On 3/22/2013 4:24 PM, WM wrote: > On 22 Mrz., 21:54, William Hughes <wpihug...@gmail.com> wrote: >> On Mar 22, 7:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:> On 22 Mrz., 16:24, William Hughes <wpihug...@gmail.com> wrote: >> >> <snip> >> >> >> >>>> Infinite sets are different from finite sets >>>> but they do not contain anything >>>> "beyond any finite set". >> >>> Of course. >> >> We have now established that there >> are sets that do not contain anything >> "beyond any finite set" but >> are different from finite sets. > > That is potential infinity. > Actually infinite sets, however, contain something beyond any finite > set. > Consider the decimal representation of pi. It is an actually infinite > sequence beyond any finite sequence.
There is nothing "beyond".
There is only the complete informational content needed to refer to pi singularly that relies on a completed infinity for the specification of the expansion.
If you read Leibniz, his logic was intensional. This is in contrast to the extensional logic of the Scholastic thinkers who preceded him. For Leibniz, the genus had been prior to the species in terms of information content. (Aristotle also followed this directionality of order).
The reason for Leibniz' position had been that the specification of a species required a more complex explanation of distinguishing features.
In case you have not noticed, the Peano axioms include:
(m+1)=(n+1) -> m=n
which is consistent with this directionality -- the extensionally greater genus is prior to the extensionally lesser species.
And, when one acknowledges what Leibniz actually wrote concerning the principle of identity of indiscernibles:
> "What St. Thomas affirms on this point > about angels or intelligences ('that > here every individual is a lowest > species') is true of all substances, > provided one takes the specific > difference in the way that geometers > take it with regard to their figures." > > Leibniz >
Then Cantor's intersection theorem,
> > "If m_1, m_2, ..., m_v, ... is any > countable infinite set of elements > of [the linear point manifold] M of > such a nature that [for closed > intervals given by a positive > distance]: > > lim [m_(v+u), m_v] = 0 for v=oo > > then there is always one and only one > element m of M such that > > lim [m_(v+u), m_v] = 0 for v=oo" > > Cantor to Dedekind >
characterizes the fact that infinity is required for the identifiability of singular reference.