Talk:Filters on N
I'm confused. Is this a page about ultrafilters (i.e. certain collections of subsets) or about the hyperreals? I'm suggesting that the current contents of the page removed or moved to a whole other page (is there a natural place for them here?) and the page rewritten to discuss what are filters, ultrafilters, and how they come into use in large cardinals (e.g. measurable cardinals, Mitchell order, etc.)--Asafk 17:37, 2 March 2012 (PST)
Maybe some misunderstanding. There was a title "Ultrafilters" with no explanation of why or what for. One of the uses of ultrafilters is to construct the hyperreals which introduce a concept of infinity which is not that of cardinality, hence en enrichment of the concept. It would be a pity to reduce the notion of infinity to cardinals.
One possibility could be to keep the content of the page strictly about filters and ultrafilters and mention that these can be used in different areas, such as hyperreals -- with a link, or whatever other links people choose to create.
By the, Asafk, considering what you say on your user page, ultrafilters mustn't be your favourite since they rely heavily on the axiom of choice...ROD
When I look at the organisation of the site, I see that the idea is to discuss infinities in general, yet the different attics aim only at cardinality type infinities. All the infinities that can be used in calculus such as hyperreals or other nonstandard infinities, don't fit in any existing attic. We would need a "branching" for these. How about "upper floor", since they are above the parlour yet below the attic... ROD
I see what you mean. However this is not directly related to ultrafilters, indeed this is a point worth noting. I could perhaps fit two answers I have written on math.SE on the difference between $\aleph$ numbers and calculus related $\infty$ and the hyperreals into the format of this site. Regardless to that, ultrafilters are not the hyperreal numbers and I think that the entries should discuss a topic rather than elaborate on its uses instead. I think we should move the contents of this to a whole other section. Perhaps open this as a topic on the main talk page - should there be a section for general discussion on infinite objects. I think this page should give the definition of ultrafilters and discuss where they are found naturally in contexts of cardinals, some mentioning of the axiom of choice and related topics, and so on.
As for your cunning remarks, I don't have a problem with ultrafilters. Indeed $\aleph_1$ can be measurable without the axiom of choice! :-) --Asafk 12:22, 3 March 2012 (PST)
I think you point to a general problem about this site. Is it about infinite cardinals and ordinals? or is it about infinities in general (in mathematics of course). Maybe this question should be decided by site administrators. The name "Cantors attic" seems to refer to cardinals and ordinals, but the portal seems to imply a broader view. I can accept both choices, but if the decision is that only cardinality type infinities are considered, then the opening page should clarify the issue.ROD