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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8
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1answer
225 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 ...
1
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0answers
29 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 ...
0
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0answers
48 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)...
3
votes
0answers
34 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
votes
1answer
48 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 \...
1
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0answers
68 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'...
0
votes
2answers
32 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 ...
-1
votes
1answer
59 views

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

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
votes
2answers
94 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
30 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
votes
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 ...
4
votes
1answer
92 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
vote
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 ...
0
votes
3answers
51 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 $...
1
vote
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
votes
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 ...
0
votes
1answer
27 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
votes
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 ...
1
vote
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
votes
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 ...
1
vote
3answers
219 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
votes
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 ...
1
vote
1answer
60 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 ...
1
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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
votes
1answer
50 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 \...
4
votes
1answer
78 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
votes
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 ...
2
votes
2answers
414 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
votes
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 ...
31
votes
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
votes
1answer
65 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
votes
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 ...
3
votes
2answers
83 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 ...
0
votes
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
votes
1answer
48 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
votes
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
votes
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 ...
1
vote
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 ${\exists}!$? ...
0
votes
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.
5
votes
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 ...
1
vote
1answer
38 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 $f$ ...
2
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2answers
59 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
1
vote
1answer
60 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\...
2
votes
2answers
76 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 ...
3
votes
1answer
70 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. ...
1
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0answers
61 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 ...
1
vote
1answer
38 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)_{...
5
votes
0answers
92 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 ...
0
votes
1answer
22 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 ...
1
vote
1answer
37 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 ...