Are Mathematical Theories Reducible to Non-analytic Foundations?

Axiomathes 23 (1):109-135 (2013)
In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process
Keywords Axiom of Choice  Finitistic  First-level idealization  Gödel’s incompleteness theorems  Individual-substrate  Infinite  Intentionality  Second-level idealization  Skolem-Löwenheim Theorem
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DOI 10.1007/s10516-012-9182-3
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Karl Popper (1935). Logik der Forschung. Journal of Philosophy 32 (4):107-108.

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