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Problems about the Axiom of Choice

 

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  • Product Description
 

The 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.

Product Specifications
SKU :COC95651
AuthorWataru Asanuma
LanguageEnglish
BindingPaperback
Number of Pages176
Publishing Year2011-05-18T00:00:00.000
ISBN978-3844323665
Edition1 st
Book TypePhilosophy
Country of ManufactureIndia
Product BrandLAP LAMBERT Academic Publishing
Product Packaging InfoBox
In The Box1 Piece
Product First Available On ClickOnCare.com2015-08-14 00:00:00