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\begin{center}{\huge  Recursive infinitary formulas}\end{center}
\begin{center}{\large Julia Knight}\end{center}
\begin{center}{\large University of Notre Dame}\end{center}
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Roughly speaking, {\it recursive infinitary} formulas are formulas of
$L_{\omega_1\omega}$ in which the infinite
disjunctions and conjunctions are over r.e. sets.   All together, the
recursive infinitary formulas have the same
expressive power as the {\it hyperarithmetical} formulas.   However,
recursive infinitary formulas form natural
hierarchies corresponding to certain measures of recursive complexity.
Definability by an appropriate recursive
infinitary formula explains why, for example, in recursive copies of a
given structure, the image of a certain
relation always is $\sum^0_3$, or always $3$-r.e., or why, anytime the
image of one relation is $\sum^0_3$, the image
of another relation must be $\sum^0_5$, etc.

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