{"product_id":"problems-about-the-axiom-of-choice-in-defense-of-platonic-realism-in-mathematics-384432366x","title":"Problems about the Axiom of Choice: In Defense of Platonic Realism in Mathematics","description":"\u003cp\u003e\u003cstrong\u003eISBN:\u003c\/strong\u003e 384432366X\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eAuthor:\u003c\/strong\u003e Asanuma, Wataru\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eCondition:\u003c\/strong\u003e New\u003c\/p\u003e\u003cp\u003eThe conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. The controversy over the Axiom of Choice (AC) is a case in point. Due to its non- constructive nature, the AC has seemingly unpleasant consequences. It leads to the existence of non- Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the latter is so called in the sense that it is a counter-intuitive theorem. To see that mathematical truths are of non- constructive nature, I draw upon Gdel's Incompleteness Theorems. The Lwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine\/Putnam's arguments assume a clear meaning. According to them, the AC depends for its truth-value upon the model in which it is placed. In my view, however, this shows a limitation of formal methods. In response to Benacerraf's challenge to Platonism, the book concludes that in mathematics, as distinct from natural sciences, Platonists see a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality.\u003c\/p\u003e","brand":"Mia Karts","offers":[{"title":"Default Title","offer_id":51931576729888,"sku":"NEW384432366X","price":116.4,"currency_code":"USD","in_stock":true}],"url":"https:\/\/miakarts.com\/products\/problems-about-the-axiom-of-choice-in-defense-of-platonic-realism-in-mathematics-384432366x","provider":"Miakarts Books","version":"1.0","type":"link"}