{"product_id":"problems-about-the-axiom-of-choice-von-wataru-asanuma","title":"Problems about the Axiom of Choice","description":"\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 Gödel's  Incompleteness Theorems. The Löwenheim-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\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783844323665\"\u003e\u003ch3\u003eIn Defense of Platonic Realism in Mathematics\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783844323665","offer_id":39495910785117,"sku":"9783844323665","price":68.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/6b0b7a83-a6ab-4ddf-8d4a-b9f0c8e5ef1a.jpg?v=1770448597","url":"https:\/\/shop.autorenwelt.de\/en\/products\/problems-about-the-axiom-of-choice-von-wataru-asanuma","provider":"Autorenwelt Shop","version":"1.0","type":"link"}