On coding uncountable sets by reals

Abstract

The forcing we use implicitly provides reshaping of the given set A.

The full basis theorem does not imply analytic wellordering

“…Note that |s(ϕ, τ )| = |u|, and hence the multitree s(ϕ, τ ) ∪ q belongs to MT(π) as well. 3 Now goes the last condition.…”

“…Another modification of Jensen's forcing construction in [11] yields such a forcing notion in L that any extension of L, containing two generic reals x = y , necessarily satisfies ω L 1 < ω 1 . See [3,15] on some other modifications in coding purposes.…”

“…We claim that γ is as required. Indeed otherwise p 0 forc #" Π ↾ γ ϕ fails for a formula ϕ ∈ Φ, thus (3…”

Countable OD sets of reals belong to the ground model

“…Do some other simple generic extensions by a real (other than Cohen-generic, Solovay-random, dominating, ans Sacks) admit results similar to Theorem 1.1 and also those similar to the old folklore lemmas 4.1 and 4.2 above? It would also be interesting to investigate the state of affairs in different 'coding by a real' models as those defined in [1,9].…”

“…† IITP RAS, Moscow, Russia, lyubetsk@iitp.ru 1 The model presented in [11] was obtained via the countable product of Jensen's minimal ∆ 1 3 real forcing [6]. Such a product-forcing model was earlier considered by Enayat [4].…”