Correction to a paper on definability of ordinals in infinite logic

Extract

We wish to make a correction to our paper On definability of ordinals in logic with infinitely long expressions (this journal, vol. 31 (1966), pp. 365–375). Let Ω be an infinite singular cardinal and Ω⁺ the smallest cardinal >Ω. For the proof of the second half of Theorem 3, we incorrectly assumed that an ordinal γ is not definable in L_Ω, if γ is not cofinal with any ordinal ≤Ω. However, Ω⁺ is indeed definable in L_Ω and Theorem 3 should read:

If Ω singular, then an ordinal α is definable in L_α, if and only if σ ≤ exp(Ω⁺, Ω⁺), where exp (β,γ) denotes ordinal exponentiation β^γ.