Questions tagged [set-theory]
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 theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.
11,847 questions
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Absent the Axiom of Regularity, could a set contain as an element its own power set?
$\newcommand\Ps{\mathcal{P}}\newcommand\mjt\textsf$It is easy to see in $\mjt{ZFC}$ that there is no $X$ such that $\Ps(X) \in X$; otherwise, $\{X, \Ps(X)\}$ has no $\in$-minimal element, ...
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Critical point of extender sequence
I am going through Mitchell and Steel's Fine Structure and Iteration Trees. In the proof of a theorem, we have an extender sequence $\vec{E}$ from the construction of an iteration tree, and we ...
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Cardinality of (0, ω) in surreal numbers [closed]
This question has come to me while thinking through the John's Conway lecture where he introduces surreal numbers.
An understanding of infinity that I took away from the Calculus I is that it is best ...
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Cardinality of the fourth powerset equals itself squared
Let $X$ be any infinite set, then $\sf{ZFC}$ proves that $|X|=|X|+1$, and that $|X|=|X|\cdot 2$, and moreover that $|X|=|X|^2$. These equalities inherently rely on Choice however, and it's consistent ...
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Definition of $\aleph_1$ [duplicate]
By definition, $\aleph_1$ is defined as being the smallest cardinal after $\aleph_0$. How are we sure that $\aleph_1$ is well-defined this way?
Just like there is no smallest real number greater than $...
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Did I just discover a new set theory paradox? [duplicate]
Say we have a set S that contains all the empty sets. By our knowledge,
$S = \{\emptyset\}$
But wait! Every set technically has $\emptyset$, so it doesn't matter if we add it as an element or not ...
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Are all subjects are written with the same ink? [closed]
I think It is actually an answerable question because solution of this question will vary with people and perspectives. But still I want your opinion.
Framing the question:
Suppose if we consider all ...
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set filter vs poset filter
Jech gives the following definitions in Chapters 7 and 14 of Set Theory (2002)
Given a set $S$, a set filter $F$ on $S$ is a subset of $\mathcal{P}(S)$ satisfying the following properties:
$S \in F$ ...
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Semmes' construction of a rapid filter
There are several expositions of Raisonnier's simplification of Shelah's proof that "all subsets of $\mathbb{R}$ are measurable" implies "$\aleph_1$ is inaccessible in $L$", over $\...
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Can you build the cartesian product for sets S,T with only extensionality, comprehension, pairing, and union?
I'm reading Kunen's foundations of mathematics and he just introduced the Replacement Axiom as a way to prove that the Cartesian product $S \times T$ exists for arbitrary sets $S, T$, and also claims $...
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countable union of sets and largest cardinality possible. [duplicate]
Let's define a sequence of sets as $A_0 = \varnothing$ and subsequently, terms are obtained by taking power set of the previous term.
If we define $$B := \bigcup_{i=0}^\infty A_i$$
What would be ...
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Exercise 15 in Bourbaki theory of sets in Chapter 3, section 1
I am trying to solve the second part of Exercise 15 (b) in Bourbaki’s theory of sets in Chapter 3, section 1. The following is the necessary definitions and notations to comprehend the problem:
Let $...
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What in the metatheory makes $\forall x$ in ZFC range over objects of a model of ZFC? [duplicate]
I am wondering what the reason is why $\forall x$, in ZFC, is understood as ranging over all objects of a model of ZFC (i.e. all sets in our universe of sets). From this answer I can see that it comes ...
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Is there a cardinality for which no structure with this cardinality has a countable dense substructure?
I have some trouble wrapping my head around some aspects of density. For starters it has always been weird to me that $\mathbb{Q}$ can be dense in $\mathbb{R}$ when one is countable and the other is ...
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Can choice be made sensitive to large cardinal axioms?
It is known that the consistency of the Axiom of Determinacy ($\mathsf{AD}$) is linked to the consistency of large cardinal axioms. For instance, the existence of infinitely many Woodin cardinals and ...
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