This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model ...
2
votes
1answer
14 views
Dissecting a proof of the $\Delta$-system lemma (part II)
This is part II of this question I asked yesterday. In the link you can find a proof of the $\Delta$-system lemma. In case 1 it uses the axiom of choice (correct me if I'm wrong). Now one can also ...
5
votes
0answers
23 views
Tightness and countable intersection of neighborhoods
The following is a problem a colleague has encountered. He would like to know whether the following conjecture is right, wrong, or neither:
Let $X$ be a topological space of countable tightness ...
10
votes
1answer
60 views
Bijection between power sets of sets implies bijection between sets?
Is it true that if $X$ and $Y$ are sets and there is a bijection between $\mathcal{P}(X)$ and $\mathcal{P}(Y)$ then there is a bijection from $X$ to $Y$ ?. I believe this should be obvious, but I ...
2
votes
1answer
42 views
Dissecting a proof of the $\Delta$-system lemma
I have a question about the following proof of the $\Delta$-system lemma:
It seems to me that $\mathscr E'$ is not needed. I would start case 1 by "If for every $a \in \bigcup \mathscr E$ the ...
2
votes
2answers
93 views
Isomorphic Free Groups and the Axiom of Choice
When I read about free group, the proof which concerns about two free groups $F(X)$ and $F(Y)$ are isomorphic only if $\operatorname{card}(X) = \operatorname{card}(Y)$ has a sentence going as follows:
...
2
votes
1answer
34 views
Two cumulative hierarchies
I'm trying to understand the purpose of
the rank of a set, and more generally
the cumulative hierarchy.
And although the comments left there are good, I find myself wanting a deeper ...
8
votes
0answers
83 views
On the large cardinals foundations of categories
It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when talking about small categories, and perhaps at ...
3
votes
3answers
85 views
Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?
I recently started studying set theory and am having quite a bit of difficulty accepting Cantor's diagonal proof for the uncountability of the reals. I also saw a topological proof via nested sets for ...
3
votes
2answers
47 views
Intersection of $\sigma$-algebras and set theory
Theorem: Given $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$, where each $E_\alpha$ is a $\sigma$-algebra on $X$. Then $E:=\bigcap_{\alpha \in \mathcal{A}}E_\alpha$ is a $\sigma$-algebra.
Proof: Take ...
1
vote
1answer
37 views
Defining Test-Objects
In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
2
votes
1answer
50 views
Factoring out a Cohen forcing
Suppose that in a forcing extension $V[G]$ by some ccc forcing $P$ there is a Cohen real over $V$. By a general argument we can factor $P$ into an iteration $P=\mathrm{Add}(\omega,1)*\dot{Q}$ for some ...
7
votes
0answers
68 views
Showing a filter on the Power set of $\mathbb{Z}$ is a one point Filter
Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
2
votes
0answers
52 views
What are the major uses of the rank of a set?
We can define the rank of set $X$ as the least ordinal $\alpha$ such that $X \in V_\alpha$, where $V_\alpha$ is the $\alpha$-th stage of the cumulative hierarchy. However, I have yet to see this idea ...
7
votes
2answers
100 views
Assuming Freiling's axiom of symmetry, can we define a specific set $X \subseteq \mathbb{R}$ such that $|\mathbb{N}|<|X|<|\mathbb{R}|$?
I learned here that GCH isn't very popular among set theorists, for a variety of reasons. I also learned about Freiling's axiom of symmetry, which (in the presence of the other ZFC axioms) is ...
4
votes
3answers
105 views
Why infinite cardinalities are not “dense”?
What tells us that the structure of the cardinals is "discrete"? I'm not using the words "discrete" and "dense" with their formal meanings.
Maybe I have this confusion because I'm using concepts ...
