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Published online by Cambridge University Press: 12 March 2014
H. Friedman proved that the following sentence SWQO(
) is true but not provable in the theory ATR0 (cf. [S]; ≔ the set of finite rooted trees, ∣T∣ ≔ the cardinality of T ∈ ).
For every k > 0 there is an n so large that for any T0, T1,…, Tn ∈ with ∀i ≤ n(∣Ti∣ ≤ k · (i + 1)) there are i < j ≤ n such that Ti is homeomorphically embeddable into Tj.
Since ATR0 is stronger than PA, SWQO() is not provable in PA, either. Furthermore, SWQO() is proof-theoretically stronger than PA. A similar sentence whose proof-theoretical strength is exactly that of PA was obtained by replacing finite trees by finite sequences of bounded natural numbers under the embeddability with H. Friedman's asymmetrical gap-condition (see [SS]). The latter sentence was modified by using the weaker symmetrical gap-condition (see the corresponding particular case in [Gl]). Further improvement is indicated in [G2, Corollary 4.5]. Namely, the assertion ∣Xi∣ ≤ k · (i + 1) as in SWQO(J), above, can be weakened to
.
(in fact, to wt(Xi) ≤ k + ε · i for an arbitrary fixed ε = l−1, l < 0, where wt(X) is the weight of the “word” X).
