# 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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### Consequences of the principle of dependent choices (DC)

It is known that if we assume the axiom of determinacy every set of real numbers is lebesgue measurable. In order to study this, I'm following Jech's Set Theory book. There, Jech says that apart from ...
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### Consequence of $V=L$

Assume $V=L$. Define $\langle A_\alpha\mid\alpha<\omega_1\rangle$ as follows: Let $A_\alpha$ be the $<_L$-least $A\subseteq\alpha$ such that $(\forall\beta<\alpha)A\cap\beta\not=A_\beta$ if ...
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### Intuitive motivation for the $\diamondsuit$-principle?

I'm having difficulty internalizing the $\diamondsuit$-principle. I understand what it says, and can see why some of the constructions that I've seen that use it work. But I don't think I understand ...
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### If ZFC has a model then the union of ZFC with the negation of inaccessible cardinals has a model

I would appreciate some help with proving the statement in the title. I'm new to model theory, so I won't understand technical terms or symbols that well. Hence a sketch of the proof will suffice. ...
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### $\Pi_2$ formula true for $L_{{\omega_1}^L}$ but not for $L$

I tried the formula [${\forall x}{ \exists} {f} $${x} is an ordinal {\land} {f}is an injective function {\land} dom f =x {\land} ran f {\subseteq} {\omega}$$]{ \lor}$ $x$ is not an ...
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### Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
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### How many sorts are there in Terry Tao's set theory?

In his 2010 post, A computational perspective on set theory, Terry Tao writes: The standard modern foundation of mathematics is constructed using set theory. With these foundations, the ...
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### $L$ and $L_{\omega_2}$ agree on all $\Sigma_1$ formulae?

I remember seeing this mentioned somewhere in a textbook I was looking at a while ago, but I can't seem to find the proof. Is anyone aware of the reference I'm looking for? Or can someone provide a ...
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### Help with a consequence of $V=L$

Assume $V=L$. Prove that there is a function $f:\omega_1\rightarrow\omega_1$, $\Delta_1$ definable over $L_{\omega_1}$, so that for every $A\subseteq\omega_1$ and every $\Sigma_1$ formula $\varphi$, ...
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### Does there exist a metric space of cardinality aleph-two that has a countable epsilon cover?

Does there exist a metric space $(M,d)$ such that $|M|=\aleph_2$ but there exists a countable $\epsilon$-cover of $M$?
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### Well-pruned $\kappa$-Suslin trees have the $\kappa$-Baire property

I'm having troubles seeing that a well-pruned $\kappa$-Suslin tree, with $\kappa$ uncountable and regular, has the $\kappa$-Baire property. I saw in Kunen's Set Theory that it can be seen by proving ...
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### Order-preserving maps on a directed-complete poset

Definitons. A poset $P$ is directed if every finite subset has an upper bound in $P$. It is directed-complete if every directed subset has a least upper bound in $P$. For any poset $P$ let $[P\to P]$ ...
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### A proof of maximality of an antichain in a complete Boolean algebra

Suppose $(\mathbf{B},\leqslant)$ is a complete Boolean algebra and let $|\mathbf{B}|=|\gamma|$. Let $C=\langle c_\alpha\mid \alpha<\gamma\rangle$ be a maximal descending chain without a lower bound:...
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### Reference: Mahlo cardinals remain Mahlo in L

The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in L. Since ...
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### Let $A \subseteq \mathbb{R}$ and $|A|=\omega$, prove $|\mathbb{R}\setminus A| = 2^\omega$. [duplicate]

Let $A \subseteq \mathbb{R}$ and $|A|=\omega$, prove $|\mathbb{R}\setminus A| = 2^\omega$. Hint: $\mathbb{R}\sim \{0,1\}^\mathbb{N}$ I'm somewhat stuck trying to prove this, could someone give ...
### If $\bigcup N_\alpha$ is stationary, then $\{ \min(N_\alpha) \}$ is stationary
It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $\kappa$ and a disjoint family ...