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W. Taylor Nagoya Math. J . Vol. 52 (1973), 31-33
A HETEROGENEOUS INTERPOLANT WALTER
TAYLOR
In this note we exhibit an interpolant for a certain valid implication |= φ —• ψ, where φ and ψ come from the infinitary language LmiΛ1. The existence of this interpolant follows from Takeuti's heterogeneous interpolation theorem [5], but unfortunately the proof in [5] is not explicit enough to allow one to find the interpolant explicitly. Takeuti's theorem asserts the existence of an interpolant in the class Lmiwi of heterogeneous formulas, which admits the rules of formation of Lωiωi plus the following additional rule: if ψeLωiωi and α[x ~y Aλx A A2y]
One may easily check that if Received March, 14, 1972. 31
^σ
(i = 1,2)
32
WALTER TAYLOR
and
then F = F\ Thus F is implicitly defined by σ, and hence by Takeuti's theorem together with the usual argument for Beth's theorem, there is a (heterogeneous) formula Φ(x, y) such that σ |= xFy Φ(#, #), with F not appearing in Φ. (Such Φ cannot be in any Lκλ, as follows from the proof of Malitz [3, Theorem 4.2].) The aim of this note is to explicitly exhibit Φ. Let C be the set of finite sequences of O's and 1/s (including the empty sequence • ) . For • Φ a = αoαx an_γan e C, we let σ = α0 an_x. For all σeC we take variables xσ and yσ. For all σeC, let Qσ stand for1 (yxσ0 e AJiaVio e A2)(v?/σl e A 2 )(3^ σl e Ax) . Now let Φ(x9y) be Qσ
