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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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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1answer
91 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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2answers
318 views

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

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 ...
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109 views

Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to $\...
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4answers
413 views

A Vector Space is a Set - Axiom or Derivation?

I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. But can a structure with the properties of a (possibly ...
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2answers
72 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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4answers
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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)\...
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1answer
55 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 ...
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4answers
1k views

What would happen if ZFC were found to be inconsistent?

If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that ...
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2answers
68 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
504 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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1answer
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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
80 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
43 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
37 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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1answer
3k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
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3answers
34 views

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

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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4answers
272 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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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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648 views
+50

The class of all classes not containing themselves

In ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox ...
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1answer
103 views

Condition for an Ultrafilter to be Ramsey.

I read in the english Ultrafilter article on Wikipedia that one can prove that a non-principal ultrafilter $D$ on $\omega$ is a Ramsey ultrafilter if and only if for every coloration $c:[\omega]^2 \...
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1answer
36 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
57 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
58 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
70 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
65 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
59 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
85 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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3answers
269 views

Proof of exchange principle in set theory

How would one use the Axiom of Foundation (that every non-empty set has an $\in$-minimal element) to prove that for any two sets $x,y$ we can find a set $x'$ so that $x'$ and $y$ are disjoint and $x'$ ...
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0answers
57 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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3answers
729 views

Misconceptions about Cantor's diagonal argument?

I started to make a demonstration of Cantor's diagonal argument on Mathematica, by reading here, I've noticed that I'd need to do a list of tuples with binary digits, and I tried to do it on ...
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1answer
65 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 ...
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0answers
83 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
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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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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0answers
151 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
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2answers
2k views

Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
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1answer
567 views

What axioms does ZF have, exactly?

While trying to find the list of axioms of ZF on the Web and in literature I noticed that the lists I had found varied quite a bit. Some included the axiom of empty set, while others didn't. That is ...
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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
73 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 ...
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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
49 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
42 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 ...