Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

set theory 65This view leads right away to many questions. Is mathematics a science, hence needingvalidation through experimentation? Is verification of truth or falsity an absolute,or is it dependent on language, social factors, and time? Is validation via computeracceptable in mathematics? Here opinions differ and no consensus has been reached oris likely to be reached in the future. My opinion is that indeed mathematics, notwithstandingits abstractedness and appearance of absoluteness, has an experimental aspect:The Experiment: Verification of truth or falsity.The Input: The supposed proof of a mathematical statement.The Hardware: Biological (i.e., a brain, firing signals along neurons).The Software: Two-valued (i.e., True–False Boolean) logic.The Output: True, False, Fail (i.e., an incomprehensible paper).This approach has limitations. Already, a definition of truth in a language cannotbe given within the given language, which must be extended in order to define truth.The value (True-False) of a proposition may depend on how we extend the language,so one must give away the notion of an absolute truth and replace it by a semanticdefinition. Finding a “good” model of mathematics (i.e., the language), compatible withthe intuition we derive from the natural world and experience, is therefore of primaryimportance. The search here continues, but in any case it has become evident that thewholesale rejection of infinity is not a good thing. In this search, the mathematician isguided by principles that go beyond mathematics, namely, a search for simplicity, order,linearity. When confronted with two different routes for approaching a new concept ora problem, he will follow Ockham’s choice, namely, the simple way. Only then will henot be mired forever in endless complications, and only then may he achieve the goalsof his research.2.7 Set TheoryThe revolution that allowed infinity to enter mathematics in a very precise, meaningfulway begins with the notion of set, introduced by Bernard Bolzano in 1847, one yearbefore his death: a naive definition of a set is as an aggregate of objects, irrespectiveof ordering.However, it was only with Georg Cantor that the mathematical foundation of settheory started. Before him, there was only one mathematical infinity, namely, thenegation of the finite, the unreachable. In his paper of 1874, which was destined tochange the course of mathematics, he noted that there were different orders of infinity.Cantor writes ℵ 0 for the infinity arising from the primitive counting as in Gamow’sbook, namely, 1, 2, 3 ...,∞. Then he shows that the continuum is a different type ofinfinity, because it cannot be counted. His argument to prove this is really remarkable,and I will repeat it here, albeit in a nonrigorous form.Let us say that a real number between 0 and 1 is an infinite sequence such as0.643546 ..., namely, 0. followed by an infinite sequence of decimal digits not allof them equal to 9 from some point onward. Suppose that we can count these real