(March 13, 2012 at 5:42 am)twocompulsive Wrote: How can one infinity which is quantitatively less than another infinity still be infinitive :All numbers? Reals? Hyperreals? Cardinals? Ordinals? Surreals? Complex Numbers? P-Adics too?
such as for example the infinity of positive numbers compared to the infinity of all numbers
I assume you mean reals, and you're talking about Cantor's diagonalization proof. I'm sure others
might to insist the cardinalities do genuinely exist, or don't in some fashion. Even if you think infinity
is an ill-defined non-concept, when we talk about unlimited sets, etc., this forms an order relation on sets:
"A =< B" iff A can be mapped injectively into B. By Cantor's logic, this relation need not be symmetric. It should
be immediately obvious that this relation must be transitive (just compose the injections). Even if you think different
'sizes of infinity' should not exist, this is a statement about sets. Even if 'the set of real numbers' is just an flourish of
language, without any content, we have a positive result: there is a sense in which some open-ended flourishes of language
are "bigger", or "more open-ended" than others. Do Cardinalities really exist? I don't know, and in a sense, it doesn't exactly
matter. Cardinalities are an incredibly useful abstraction. If their assumption does not lead to any contradiction, the results about
"existing" numbers that I can access via cardinalities will still be true. On the other point: in what sense does a non-maximal infinity
fail to be 'infinitive'? There is no reason a priori why the unlimited should also be absolute, beyond the fact that our existence is finite.
