Talk:Ultrafilter

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 Field: Foundations, logic, and set theory

axiom 2 is dependent[edit]

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)

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)

subsets[edit]

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)

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)

Ultrapower construction of "hyperreals"[edit]

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)

Nets and Universal Nets[edit]

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)

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)
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)
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)
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)

Axiom of Choice[edit]

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)

The Fréchet filter is not an ultrafilter. — Emil J. 11:25, 25 November 2009 (UTC)
More precisely, the frechet filter is the intersection of all free ultrafilters. 109.253.189.86 (talk) 13:57, 29 June 2011 (UTC)

Ramsey filter[edit]

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)

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)

door topology[edit]

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