Abstract
This chapter presents some general results that hinge on the notion of cardinality.
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Notes
- 1.
Georg Ferdinand Ludwig Philipp Cantor (1845–1918), a German mathematician, developed the theory of transfinite numbers, spelling out its fundamental concepts and proving its most important theorems. Although today set theory is widely accepted as a fundamental part of mathematics, this was not always the case, and Cantor had to argue for his view. Some theologians criticized the theory of transfinite numbers because they saw it as a challenge to the absolute infinity of God. Cantor contended that we can conceive God as the Absolute Infinite, a number which is bigger than any conceivable or inconceivable quantity. In fact, he believed that God himself communicated to him the theory of transfinite numbers.
- 2.
Leopold Löwenheim (1878–1957), a German mathematician, proved a first result in Löwenheim [43]. Albert Thoralf Skolem (1887–1863), a Norwegian mathematician, provided a simplified version of Löwenheim’s result in Skolem [59]. These results were first outlined in their full generality in Maltsev [46]. The upward result is attributed to Tarski. Surely it cannot be ascribed to Skolem, who did not believe in the existence of nondenumerable sets.
- 3.
Willard Van Orman Quine (1908–2000), an American philosopher, is widely considered one of the dominant figures in analytic philosophy in the second half of the twentieth century. He produced original and important works in several areas, including logic, ontology, epistemology, and the philosophy of language. He presents his arguments about the inscrutability of reference in Quine [54], among other works.
References
Löwenheim, L. (1915). Über Moglichkeiten im Relativkalkül. Mathematische Annalen, 76, 447–470.
Maltsev, A. I. (1936). Untersuchungen aus dem Gebiete der mathematischen Logik. Novaya Seriya, pp. 323–336.
Quine, W. V. O. (1968). Ontological relativity. Journal of Philosophy, 65, 185–212.
Skolem, T. (1920). Logisch-kombinatorische über die Untersuchungen Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen. Videnskapsselskapet Sktrifter, I. Matematisk-naturvidenskabelig Klasse, 4:1–36.
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Iacona, A. (2021). Theories and Models. In: LOGIC: Lecture Notes for Philosophy, Mathematics, and Computer Science. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-030-64811-4_18
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