Abstract. In this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure A that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V , as does any structure B generically Muchnik reducible to a structure A of cardinality ℵ1. The former positive result yields a new proof of Harrington's result that counterexamples to Vaught's conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

In this paper we investigate the reverse mathematics of higher-order analogues of the theory ATR0 within the framework of higher order reverse mathematics developed by Kohlenbach [Koh05]. We define a theory RCA 3 0 , a close higher-type analogue of the classical base theory RCA0, and show that it is essentially a conservative subtheory of Kohlenbach's base theory RCA ω 0 . Working over RCA 3 0 , we study higher-type analogues of statements classically equivalent to ATR0, including open and clopen determinacy, as well as two choice principles, and prove several equivalences and separations. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel forcing; in the presentation of this result, we develop a new, more flexible framework for Steel-type forcing.

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We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of I-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of pN,`,¨q.2010 Mathematics Subject Classification. 03D25.

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