On 2/20/2014 1:09 PM, email@example.com wrote: > On Thursday, 20 February 2014 18:39:28 UTC+1, fom wrote: > >> To the chagrin of many, including yourself, >> >> Cantor argues successfully for a metaphyical >> >> existence of limits as numbers and for a >> >> transfinite arithmetic. > > Cantor's diagonal argument works exclusively in the domain of terminating sequences which he himself as proven to be countable. Therefore the notion of uncountability, as far as it is based on this argument (the others can be disproven too), is far from being successful (although he has dazzled many mathematicians) but is simply self-contradictory. >
You keep wanting to blame Cantor. The others -- that is, the ones who "fixed" set theory -- basically ignored certain critical notions advocated by Cantor. I do not think this applies to Zermelo. But, his views did not hold sway. Russell, Goedel, and Skolem (possibly Fraenkel) are more responsible for much of the oddity than Cantor.
If you know the adage: Too many cooks spoil the soup.
Also, I did not say that Cantor succeeded in convincing his critics. But, I should have.
I think the phrase "for a metaphysical existence of limits", is poorly chosen, even if it is probably correct. I should have said that he successfully argued for an "arithmetic of limits" on the basis of metaphysical arguments which certain others found persuasive.