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We show that it is possible to define a realizability interpretation for the$\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. Thissupplements a previous result of Schwichtenberg regarding bar recursion attypes 0 and 1 by showing how to avoid using bar recursion altogether. Ourresult is proved via a conservative extension of System T with an operator forcomposable continuations from the theory of programming languages due to Danvyand Filinski. The fragment of Analysis is therefore essentially constructive,even in presence of the full Axiom of Choice schema: Weak Church's Rule holdsof it in spite of the fact that it is strong enough to refute the formalarithmetical version of Church's Thesis.

An interpretation of the Sigma-2 fragment of classical Analysis in System T