Author:
HARRISON-TRAINOR MATTHEW,KRETSCHMER MILES
Abstract
Abstract
We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula
$\varphi $
(in
$\mathcal {L}_{\omega ,\omega }$
) is equivalent to a formula of the infinitary language
$\mathcal {L}_{\infty ,\omega }$
with n alternations of quantifiers. We prove that
$\varphi $
is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.
Publisher
Cambridge University Press (CUP)
Reference7 articles.
1. Infinite Forcing in Model Theory
2. A separation theorem for
${\varSigma}_1^1$
sets;Louveau;Transactions of the American Mathematical Society,1980
3. Invariant sets in topology and logic
4. An interpolation theorem for denumerably long formulas