On 3/18/2013 4:57 PM, WM wrote: > On 18 Mrz., 20:32, Virgil <vir...@ligriv.com> wrote: > >>> On the contrary. I am looking for any applications of infinite set >>> theory. But hitherto I could not find anything serious. >> >> Standard calculus is based on the infiniteness of the real number line >> and the infiniteness of the Cartesian plan. > > Yes, but many matheologians are unable to recognize that infinity is > not finished and that. and that calculus has been existing long before > Cantor.
Yes. Since Newton and Leibniz.
Berkeley had been the original critic of Newton's introduction of the infinite into mathematics. He, legitimately, compared the convergence of ratios to a set of accounting books out of balance equally and oppositely on the different sides of the ledgers.
Today that is referred to in the definition of differentials as the error approaching to 0 faster than the function approaching its limit.
There is just one minor issue with that view. It is based on prior knowledge of the derivative function and it is said in relation to a fixed point of the domain.
The debates begun by Berkeley occurred during the period when numerical techniques were being developed to approximate the solutions to problems of the infinitesimal calculus. This, in turn, set up the conditions for "solving" the problem of infinitesimals arithmetically. That would, if I recall correctly, be attributed to Lagrange and Cauchy.
Whoops! New problem: what *exactly* does it mean to be continuous?
Well, thanks to Lobachevsky, one could not turn to familiar geometry anymore.
So, Dedekind showed us how to represent the continuity of our functions through an arithmetical continuity on our domains.
Then, as noted before, there is Leibniz' statement of the principle of identity of indiscernibles linking nested logical domains with geometric intuition. I could be wrong, but it seems a lot like Cantor's intersection theorem stating that a nested sequence of closed non-empty sets with vanishing diameter is non-empty.
In other words, the only reason you can complain about Cantor today is because Berkeley complained about Newton in the past.