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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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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1answer
30 views

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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1answer
51 views

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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1answer
23 views

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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3answers
28 views

Why isn't there a need to specify a single $x$ in the axiom schema of replacement?

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 ${\exists}!$? ...
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0answers
34 views

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.
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4answers
262 views

Equivalent form of continum hypothesis

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 ...
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1answer
35 views

What's the meaning of the axiom schema of replacement?

The axiom schema goes: We have $∀y(∃x:(∀z(P(y,z)⟺(x=z))))$. Then we state as an axiom $∀w(∃x:(∀y((y∈w)⟹(∀z:(P(y,z)⟹(z∈x))))))$. I've seen it expressed in English as For any function $f$ ...
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2answers
56 views

Why intersection is not an axiom in naive set theory by halmos

Why intersection is not an axiom in naive set theory? though, union was given as an axiom
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1answer
56 views

Equivalents forms of $\diamondsuit$

I'm trying to see that assuming $\diamondsuit$ the following holds: Exists $\{A_\alpha\}_{\alpha<\omega_1}$ such that $A_\alpha\subset\alpha\times\alpha$ and for every $A\subset \omega_1\times\...
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2answers
69 views

Existence of model of ZF in which every uncountable cardinal is the cardinality of some power set?

Does there exist a model of ZF in which any uncountable cardinal number is equal to the cardinality of the power set of some set ? ( In ZFC it is not possible as is shown by the answers to this ...
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1answer
63 views

When do minimal subcovers always exist - without choice?

In my answer to Doubt in the definition of a compact set, I sketched a proof of the following fact: Suppose $X$ is a topological space such that every open cover of $X$ has a minimal subcover. ...
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0answers
57 views

Cardinallity increasing constructions

If we start with $\mathbb{Z}$ we can through localization get $\mathbb{Q}$, but that has the same cardinallity as $\mathbb{Z}$, so it doesn't increase cardinality for infinite sets, which is what I am ...
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1answer
36 views

Convergence of real numbers along an ultrafilter

Suppose we have an uncountable set $I$ and a non-principal ultrafilter $U$ on it. I am interested whether it is possible to conclude that if cardinality of $I$ is big enough, then every tuple $(x_i)_{...
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0answers
83 views

Is 4 really that significant?

I have seen the ZFC theorem, (($2^{\aleph_n}$ $<$ $\aleph_\omega$ for all $n$ $\epsilon$ $\mathbb N$ ) $\rightarrow$ $2^{\aleph_\omega}$ $<$ $\aleph_{\omega_4}$). My question is whether this ...
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1answer
21 views

A family of subsets $\{A_i\subset \mathbb{N}\}_{i\in \mathbb{R}}$ s.t. $A_i\Delta A_j$ is infinite and co-infinite for $i\not=j$?

In a paper I am reading (lemma 6) the author uses without proof that there exists a family of subsets $\{A_i\subset \mathbb{N}\}_{i\in \mathbb{R}}$ s.t. $A_i\Delta A_j$ is infinite and co-infinite for ...
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1answer
35 views

A club in an $\omega_1$ tree

Given an $\omega_1$ tree $T$ one needs to prove $\{\alpha\in Lim:\ T{\restriction_\alpha} = \alpha \}$ is a club in $\omega_1$. Why would such a set even be non-empty? What if $T$ is composed of two ...
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0answers
54 views

A poset oriented proof for the intermediate model theorem.

The intermediate model theorem: If $M\subseteq N\subseteq M[G]$ are all models of $\sf ZFC$, with $G$ generic over $M$, then $N$ is a generic extension of $M$, and $M[G]$ is a generic extension of ...
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1answer
62 views

Every transitive $\in$-linearly ordered set is $\in$-well ordered without axiom of foundation

I try to prove that every ordinal number $(\alpha,\in)$ is well-ordered, where an ordinal number is defined as a transitive $\in$-linearly ordered set. So all I have to show is, that every non-empty ...
3
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1answer
65 views

Product of Ultrafilters: How to show $p,q <_{RK} p\otimes q$?

Let $p,q$ be two free ultrafilter on $\mathbb{N}$, i.e. elements of $\mathbb{N}^* = \beta\mathbb{N}\setminus\mathbb{N}$. The Rudin-Keisler order is defined as follows: $p \leq_{RK} q$ iff there is a ...
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1answer
32 views

Inequality - dominating, bounding number

I am stuck with one inequality from Cichon Diagram: $\mathfrak{b} \leq \mathfrak{d}$. Maybe it is easy, but I have no idea how to proof it. where: $\mathfrak{b} : = \mathfrak{b}(\mathbb{N}^{\mathbb{...
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0answers
34 views

Countable anti-chain in an Aronszajn tree

The problem I'm facing is to show there exists a countable anti-chain in an Aronszajn tree $T$. I thought of something, and I wanted to ask you if my proof is correct. So, the idea is to proceed ...
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0answers
81 views

Are these strengthenings of a rank-into-rank cardinal axiom known to be inconsistent with $ZFC$?

I am just getting acquainted with "very strong" large cardinal axioms, and it seems there is a consensus that among large cardinal axioms, the rank-into-rank cardinal axioms are at the threshold of ...
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1answer
79 views

Prove $x \notin x$ without regularity?

In $\mathsf{ZF}$, can we prove that no set is an element of itself without using regularity? My guess is that it is not possible, but I have no idea how to prove this.
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2answers
1k views

Is there a model of ZFC inside which ZFC does not have a model?

Assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC has no model? Also, assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC is inconsistent?
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0answers
69 views

Well-ordering of the reals in ZF with constructibility?

The question Do we know that we can't define a well-ordering of the reals? states: There exist pointwise definable models of ZFC where every set is definable without parameters: it is the ...
2
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1answer
53 views

A question about the relative sizes of Measurable and Supercompact Cardinal Numbers.

Is the least Supercompact Cardinal Number necessarily greater than the least Measurable Cardinal Number?
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1answer
67 views

Can we take definability and existence as primitive notions of a theory?

One of my friend tries to develop an alternative viewpoint of Set Theory. For this he has taken the terms binary relation, set, existence and definability as primitive notions of his Set Theory. After ...
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0answers
47 views

Density of $P(\kappa) / [\kappa]^{<\kappa}$

Suppose $\kappa$ is an infinite cardinal. Can there exist a family $\mathcal{F} \subseteq [\kappa]^{\kappa}$ such that $|\mathcal{F}| = \kappa$ and for every $X \in [\kappa]^{\kappa}$, there exists $F ...
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2answers
49 views

How does the axiom schema of replacement work?

According to this website, the first partion of this axiom schema is Let $P(y,z)$ be a propositional function, which determines a function. That is, we have $∀y(∃x:(∀z:(P(y,z)⟺(x=z))))$. ...
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1answer
48 views

Function $F(n)=n+n$ is not $\Delta_0$

Define $F(n)=n+n$, for $n<\omega$, and $F(n)=0$, for $n\not\in\omega$. I have to show that this is not a $\Delta_0$-function but it's the composition of two $\Delta_0$-functions. I have one hint; ...
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1answer
28 views

Absoluteness of exponential function and forcing

I recall reading that the exponential function $ \alpha^{\beta}$ is absolute for transitive models of ZFC. Is it true that if we have $ 2^{ \alpha } < \beta $ in the ground model $V$, then $( 2^{ ...
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1answer
41 views

Ultrafilter invariant semigroups

This may be quite basic but I was unable to figure out the answer on an uncountable set. Suppose that $S$ is an infinite set and $U$ a non-principal ultrafilter on $S$. Does there exist a commutative ...
2
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1answer
41 views

Constructing a club that satisfies certain conditions

I have a sequence of countable sets $(A_{\gamma} \colon \gamma < \omega_1)$ for which I know that if I take any $x\in\bigcup\limits_{\gamma<\omega_1} A_{\gamma}$ then the set $$\{\beta < \...
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1answer
56 views

Can someone reconcile the two definition of Suslin's condition?

I am given two definitions of the so called "Suslin's condition" and I need to reconcile them. I am an undergrad and this is just for exploration. Definition 1. A partially ordered set $X$ is said ...
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2answers
58 views

A specific example of transfinite induction

I am trying to understand better how transfinite induction can be applied in different concrete problems. Here is an example that seems relevant, but I am stuck on it. Consider a couple of points $p, ...
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1answer
59 views

Proving an existence of a cardinal when making assumptions on exponentiation

Let's assume $2^{\aleph_3}=\aleph_4$ and $\left(\aleph_{\omega_1}\right)^{\aleph_1}\neq\left(\aleph_{\omega_1}\right)^{\aleph_2}$. Prove that $$\exists_{\alpha\in Lim}\left( \left(\aleph_{\alpha}\...
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7answers
542 views

Existence in variants of the Axiom of Choice

Let $\{A_i: i \in I\}$ be a nonempty family of nonempty sets. Why is it allowed to prove the Axiom of Choice using the Well Ordering Principle as follows: There is a well-ordering of $\cup_{i \in ...
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1answer
80 views

Well-orderings of $\mathbb R$ without Choice

The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
2
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1answer
81 views

Elementary suborders of Cohen forcing

My question is basically whether being a Cohen poset is a first-order statement within the order itself. More specifically, let $\mathbb{P}$ be an elementary suborder of $\mathrm{Add}(\omega,\lambda)...
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3answers
126 views

$(\omega +3)\cdot\omega=\omega\cdot\omega$ [duplicate]

Show that $(\omega +3)\cdot\omega=\omega\cdot\omega$. Is this just $(\omega +3)\cdot\omega=(\omega +\omega)\cdot\omega=\omega\cdot\omega$? Also, could someone suggest a good book for set theory?
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2answers
47 views

Functions from $ \kappa $ to $ 2 $ ordered lexicographically

I'm wondering how to approach the following exercise: Suppose $ \kappa \geq \omega $. Consider the set of all functions $ \kappa \rightarrow 2 = \{0,1\} $ ordered lexicographically. Show that ...
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0answers
66 views

Reference about $\sigma$-linked posets and related notions

In this link, the following list appears: Some chain conditions [of posets], listed from easiest to satisfy to hardest to satisfy: ccc powerfully ccc productively ccc $\sigma$-...
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2answers
129 views

There exists a partial ordering of $ \mathbb{R} $ with no uncountable chains or antichains

here's an exercise I'm stuck upon. I'm not sure how to approach this: There exists a partial ordering of $ \mathbb{R} $ with no uncountable chains or antichains While looking for some help, I ...
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0answers
51 views

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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1answer
97 views

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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1answer
40 views

Models of Comprehension Schema

Let $M_\alpha$ for $\alpha\in ON$ be transitive sets and let $M=\bigcup_{\alpha\in ON}M_\alpha$. Suppose that (i) for every $\alpha<\beta$, we have $M_\alpha\in M_\beta$ and (ii) for every limit $\...
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1answer
47 views

Principle equivalent to $\diamondsuit_\kappa$

Let $\kappa>\omega$ be regular. The principle $\diamondsuit_\kappa$ is as follows: There exists a sequence $\langle X_\alpha\mid \alpha<\kappa\rangle$ such that each $X_\alpha\subseteq\alpha$ ...
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1answer
25 views

Help with exercise from Kunen (II. Ex. 51)

Show that the following are equivalent: (1) $\diamondsuit$ (2) The existence of $B_\alpha\subseteq\alpha\times\alpha\ (\alpha<\omega_1)$ so that the set $\{\alpha<\omega_1\mid B\cap (\...