Tagged Questions

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 ...

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Club class of inaccessibles

I am currently looking at what Drake calls the Axiom Schema F, "Every normal function defined for all ordinals has a regular fixed point". In ZFC+(Axiom F), does it hold that there is a club class of ...
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Is a vector space isomorphic to the kernel $\oplus$ image of a map out of it?

Let $f:V\to W$ be a linear map of finite-dimensional vector spaces. By simply counting dimensions and using rank-nullity, it is clear that $V\cong \mathrm{im}\,f\oplus\mathrm{ker}\,f$. I want to know ...
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What is the cardinal of non measureable set?

we know that $|P( \Bbb{R} )|=|L (\Bbb{R} )|$ ( $L (\Bbb{R} )$ is the set of all Lebesgue measureable set ) note that $L (\Bbb{R} ) \subsetneq P( \Bbb{R} )$. What is the cardinal of non ...
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Partial order on the set of ordinal functions

Let $\kappa$ be a regular ordinal. Say that for $f,g: \kappa \rightarrow \kappa$, $f \leq g$ iff $f(\alpha) \leq g(\alpha)$ for sufficiently large $\alpha$. Now define $(X_{\kappa},\preceq)$ as the ...
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How to prove that the universal sets “V ” can be well-ordered

I have a idea for prove the principle in my topic.Due to my limited knowledge.I will use a simple example to show my thought briefly: First of all,Suppose a set Vα∈V("α" is an abitrary ordinal number)...
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Consistency strength of a proposition about models of ZFC

I made a proposition about models of ZFC, which says that every countable model of ZFC is really countable in larger countable model: For every countable transitive model $M$ of ZFC there is a ...
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Measure of an antichain of the Random forcing

I'm dealing with the random forcing $\mathbb{P}=\mathbb{M}\mathbb{B}(X,\mu)$ defined in Kunen's 2013 book. Recall that the conditions of that forcing are the equivalence classes of positive measureble ...
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How do you call the operation $\Lambda(X) = \{\alpha\in X| |X\cap\alpha|=\alpha\}$ on the ordinals?

I found this operation $\Lambda(X) = \{\alpha\in X| |X\cap\alpha|=\alpha\}$ with $X\subset On$ in the book "The higher infinite" of Akahiro Kanamori and can't really imagine what it does. So could ...
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Property ccc of the random real forcing

I'm dealing with the random forcing $\mathbb{P}=\mathbb{M}\mathbb{B}(X,\mu)$ defined in Kunen's 2013 book. Recall that the conditions of that forcing are the equivalence classes of mesurable sets ...
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Why are the Separation axioms 'too weak to develop set theory with its usual operations and constructions'?

I was reading Jech's Set theory; there after introducing Russell's Paradox, he asserted: The safe way to eliminate paradoxes of this type is to abandon the Schema of Comprehension and keep its ...
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Cardinality of the base of a ring of sets

Concretely, my question is: What is the size of the minimal base of a ring of sets? In my understanding the base is the set of elements that can be used to "deduce the rest". E.g., if I have a ring ...
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About the existence of the diagonal set of Cantor

The classic proof of the Cantor set start with the assumption that the set $$B=\{x\in A:x\notin f(x)\}$$ exists, where $f: A\to\mathcal P(A)$ is a bijective function. I understand the proof but I ...
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Can one have a theory that includes its own consistency as an axiom?

Consider the theory with the following axioms: The axioms of ZFC The "axiom of consistency": "This theory, including this axiom and all of the theory's other axioms, is consistent." Phrased ...
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Filter, which does not have the Baire property

I do not understand the following proof of the following theorem. It is part of Theorem 4.1.2 (Talagrand theorem) in the book Tomek Bartoszyński and Haim Judah: Set theory. On the structure of the ...
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Jech 3rd Edition Section 12 page 162 Models of Set Theory

Jech page 162 states : Let Form denote the set of all formulas of the language {$\in$}. As with any actual (metamathematical) natural number we can associate the corresponding element of N (i.e. the ...
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Prikry Hypothesis implies Kurepa Hypothesis

I want to show that $\textbf{HP}_\kappa$ implies $\textbf{HK}_\kappa$, Prikry hypothesis and Kurepa hypothesis in $\kappa$, respectively. Where \textbf{HK}_\kappa: \text{ There's a ...
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A Puzzle on Infinity: How to guess the color of hats? [duplicate]

Infinitely many (i.e. $\omega$ - many) people each have either a white hat or black hat on their heads. Each person can see everyone's hats except their own. Each person simultaneously announces a ...
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How do you formally state the axiom of constructibility?

The Axiom of Constructibility states every set is constructible. My question is, how would one formally state this, since it seems to involve quite a bit of metamathematics? In particular, if one ...
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Polynomial ring with arbitrarily many variables in ZF

For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom ...
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How does the second part of the axiom schema of replacement imply that the image of the function created in the first part is a set?

Thanks to the post here, I understand how the first part of that axiom schema, ${\forall}y({\exists}x:({\forall}z(P(y,z){\iff}(x=z))))$, defines a function. The idea replacement conveys is that the ...
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An infinite set of axioms in ZF? What does that mean?

Before write this question, I lookeded around enough in this forum for a possible answer and although there are many similar questions, I couldn't find one answer which understand or satisfies me. I ...
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Why do we need ultrafilter for construction of hyperreal numbers?

While trying to get some hold on the hyperreal numbers I found that their construction using the already existing real number system $\mathbb{R}$ requires the use of objects called free ultrafilters ...
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Forcing with a “shrinked” poset

Assume, in $V$, that we have some forcing $P$, a $P$-name $\tau$ and some sentence in the forcing language $\varphi(\tau)$ (it may include other names but we focus on $\tau$). Now we take some ...
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Why isn't there a need to specify a single $x$ in the axiom schema of replacement? [closed]

The axiom schema of replacement needs a function, defined by ${\forall}y({\exists}x:({\forall}z(P(y,z){\iff}(x=z))))$, where $f(y)=x$. My question is: why isn't ${\exists}$ replaced by ${\exists}!$? ...
Prove equinumerosity between $2^\mathbb{N}$ and R total orderings [duplicate]
T=$\left\{R\vert R \text{ is a total order over } \mathbb{N}\right\}$ Prove that T and $2^\mathbb{N}$ are equinumerous.
The Continuum Hypothesis states that $$2^{\aleph_0}=\aleph_1$$ And Cantor put it equivalently as: "There is no uncountable subset $A$ of $\mathbb{R}$ such that $|A| <\mathbb{R}$." Why are ...