
Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 22, 2013 2:35 PM


On Saturday, June 22, 2013 11:00:44 AM UTC7, muec...@rz.fhaugsburg.de wrote: > On Saturday, 22 June 2013 19:24:01 UTC+2, Virgil wrote: > > > Without having assumed some one of the many forms of the axiom of choice, neither Zermelo, nor anyone else has ever claimed to have proven that every set can be well ordered. > > > > The axiom was not assumed as the typical nonsense of present day "logic" but as true. > > > > Listen to Bertrand Russell: > > An /existent/ class is a class having at least one member. [1, p. 47] {{Surely you are joking Mr. Russell? The class without any member is not among the existent classes?}} > > > > > > Whether it is possible to rescue more of Cantor's work must probably remain doubtful until the fundamental logical notions employed are more thoroughly understood. And whether, in particular, Zermelo's {{Auswahl}} axiom is true or false {{ja, konnte ein Axiom für sich genommen denn zu Ihrer Zeit noch wahr oder falsch sein, Herr Russell? Suchte und vermutete man in der Mathematik damals etwa noch Wahrheit und Sinn?}} is a question which, while more fundamental matters are in doubt, is very likely to remain unanswered. The complete solution of our difficulties, we may surmise, is more likely to come from clearer notions in logic than from the technical advance of mathematics; but until the solution is found we cannot be sure how much of mathematics it will leave intact. [1, p 53] {{Ach, hätte doch Herr Cantor niemals beschlossen, Mathematiker zu werden!}} > > > [1] [Bertrand Russell: "ON SOME DIFFICULTIES IN THE THEORY OF > > TRANSFINITE NUMBERS AND ORDER TYPES", Proc. London Math. Soc. (2) 4 (1906) 2953, Received November 24th, 1905.  Read December 14, 1905.] >
In the last hundred years there have been great advances in Logic and its application to Mathematics. We, Mathematics, now know that there can be more than one consistent formal system in Mathematics, even if they contradict each other. For example, ZFC and ZF + AD. Mathematicians us both of these systems, since they relate to different problems. Neither one is "Right" or "Wrong".
> > You see: Not the present Fools and Cranks of logic are speaking here, but a profound expert of logic. >
Whose ideas have been amended and surpassed by over a century of research.
> > > Regards, WM
ZG

