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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1answer
68 views

Equivalent formulations of the Axiom of Replacement

Let $a$, $b$, $S$ denote sets, and let $\varphi$ be a statement in two set variables. Here are two ways of expressing the Axiom of Replacement which I have seen used: 1) If $$\text{for all } a \in S, ...
1
vote
1answer
47 views

Models of ZFC as Ordered Fields

Question: Let $M$ be a set such that $\langle M,\in\rangle\vDash ZFC$. Consider the partial order $\langle M,\subseteq\rangle$. Using order extension principle there is a linear order $\subseteq^{*}$ ...
1
vote
1answer
102 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
2
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2answers
110 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
6
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0answers
175 views

Is Kunen's claim about non-equivalent forms of Axiom of Choice, true?

Consider the following forms of the axiom of choice: $AC_1:\forall F\neq \emptyset~~~(\emptyset\notin F~\wedge~\forall x,y\in F~~~(x\neq y\rightarrow x\cap y= \emptyset))\rightarrow \exists C~\forall ...
2
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0answers
79 views

Are there any legitimate examples of applications of set theory to Physics?

Sets are of course ubiquitous, so in this sense set theory is applicable to physics. But I'm thinking more like techniques like forcing, or constructs like cardinals or ordinals. Intuitively it ...
2
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3answers
210 views

Is there a largest large cardinal?

In ZFC, a cardinal is an isomorphism class of sets. However ZFC doesn't explicitly have classes; NBG, which is a conservative extension of ZFC does. There is no largest cardinal by Cantors Theorem ...
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0answers
37 views

A question about $KP + V = L$ and $KP$ set theory.

In reading Rathjen (Choice principles in constructive set theories) and Jager (On Feferman's OST) I've come across two facts that are taken as obvious/well known, and probably are, but for which I ...
1
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1answer
90 views

Which axioms of Zermelo set theory fail for $U$, and how can we prove this?

By the phrase "$O$ is an outer model of M" let us mean that $M$ is an inner model of $O,$ according to $O$. So in particular, they have the same ordinals. Now I learned from Andres here that: ...
2
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0answers
93 views

Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
0
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5answers
438 views

Unprovability results in ZFC

I am looking for real-world examples of unprovable statements in ZFC. So, not contrived logical formulae but statements that are of importance for ordinary mathematicians. Could you please point to ...
0
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0answers
49 views

Show that $(V_\omega,\in) \models \mathrm{Power\ set}$

Let "$\mathrm{Power\ set}$" denote the Power set axiom. I'd like if some of you could tell me if the solution of the following exercise is correct. I want to prove that $(V_\omega,\in)\models ...
16
votes
1answer
696 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
2
votes
1answer
99 views

What does it mean to say that a forcing “collapses cardinals”?

I hear the following terminology a lot: "So-and-so forcing collapses cardinals." Does this just mean that certain cardinals in the ground model are no longer cardinals in the forcing extension? If ...
5
votes
1answer
64 views

Equivalence between AF and $AF^{*}$

I have a problem with the proof of this: Let $AF^{*}$ be the axiom schema $(\forall x(\forall y \in x \varphi(y) \to \varphi (x))) \to \forall x \varphi (x)$, where the variable y does not occur in ...
9
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0answers
185 views

Paradoxical models of $\sf ZF$ without choice [closed]

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold ...
4
votes
2answers
246 views

Does the Laver real determine the generic filter?

Let us concern the Laver forcing $ \mathbb{L} $. Let $ G $ be $ \mathbb{L} $-generic over a c.t.m. $ M $ for ZFC. Let $$ x_G := \bigcup \{ \operatorname{stem}(p) : p \in G \} $$ be the Laver real ...
2
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1answer
27 views

Is the class of principal $G$-bundles over $M$ a set?

Let $G$ be a Lie group and $M$ a manifold. Question: Is the class of principal $G$-bundles over $M$ a set? This question came up when I was thinking about the classifying stack of $G$. It is the ...
3
votes
1answer
87 views

Proving König's lemma (technical problems)

the aim of my exercise is to give a proof of the König's lemma. So, let $\kappa, \lambda$, be cardinals such that $cf(\kappa)\leq \lambda$. My professor's suggested us to prove that there exists a ...
3
votes
1answer
28 views

Show that any transitive class is contained in $WF$

I need an help with this exercise. It states: Work in $ZF^-$. Let C a transitive class. Show that if $\in$ is well-founded on $C$, then $C\subseteq WF$. I thought to argue by contradiction: let $x\in ...
6
votes
1answer
92 views

Is every set countable according to some outer model?

Let $M$ denote a set-sized model of $\mathrm{ZFC},$ not necessarily well-founded. Is it true that for any $x \in M,$ there is an outer model $O$ of $M$ such that $(O \models x \mbox{ is countable})$?
1
vote
1answer
74 views

Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
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0answers
70 views

Is there a set $A$ such that $A=\{A\}?$ [duplicate]

So I am kind of curious about the type of things sets can't satisfy (like the "set" of all sets). So, is there a set $A$ such that $A=\{A\}$ ?
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1answer
65 views

Forcing Preservation on Arbitrary Set Theoretic Formulas

Fixing a ground model $M$, a forcing notion $\mathbb{P}$ is called cardinal preserving iff for all $\mathbb{P}$-generic filter $G$ over $M$ we have: $$\forall a\in M~~~(M\models ...
2
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1answer
58 views

Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z)) $$ What ...
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0answers
49 views

Let $\Phi$ denote the statement that $\mathrm{GCH}$ holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition. Definition. Call a set of axioms $\Phi$ in ...
3
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1answer
47 views

Is this an equivalent characterization of rapid filters?

A filter $\mathcal F$ on $\omega$ is called rapid filter if for every function $f\in\omega^\omega$ there exists $X\in\mathcal F$ such that $|X\cap f(n)|\le n$ for $n\in\omega$. In Lemma 4.6.2 in the ...
4
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0answers
64 views

How to think about iterated ultrapowers?

I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, ...
9
votes
2answers
196 views

Examples of theorems that haven't been proven without AC in practice but can be proven without it in principle

It is possible to prove theorems of the form "if $\phi$ is provable in ZFC, then $\phi$ is provable in ZF". For example, let $\phi$ be a statement that is absolute between $V$ and $L$. If $\phi$ were ...
0
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0answers
48 views

Absolute formula

I'm trying to study the model of Set Theory and I have a problem. If I know that a formula is absolute for a class, could I infer that the formula is true in my class? Namely, I'm trying to prove in ...
2
votes
2answers
66 views

On those behaviors of continuum function which imply the axiom of choice

It is a folklore fact that within $\text{ZF}$ the generalized continuum hypothesis ($\text{GCH}$) implies the axiom of choice ($\text{AC}$), namely: $$ZF+\forall \kappa\in ...
5
votes
2answers
78 views

Meaning of “There exists a proper class of…”

How a statement of the form "There exists a proper class of..." can be formalized in $\sf ZFC$? It sounds a bit like an oxymoron to me, because it's the very essense of a proper class that it does not ...
2
votes
1answer
25 views

Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
5
votes
2answers
71 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
13
votes
2answers
530 views

Are there any infinite sets that are not known to be either countable or uncountable?

Are there any known examples of sets that are definitely infinite, but where we don't know whether or not they're countable? I haven't heard of anything like this before, but it seems that there ...
5
votes
0answers
66 views

Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
0
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1answer
36 views

About a representation on cardinals

If k is a limit cardinal, necessarly one have that $ ZFC ^ {R_{K}} $ ? And, if k is only ordinal, does it have any sense?
0
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1answer
39 views

A set which is $\in$-transitive and well ordered by $\in$ is an ordinal

A set $x$ is said to be $\in$-transitive if $\forall y$ $\forall z$($ y \in x$ and $ z \in y \Rightarrow z \in x$). A set $x$ is said to be an ordinal if $x$ and every member of $x$ is ...
9
votes
1answer
107 views

The cardinality of the set of all linear order types over $\omega$ is $2^{\aleph_0}+\aleph_1$ in ZF+AD?

In ZFC, cardinality of set of linear orders over $\omega$ is $2^{\aleph_0}$. By the argument given by here, we can prove (without the choice) the number of linear orders over $\omega$ is at least ...
2
votes
0answers
34 views

Strategic closure of quotients

Is there an example of a poset $P$ that is a regular suborder of $Q$ such that $Q$ is $\omega_2$-strategically closed, but the quotient forcing $Q/P$ fails to be $\omega_1$-strategically closed? To ...
9
votes
1answer
207 views

$\mathcal U$ Grothendieck universe. Is $\mathcal{P(U)}$ a model for NBG?

Suppose we are in ZFC, let $\mathcal U$ be an uncountable Grothendieck universe and consider the set of its parts $\mathcal{P(U)}$. (I will index axioms as $(\mathcal U.n)$) Note that if $x \in ...
1
vote
2answers
32 views

The product of cardinals

Let $\gamma_i$ be infinite cardinal numbers for $i=1,2,3$ such that $\gamma_i<\gamma_3$ for $i=1,2$. Is it true that $\gamma_1.\gamma_2<\gamma_3$?
1
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1answer
37 views

Model for replacement

If $R_{k} $ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...
0
votes
1answer
47 views

About the definible sets $L_\alpha$

Let $\alpha$ be an ordinal number. Is that true that $\alpha$ = $\beth_\alpha $ is equivalent to the statement $|L_\alpha|=|R_\alpha|$, where $L_\alpha $ is the $\alpha$-th stage of the constructible ...
0
votes
1answer
46 views

Ordinal arithmetic question

The question is: find $\delta \le \omega_1$ and $\rho \lt \omega$ such as $\omega_1 = \omega \cdot \delta + \rho$. My guess is that $\delta = \omega_1$ and $\rho = 0$, abut I don't know how to prove ...
7
votes
1answer
58 views

Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
2
votes
1answer
71 views

Existence of countable transitive models.

I have read that $\mathrm{ZF}$ has a transitive model iff it has a countable transitive model. I am interested in generalizations of this result. In particular: Question. Let $\varphi$ and $\Psi$ ...
103
votes
1answer
3k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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0answers
21 views

Choosing from two index families of sets

Let $a$ be a (nontrivial) ultrafilter and $n$ be an infinite set. Let also $U$ be an infinite set. Define $n$-ary relation $\phi$ on $\mathscr{P}U$ by the formula $L\in\phi \Leftrightarrow \forall ...
3
votes
2answers
170 views

Random reals and Martin-Löf randomness

My questions are about the relationship between the following notions of randomness: A real $r$ is random over the model $M$ if $r\notin B$ for every null Borel set $B$ coded in $M$. A real $r$ is ...