Abstract
According to the mainstream in the 20th century, the foundations of mathematics were identified with logic and set theory. Indeed, results concerning philosophically most interesting questions are often negative: the first order axiomatic set-theoretical universe is deductively incomplete, inevitably non-standard, and we have no clear idea of what the intended models of set theory are (part I). So, the foundational view of mathematics itself might be suspect. But in the spirit of Poincaré, one should look for an other solution. He remarks that the varieties of classical first order theories is unable to deal with the most common modes of mathematical reasoning such as complete induction and model building. For such a purpose, Hintikka's IF-Logic seems to be an adequate way-out.
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Heinzmann, G. (2009). Some Coloured Remarks on the Foundations of Mathematics in the 20th Century. In: Rahman, S., Symons, J., Gabbay, D.M., Bendegem, J.P.v. (eds) Logic, Epistemology, and the Unity of Science. Logic, Epistemology, And The Unity Of Science, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2808-3_4
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