Ultrafilter Richness: Explore Options Beyond AC

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In summary, the conversation discusses the two options for ultrafilters: principal ultrafilters built from singletons and nonpricipal filters containing the cofinite filter. The speaker questions the limitations of these options and wonders if giving up AC or the ultra condition would lead to more exotic objects. The expert mentions that using an ultrafilter construction while still keeping AC can result in measurable cardinals. The speaker clarifies that they are looking for objects beyond principal ultrafilters and finite set subalgebras. They also mention the possibility of non-principal ultrafilters containing objects with complements greater than finite sets. They express a desire to fully understand the conceptual reach and limits of ultrafilters before investing time in them.
  • #1
BDV
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0
Hello,

With the axiom of choice, we are left with two options for ultrafilters:
a) principal ultrafilters, built from a singleton {x}.
b) nonpricipal filters of which all contain the cofinite filter, ergo complements of finite sets subalgebras.

Isn't this kind of flimsy? To get to more exotic/exciting objects does one:
give up AC
or
gives up the ultra in ultafilter (the A or X\A is in F condition)?
 
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  • #2
Why flimsy? For example, using an ultrafilter construction and without abandoning AC, you end up with measurable cardinals, which can do all sorts of interesting things. What sort of "exotic" objects did you have in mind?
 
  • #3
I meant more exotic than the principal ultrafilters and complements of finite set subalgebras.

I realized last evening that non-principal ultrafilters may also contain objects with complements greater than finite sets. So there may be some richness right there.

I just wanted to get a good feel of the conceptual reach/limits of ultrafilters before I sink a significant amount of work in them. Ars longa, vita brevis.
 

1. What is Ultrafilter Richness?

Ultrafilter Richness is a concept in mathematics that explores options beyond the Axiom of Choice (AC). It is a way of examining mathematical structures and theories without relying on the Axiom of Choice, which is a controversial assumption in mathematics.

2. Why is it important to explore options beyond AC?

The Axiom of Choice has been a subject of debate in mathematics for many years. By exploring options beyond AC, we can better understand the strengths and limitations of different mathematical theories and potentially discover new and more efficient ways of proving mathematical theorems.

3. How does Ultrafilter Richness relate to other mathematical concepts?

Ultrafilter Richness is closely related to the study of set theory, particularly in the area of infinitary combinatorics. It also has connections to other areas of mathematics such as topology and functional analysis.

4. What are some examples of Ultrafilter Rich structures?

Some examples of Ultrafilter Rich structures include Boolean algebras, topological spaces, and groups. These structures have been studied extensively in the context of Ultrafilter Richness and have led to new insights and results in mathematics.

5. How does Ultrafilter Richness impact the foundations of mathematics?

Ultrafilter Richness has the potential to impact the foundations of mathematics by providing alternative approaches to proving mathematical theorems. It also challenges the traditional reliance on the Axiom of Choice and opens up new avenues for research in various mathematical fields.

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