Finite state automata and monadic definability of singular cardinals

We define a class of finite state automata acting on transfinite sequences, and use these automata to prove that no singular cardinal can be defined by a monadic second order formula over the ordinals. A formula is monadic second order (monadic for short) if each of its variables is assigned a type, either the type "first order" or the type "second order." When interpreting the formula over a structure with universe A, the first order variables are taken to range over elements of A, and the second order variables are taken to range over subsets (or subclasses) of A. For more on monadic theories we refer the reader to Gurevich [4]. Let us here note that monadic formulae do not allow, at least not directly, talking about sets of pairs of elements of A. In particular they need not introduce G

• Type: Journal
• Year: 2008
• Where: JSYML
• Authors: Itay Neeman
• Keywords: JSYML 2008, Order Formula, Singular Cardinal, State Automata