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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Cofinality of the bounding number and the dominating number

Let's define a relation $\leq^*$ on $\omega^{\omega}$ by $$f\leq^* g \iff \exists m\in\omega\ \forall n>m\ f(n)\leq g(n),\ \ f,g\in\omega^{\omega}.$$ Then let $$\mathfrak{b}=\min\{|\mathcal{F}|:\ \...
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1answer
22 views

Is there exists unbounded collections rather than be restricted into the world “class”?

In set theory,there is a question states that whether the class which contains all the classes exists? The answer is "NO" obviously.Otherwise it will lead to paradox.The reason is that there is a ...
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0answers
35 views

Reference request: Models of set theory and relativization

I am self-studying set theory and have been using Jech's text. When reading chapter 12, things are ok until I get to the section "Models of Set Theory and Relativization." I do not understand a ...
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0answers
38 views

Kurepa trees and inaccessible cardinals in $L$

Given a regular uncountable cardinal $\kappa$ we say that a $\kappa-$tree is $\kappa-$Kurepa if it has at least $\kappa^+$ branches. If $\kappa=\omega_1$ we simply say that $T$ is Kurepa. In this ...
2
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1answer
32 views

Equivalence between CH and existence of certain sets in $\mathbb{R}^2$

The problem is to prove the following equivalence: $$2^{\aleph_0}=\aleph_1 \iff \exists\ A,B\in\mathbb{R}^2:\\ \textrm{a)}A\cup B=\mathbb{R}^2 \\\textrm{b)}\ \forall\ \textrm{lines } l\ \textrm{...
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2answers
36 views

Set theory: how to interpret multiple quantifiers

How does one interpret this ZFC Union axiom? I can't quite understand what is meant after "There exists some elements y for all elements z"? I'm also wondering if the x is a typo. $\exists y \...
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1answer
47 views

Could “ω is an ordinal” be proved without axiom of induction?

Let ω (or all natural numbers) be the set defined by the axiom of infinity in ZF system, ordinal be "transitive and well ordered by ∈", and a set is transitive if all its elements are also its subset. ...
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0answers
55 views

Large family of certain subsets of a regular cardinal

$\kappa$ is an infinite, regular cardinal. There's not much to start from as you see. I'm asked to prove there exists a family $\mathcal{A}\in\mathcal{P}(\kappa):$ $i)\ |\mathcal{A}|=\kappa^+$ $ii)...
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1answer
49 views

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 ...
2
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1answer
65 views

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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1answer
248 views

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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0answers
43 views

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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0answers
56 views

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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0answers
41 views

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 ...
0
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1answer
53 views

Uncountable family of maximal antichains whose union is countable

Given a ccc preorder $\mathbb{P}$ and a family $\mathcal{A}=\{A_\alpha:\alpha<\omega_1\}$ of pairwise distinct maximal antichains of $\mathbb{P}$, is it possible that $|\bigcup\mathcal{A}|\leq \...
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0answers
71 views

Brief overview of the foundations of math?

I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don'...
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2answers
35 views

When is the union of a set of ordinals a limit ordinal?

Let $Y$ be a set of ordinals such that $\bigcup Y \not\in Y$. Then $\bigcup Y$ is a limit ordinal. Does this hold? It's taken from Drake's Set Theory, An Introduction to Large Cardinals, page 114, I ...
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1answer
62 views

How to prove axiom of limitation of size with axiom of replacement and global axiom of choice [closed]

In the book<>written by Charles C.Pinter it said that in order to prove "the axiom of replacement together with strengthened version of axiom of choise imply axiom of limitation of size"(X is a ...
5
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2answers
98 views

Meta proof-searching

Suppose you have a particular theory (ex: $ZFC$) in which you want to prove a statement $\phi$. One can attempt to find a proof of $\phi$ that can be verified, but another tactic can be to find a ...
2
votes
1answer
32 views

Counting the number of names for elements of a certain name.

I'm self studying the proof of consistency of MA on Jech's Set Theory (Theorem 16.13, p. 272). There is a step which I can't understand. To simplify the notation, I will try to "extract" the relevant ...
2
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1answer
54 views

Translation of the Axiom schema of Separation into purely category-theoretic terms.

It is well-known the the category $\mathcal Set$ is a topos. In this topos, how would one translate the Axiom schema of Separation into purely category-theoretic terms? I ask this question because ...
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1answer
96 views

Implications of existence of two inaccessible cardinals?

Many years ago in an oral exam I was asked, what could be concluded from the existence of an inaccessible cardinal in ZFC? I knew that would provide a model for ZFC and imply the consistency of ZFC. ...
1
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1answer
135 views

Did Cantor both prove and disprove the Continuum hypothesis?

I have been listening to the podcasts of A Brief History of Mathematics on BBC Radio 4. In the episode on Georg Cantor the narrator, Prof. Marcus du Sautoy, says that one day Cantor proved that there ...
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3answers
52 views

Injective function between $\kappa^{\omega}$ and $[\kappa]^{\leqslant \omega}$

Is there an injective function $\varphi :\kappa^{\omega} \rightarrow [\kappa]^{\leqslant \omega}=\{ A\subset \kappa :|A|\leqslant \omega\}$ such that $\varphi (\alpha) \backslash \varphi (\beta)$ and $...
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1answer
26 views

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 ...
2
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0answers
40 views

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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1answer
28 views

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 ...
2
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2answers
76 views

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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2answers
51 views

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 ...
2
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2answers
365 views

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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3answers
230 views

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 ...
3
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1answer
102 views

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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1answer
64 views

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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1answer
61 views

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 \begin{equation} \textbf{HK}_\kappa: \text{ There's a ...
3
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1answer
51 views

If a theory $A$ can prove $B$ consistent relative to $A$, and $A$ is consistent, does $B$ have to be consistent?

Let's say we have two sets of axioms $A$ and $B$ such that $\mathsf{ZF} \subseteq A \subseteq B$, and from $A$ we can prove that if $A$ is consistent, then $B$ is consistent as well (that is, $A \...
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1answer
79 views

Appealing axioms incompatible with large cardinal axioms

I'm interested to know what are some 'appealing' axioms that are inconsistent with ZFC plus some large cardinal axiom. I saw the question On the contradictory nature of large cardinals & choice-...
3
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1answer
42 views

Definition of length of a sequence

In definition $3.11$, the author defined the following rank: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of ...
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2answers
429 views

What would be the consequences if ZFC proved its own inconsistency, nonconstructively? [closed]

Let's say a nonconstructive proof was given in ZFC that ZFC was inconsistent. Note that this doesn't automatically make ZFC inconsistent. Given a consistent theory $X$, $X + \neg \text{Con}(X)$ is ...
2
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1answer
109 views

How is the set of all even numbers definable from $\omega$?

This Set Theory textbook (page 89) defines definable sets as follows: Definition 6.8. Given a set $a$ and a formula $\Phi$ we define the formula $\Phi^a$ to be the formula derived from $\Phi$ by ...
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4answers
2k views

Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.

In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\...
2
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1answer
66 views

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 ...
2
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1answer
519 views

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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2answers
85 views

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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1answer
14 views

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 ...
2
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1answer
49 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 ...
3
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1answer
93 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 ...
2
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1answer
47 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
37 views

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 ${\exists}!$? ...
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0answers
36 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
280 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 ...