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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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 ...
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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 ...
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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 ...
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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 ...
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2answers
531 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 ...
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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?
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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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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 ...
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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 $\gamma_1.\gamma_2<\gamma_3$?
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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 ...
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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 ...
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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 ...
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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 ...
7
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2answers
159 views

Is $\mathrm{ZFC}^E$ outright inconsistent?

From $\mathrm{ZFC},$ define a new theory $\mathrm{ZFC}^E$ by adjoining a constant symbol $E$ together with axioms to the effect that: $E$ is countable and transitive $(E,\in)$ is an elementarily ...
2
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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 $\Psi$ ...
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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 ...
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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 ...
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1answer
78 views

Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
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1answer
78 views

Is there a formalisation of set theory where unions can be taken over some classes that are not a priori known to be sets?

Of course, fully unrestricted unions in ZFC will immediately lead to inconsistency, but is there a variation where restrictions on allowed classes are given explicitly at least, in terms of the ...
7
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2answers
144 views

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$?

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$? I think the question speaks for itself, but let me try and satisfy the "quality standards" algorithm by padding it. Yes, I ...
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0answers
51 views

Can existence of aleph one be proved without the power set axiom? [duplicate]

Cantor's construction of $\aleph_1$ is to notice that all ordinals constructed after $\omega$ are countable, take the union of all countable ordinals, and then show that this union can not be ...
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1answer
85 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
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2answers
130 views

CH imples the existence of a function

I was studying an article and the author stated that CH implies that there exists a function from $\omega_1 \setminus \omega$ onto the set of all countable subsets of $\omega_1$ such that for each ...
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2answers
42 views

Cardinal of $V_{\omega+\alpha}$

I do not understand why $card(V_{\omega+\alpha})=\beth_\alpha$. The steps in the recursion for $0$ and succesor ordinals are quite easy, but I do not manage to prove it for limit ordinals. I see that ...
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2answers
106 views

Universe cardinals and models for ZFC

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal ...
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2answers
89 views

What do you need to know about ordinals to understand Zorn's Lemma's proof?

I'm trying to understand the proof of Zorn's Lemma but the one which does not use ordinals (Halmos' proof) is extremely long and I really feel I get lost somewhere along the way. On the other hand, ...
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1answer
69 views

Is there such a thing as “completion with respect to the axiom of choice?”

Completions abound in mathematics. For example: the completion a metric space with respect to Cauchy sequences the algebraic closure of a field the Stone-Čech compactification of a topological ...
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3answers
147 views

Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
2
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2answers
67 views

How do we define recursive class functions like $\alpha \mapsto V_\alpha$ and $\alpha \mapsto L_\alpha$ in the language of set theory?

It is well-known that ZFC proves that everything is an element of $V$. Symbolically, $\forall x(x \in V).$ However, I can't figure out how to translate this into the language of ZFC. We know that $V$ ...
2
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1answer
88 views

Sentence $\varphi$ of set theory that is satisfied by all well-founded models of ZFC, but which is not a theorem of ZFC.

I think I read somewhere the following. If a first-order sentence $\varphi$ in the language of set theory holds for every well-founded model of ZFC, then nonetheless: $\varphi$ may fail for a ...
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69 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
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1answer
45 views

Well-Ordering Theorem without Axiom of Regularity

Personally, I am not fond of the Axiom of Regularity. Some alternative models in set theory use the negation of the Axiom of Regularity as an axiom (non well-founded theories). I am curious if the ...
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1answer
38 views

Inductive posets, exercise from Notes on set theory.

OK, this book is driving me nuts. This is exercise 16 from chapter 6. For each set $E$, the set $P = E* ∪ (ℕ → E)$ of finite and infinite sequences from E is an inductive poset, under $⊆$. For ...
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1answer
87 views

Has anyone considered axioms to the effect that: “The axiom of constructibility fails very very badly?”

If I'm not mistaken, the axiom of constructibility basically says that the universe has no (non-trivial) inner models. Has anyone considered axioms of the opposite flavour, basically asserting that ...
5
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1answer
142 views

Example of an interesting theorem that fails in intuitionistic set theory but is classically valid?

I'm interested in intuitionistic set theories at the moment. I know that lots of principles imply LEM and so fail intuitionistically, and also a few basic principles - linear ordering of ordinals, for ...
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2answers
48 views

Finite set with unknown number of elements

Does there exist anything like a finite set whose cardinality cannot be established by means of an algorithm or mathematical proof? I think ZFC should have no problem in accomodating such an object, ...
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0answers
82 views

Jech, “Set theory” exercises 12.11 - Is my proof right?

I try to prove the Jech's "Set theory", exercises 12.11: 12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$. My attempt. Since ...
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0answers
28 views

Addition on well ordered sets not-commutative by showing $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$

My goal is to show that addition on well ordered sets are non-commutative by showing that, $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$ Some definitions (let A and B be ...
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2answers
53 views

countably closed forcing cannot add a branch to a $\aleph_2$-tree if $\neg\mathsf{CH}$

I'm reading this survey. In it the author states the following result (fact 5.3) which is attributed to Silver: If $2^{\aleph_0}>\aleph_1$, countably closed forcing cannot add a new branch to ...
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1answer
69 views

Conservativity of $\mathrm{ZFC}+\varphi$, where $\varphi$ contradicts CH.

It is well-known that ZFC with the continuum hypothesis is a $Π^2_1$-conservative extension of ZFC. General question. What is known about the conservativity of $\mathrm{ZFC}+\varphi$ over ...
3
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1answer
64 views

Truth values in Boolean valued models

In Devlin's "The Joy of Sets" the author introduces the Boolean valued model $V^{{\mathcal B}}$ based on a given Boolean algebra ${\mathcal B}$ and describes how to assign a truth value ...
2
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1answer
34 views

Do Different Generic Filters Give Different Generic Extensions?

Let $\mathbb{P}$ be a forcing. If $G \subseteq \mathbb{P}$ and $H \subseteq \mathbb{P}$ are two $\mathbb{P}$-generic filters over $V$ and $G \neq H$, does this imply that $M[G] \neq M[H]$. If $G$ ...
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99 views

Cardinality of an algebra

Suppose that $B$ is the Boolean algebra of all Lebesgue measurable sets in $I=[0,1]$ modulo Null sets. I am asking (1) What will be the cardinality of $B$. Does it have to be $|B|=\mathfrak ...
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2answers
89 views

Consistency of ZFC and the key assumption [closed]

I recently read this answer to a MathOverflow question that got me thinking. Very roughly, here's what the author of that answer says: Gödel's second incompleteness theorem implies that if there ...
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1answer
41 views

Object set of a clonal category.

I read the statement that "a clonal category has a small set of objects", which I don't quite agree about. In the definition of clonal category, at least as it is given in that context, it is required ...
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0answers
121 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
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0answers
54 views

Is the “set” of all algebraic extensions a set? [duplicate]

Consider a field $K$. Now, consider the class of all algebraic extensions of $K$. Is this a set? Since I think it isn't, how to prove it isn't? If the class were of all extensions of $K$, I think I ...
5
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2answers
171 views

About proving that the Continuum Hypothesis is independent of ZFC

In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the ...
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1answer
82 views

Hall's marriage Theorem and Tychonoff Theorem

I was reading this paper. In particular the second point. He proves the Hall's marriage Theorem for infinite family using the Tychonoff theorem on topological product of compact $T_2$ spaces and the ...
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1answer
52 views

proving set intersection is possible with infinite collections

I'm currently going through Enderton's Elements of Set Theory. I have two questions, but they need some context. Here are the subset axioms introduced in the book: For each formula _ not ...