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Xn )}) 50 4 Pure Logic as the formula which is obtained from F by replacing all occurrences (Ut1 . ,un (t1 , . . ,tn ). Sometimes we write just FU (A) or even shorter F(A) if it is clear which variable is to be replaced. We have introduced L as a second-order language. We call FV(F) the free firstorder variables of F, BV(F) the bounded first-order variables of F and analogously FV2 (F) and BV2 (F) the free or bounded second-order variables of F. A term t with FV(t) = 0/ is closed. A formula F is a sentence if FV(F) = / A formula F is called first-order if BV2 (F) = 0.

C) If λ and κ are cardinals and there is no cardinal in the interval (λ , κ ) then κ is regular. (d) The class of regular ordinals is unbounded. 1 Definition For an ordinal α let Onα := {β ∈ On α ≤ β } denote the class of ordinals ≥ α . Let α + ξ := enOnα (ξ ) and call α + β the ordinal sum of α and β . Since Onα is obviously club in any regular κ > α , the function λ ξ . 19. 2 Observation The function λ ξ . 2 that α + ξ extends the addition of natural numbers into the transfinite. We easily check the following properties of ordinal addition.

8 Theorem (Cantor normal-form) For all ordinals α = 0 there are uniquely determined ordinals α1 , . . , αn such that α =NF α1 + · · · + αn . 3 Writing ω ξ as an exponential is not by accident. (cf. 3 Fundamentals of Ordinal Arithmetic 31 Proof We prove the existence by induction on α . If α ∈ H then α =NF α . Otherwise we have α = ξ + η with ξ , η < α . By induction hypothesis we get ξ =NF ξ1 + · · · + ξm and η =NF η1 + · · · + ηn . Then α =NF ξ1 + · · · + ξ j + η1 + · · · + ηn where 1 ≤ j ≤ m is the biggest index such that ξ j ≥ η1 .