theorem.
Gaisi Takeuti · Journal of the Mathematical Society of Japan · 1957
In 1922, Th.Skolem proved the following famous theorem: If there exists a model of any cardinal number for a system of axioms (satisfying certain conditions), then there exists also a countable model for the system.The aim of the present paper is to formulate and prove a corresponding theorem from the finite stand point.Our theorem reads as follows:MAIN THEOREM.If $A,$ $B,$ $C,$ $D,$ $E$ in Godel [2] are consistent, then $A,$ $B,$ $C,$ $D,$ $E$ and the following axioms are consistent.where $f_{0}$ is a function, which is not contained in axioms $A-E$, and $\omega$