Talk:Ultrafilter

axiom 2 is dependent

Most natural, perhaps, to regard U as a boolean homomorphism, as is said in the article. Then axiom 2 follows from the rest.

A \subset B iff A \cap B^c = \{\} which implies f(A)\wedge \neg f(B) = 0, i.e., f(B)\imples f(A) from axioms 1,3,4

MotherFunctor (talk) 21:43, 17 February 2009 (UTC)[reply]

You're right that axiom (2) follows from (1),(3),(4). I think the advantage of this presentation is that (1)-(3) define a filter, and (4) is then the property that distinguishes ultrafilters. Perhaps this should be made more explicit. EdwardLockhart (talk) 08:46, 18 February 2009 (UTC)[reply]

subsets

Concerning the little argument about subset/subseteq, see the longstanding consensus described at Wikipedia:WikiProject Mathematics/Conventions#Notational conventions. In both instances subsetneq instead of subseteq would have been formally correct but confusing. Therefore subset would confuse both those readers who read subset as strict inclusion and those who are aware that both readings are possible and who would have to think about which is meant here because it's not sufficiently obvious that it doesn't make a difference. --Hans Adler (talk) 13:30, 19 February 2009 (UTC)[reply]

Thanks for the reference. The first part of the original text was

U\subset F\implies U=F

. That seems about as obvious a case as could be imagined where subseteq and subsetneq are equivalent. The other one looks obvious as well (since the case A=X is the same as A={}). If we think that's not the case, I guess I'll revert the first and leave the second alone. EdwardLockhart (talk) 16:41, 19 February 2009 (UTC)[reply]

Ultrapower construction of "hyperreals"

I think the following is highly misleading:

For example, in constructing hyperreal numbers as an ultraproduct of the real numbers, we first extend the domain of discourse from the real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead we define the functions and relations "pointwise modulo U", where U is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic.

On a minor quibble, the term "ultraproduct of the real numbers" doesn't make sense: product with what? It should be "ultrapower", i.e. a product of (infinitely many) copies of the real numbers.

More importantly, the individuals (i.e. the elements of the domain of discourse) in the ultrapower model are not sequences of real numbers. They are equivalence classes of sequences of real numbers modulo the ultrafilter. This means that the rest of the description needs to be revised. Rdbenham (talk) 04:45, 10 April 2009 (UTC)[reply]

Nets and Universal Nets

How can an article on ultrafilters be considered comprehensive if it doesn't even mention the related (competing?) notion of a universal net? Rwilsker (talk) 14:14, 16 June 2009 (UTC)[reply]

This is not a featured article, so there is no implicit claim that it is more or less complete. From the point of view of mathematical logic, ultrafilters are an important tool while (universal) nets simply do not play a role. Why don't you write a section on universal nets if you care about them? --Hans Adler (talk) 14:42, 16 June 2009 (UTC)[reply]

As I understand it there is a "duality" between nets in general and filters in general, including a formal "bridge" between the notions. Also, at least some (all?) theorems have an analogue "across the bridge". This probably [decidedly if you ask me:)] deserves mention in the filter/net page or both. YohanN7 (talk) 22:24, 2 September 2009 (UTC)[reply]

It is mentioned well enoguh in filter that filters were developed as an alternative notion to nets. YohanN7 (talk) 23:02, 2 September 2009 (UTC)[reply]

Nets are important in topology, but ultrafilter is a much broader notion not restricted to the setting of topology. -- Walt Pohl (talk) 20:04, 10 November 2009 (UTC)[reply]

Axiom of Choice

The prove that there are free ultrafilters involve the axiom of choice, so we cannot make an explicit general construction of these ultrafilters. Nonetheless, we can show some specific examples. The Frechet filter is an example of a free ultrafilter. —Preceding unsigned comment added by 163.1.180.84 (talk) 03:51, 25 November 2009 (UTC)[reply]

The Fréchet filter is not an ultrafilter. — Emil J. 11:25, 25 November 2009 (UTC)[reply]

More precisely, the frechet filter is the intersection of all free ultrafilters. 109.253.189.86 (talk) 13:57, 29 June 2011 (UTC)[reply]

Ramsey filter

The definition of a Ramsey filter seems to require an unstated condition, namely that the C_n are nonempty. 109.253.189.86 (talk) 13:19, 29 June 2011 (UTC)[reply]

Yes, they certainly must be, but I believe that is usually implicit in saying a partition. The current article for partition of a set agrees with this convention, albeit with a citation from a combinatorics text. Wgunther (talk) 15:40, 19 July 2011 (UTC)[reply]

door topology

Can someone comment on the relation to door space? Tkuvho (talk) 15:03, 12 February 2012 (UTC)[reply]

Confusing tag

I added the confusing tag because the article intro is unintelligible to non-mathematicians. Lay readers should be given an intuition of the topic, as well as a sense of its history and importance to mathematics, before the symbols and jargon are explored. As an example, see infinitesimal. 69.130.248.32 (talk) 04:03, 21 September 2014 (UTC)[reply]

I've rewritten the introduction to make it more accessible. Let me know what you think. Grabigail (talk) 17:39, 25 September 2014 (UTC)[reply]

Names of axioms

Crasshopper, I consider the names you put on the axioms to be confusing. Perhaps some names might be appropriate, but not those. Also, I think we may need a source for those axioms. — Arthur Rubin (talk) 06:16, 25 December 2014 (UTC)[reply]

I would prefer that if you don't like them you replace them with better ones rather than deleting them. Reason being that someone who wants to scan the page should be able to get an idea from the English description and then invest more time to read the symbols for the exact meaning. Crasshopper (talk) 05:05, 9 January 2015 (UTC)[reply]

It might indeed be convenient if they had names. But we don't invent names, just for convenience in our articles. That's not what an encyclopedia does.

If you write a textbook, go ahead and come up with names, if you think it helps the exposition. If those names become widely accepted, then at some point, they can be reflected in Wikipedia. But not now. That's just not what we do. --Trovatore (talk) 08:29, 9 January 2015 (UTC)[reply]

Notation

Could the meaning of the symbol "⊊" please be stated on this page? Without that, it's impossible to understand the definition. Nathaniel Virgo (talk) 15:13, 25 February 2020 (UTC)[reply]

Done I linked the first occurrence of that symbol to its definition - Jochen Burghardt (talk) 17:18, 25 February 2020 (UTC)[reply]

My feeling is that linking symbols is pretty useless and can be visually distracting. Maybe we should call it out in prose instead? --Trovatore (talk) 18:58, 25 February 2020 (UTC)[reply]

"Proper" filters

I suggested on talk:filter (mathematics) that we consider adopting the convention that filters are "proper" by definition. I think this is a reasonably common convention in the wild, would save a lot of verbiage related to trivialities, make the article more readable, and not lose any useful generality. --Trovatore (talk) 00:25, 15 March 2020 (UTC)[reply]

Section on applications

Should the section be moved to Ultrafilter (set theory)? The section seems to deal with that notion. Paolo Lipparini (talk) 19:53, 4 January 2023 (UTC)[reply]

What preposition should we use by default?

Currently, this article seems to say "an ultrafilter on a given partially ordered set $P$ ", which can cause confusion with the phrase "an ultrafilter on a set $X$ ", where $P$ would actually be the power set of $X$ ordered by set inclusion.
Note 1 in the "Notes" section mentions that "Some authors [citation needed] use '(ultra)filter of a partial ordered set' vs. 'on an arbitrary set'".
@PatrickR2: You used "ultrafilter in a partially ordered set" on Talk:Ultrafilter (set theory), so I followed suit there, although I don't know if that's just a personal habit or whether a more notable source also uses "in".

Should we keep using the potentially confusing "on" as the default (especially as we try to clarify the relation between this article and Ultrafilter (set theory)), or should we switch to another preposition across the board? Bbbbbbbbba (talk) 06:05, 28 May 2023 (UTC)[reply]

Certainly "ultrafilter on a set" seems the most common usage for that case. I wrote "ultrafilter in a poset" without thinking too much, just wanting a different proposition to keep things straight. But looking at Jech, Set theory, and various posts in https://mathoverflow.net, it seems that "ultrafilter on a poset" is also used most frequently for that second case. ("ultrafilter of a poset" was only used in Davey and Priestley?) So I would just leave things alone, relying on context to know which case is being considered. It should not cause confusion for readers who are at the level of appreciating any of the two ultrafilter articles. PatrickR2 (talk) 16:49, 28 May 2023 (UTC)[reply]