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Alternatives to Zermelo's assumption..

New York,: New York (1927)

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  1. Zermelo's Cantorian theory of systems of infinitely long propositions.R. Gregory Taylor - 2002 - Bulletin of Symbolic Logic 8 (4):478-515.
    In papers published between 1930 and 1935. Zermelo outlines a foundational program, with infinitary logic at its heart, that is intended to (1) secure axiomatic set theory as a foundation for arithmetic and analysis and (2) show that all mathematical propositions are decidable. Zermelo's theory of systems of infinitely long propositions may be termed "Cantorian" in that a logical distinction between open and closed domains plays a signal role. Well-foundedness and strong inaccessibility are used to systematically integrate highly transfinite concepts (...)
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  • Step by recursive step: Church's analysis of effective calculability.Wilfried Sieg - 1997 - Bulletin of Symbolic Logic 3 (2):154-180.
    Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...)
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  • Long Borel hierarchies.Arnold W. Miller - 2008 - Mathematical Logic Quarterly 54 (3):307-322.
    We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω2. We also show that assuming a large cardinal hypothesis there are models of ZF in which (...)
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  • Alonzo church:his life, his work and some of his miracles.Maía Manzano - 1997 - History and Philosophy of Logic 18 (4):211-232.
    This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...)
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  • It is difficult to admit that the word if acquires, when written⊃, a virtue it did not possess when written if. Principia provided no very convincing answer to Poincaré. Indeed the fact that the authors of Principia saw fit to place their first two “primitive propo-sitions”. [REVIEW]Martin Davis - 1995 - Bulletin of Symbolic Logic 1 (3).
     
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