On 31 Jan., 23:35, david petry <david_lawrence_pe...@yahoo.com> wrote:
> It is eminently reasonable to believe that the purpose of mathematics is to provide a rigorous and practically useful conceptual framework for reasoning quantitatively about real world phenomena.
Just Cantor's aim. He devised set theory for application to reality. In a letter to Hilbert he wrote about his plan of a paper on set theory and its applications: The third part contains the applications of set theory to the natural sciences: physics, chemistry, mineralogy, botany, zoology, anthropology, biology, physiology, medicine etc. It is what Englishmen call natural philosophy. In addition we have the so called humanities, which, in my opinion, have to be called natural sciences too, because also the mind belongs to nature. (Original German text in Matheology § 039)
The objects that exist in a conceptual framework are concepts; the universe of mathematical objects is a collection of concepts. Concepts can be encoded in language. Languages are countable. The Cantorian claim that "uncountable" infinite collections exist is tantamount to the claim that mathematics should assert the existence of things that are not within the mathematical conceptual framework. That is a truly extraordinary and even bizarre claim that requires truly extraordinary evidence, and such evidence is lacking, and prominent and well respected mathematicians have pointed out that such evidence is lacking.
This is due to a reversal of implication. Every finite definition implies an infinite decimal, but not vice versa. Numbers _are_definitions. 1/4 or pi or 0.333... or 2.718281828andnotsoon are abbreviations of definitions. These definitions yield infinite sequences of digits, some of which eventually get constant, others do not. In any case we can never in mathematics, i.e., in the mathematical discourse, obtain a number from an infinite decimal because we never see the end and, hence, never can be sure what will follow. All we can use in mathematics is the finite definition of a number. Therefore undefinable numbers are not part of mathematics.
In Cantor's argument, these facts are inversed. There is an infinite list of infinite sequences. Neither of them exists in mathematics without a finite definition. But with a finite definition, every diagonal is defined too. Cantor's argument only shows that there is no completed "actual" infinity, i.e., there is no complete list. But if we assume completed infinity, then Cantor's argument shows that the _definable_ reals are uncountable, because every definable Cantor-list yields a definable diagonal.