Consistency with the formal Church's thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by the second author and G. Sambin in 2005.Here we show that this is the case for the intensional level of the twolevel Minimalist Foundation, for short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT.The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church's thesis of a fragment of intensional Martin-Löf's type theory, called MLtt 1 , where mTT can be easily interpreted. Then to show the consistency of MLtt 1 with CT we interpret it within Feferman's predicative theory of non-iterative fixpoints ID 1 by extending the well known

Here, we present a category which can be considered a predicative variant of Hyland's Effective Topos for the following reasons. First, its construction is carried in Feferman’s predicative theory of non-iterative fixpoints . Second, is a list-arithmetic locally cartesian closed pretopos with a full subcategory of small objects having the same categorical structure which is preserved by the embedding in ; furthermore subobjects in are classified by a non-small object in . Third happens to coincide with the exact completion of the lex category defined as a predicative rendering in of the subcategory of of recursive functions and it validates the Formal Church’s thesis. Hence pEff turns out to be itself a predicative rendering of a full subcategory of .

MR ID 807435), ORCID: 0000-0002-4957-4799) is an associate professor of mathematical logic at the University of Padova. He received is Ph.D. in mathematics from the University of Palermo in 2007. His research topics include pointfree topology and constructive mathematics. He is also interested in mathematics education and is actively involved in the training of perspective teachers.

We give a constructive account of the frequentist approach to probability, by means of natural density. Then we discuss some probabilistic variants of the Limited Principle of Omniscience.

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