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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14
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3answers
110 views

Axiom of Choice: Where is falsity in argument and understanding to put is as Axiom?

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
35 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 ...
11
votes
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 ...
3
votes
2answers
53 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 ...
2
votes
1answer
415 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 ...
0
votes
1answer
12 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 ...
3
votes
1answer
69 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
39 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 ...
1
vote
1answer
28 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 ...
18
votes
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 ...
1
vote
3answers
29 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 ${\exists}!$? ...
5
votes
4answers
264 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 ...
0
votes
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.
16
votes
1answer
616 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 ...
3
votes
1answer
100 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 \...
1
vote
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 $f$ ...
2
votes
2answers
56 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
56 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
69 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
64 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
vote
0answers
57 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
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)_{...
5
votes
0answers
83 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
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 ...
1
vote
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 ...
7
votes
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 $x'$ ...
3
votes
0answers
54 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 ...
2
votes
3answers
726 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 ...
1
vote
1answer
64 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 ...
3
votes
1answer
65 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 ...
2
votes
0answers
82 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 ...
0
votes
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{...
2
votes
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 ...
0
votes
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.
3
votes
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 ...
21
votes
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^{\...
16
votes
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 ...
26
votes
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?
5
votes
0answers
70 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 ...
2
votes
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?
0
votes
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 ...
2
votes
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 ...
1
vote
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 $∀y(∃x:(∀z:(P(y,z)⟺(x=z))))$. ...
2
votes
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; ...
0
votes
1answer
41 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 ...
1
vote
1answer
28 views

Absoluteness of exponential function and forcing

I recall reading that the exponential function $ \alpha^{\beta}$ is absolute for transitive models of ZFC. Is it true that if we have $ 2^{ \alpha } < \beta $ in the ground model $V$, then $( 2^{ ...
2
votes
1answer
41 views

Constructing a club that satisfies certain conditions

I have a sequence of countable sets $(A_{\gamma} \colon \gamma < \omega_1)$ for which I know that if I take any $x\in\bigcup\limits_{\gamma<\omega_1} A_{\gamma}$ then the set $$\{\beta < \...
11
votes
2answers
261 views

Can a basis for $\mathbb{R}$ be Borel?

Work in ZF (so no choice). Then it is consistent that there is no (Hamel) basis for $\mathbb{R}$ as a $\mathbb{Q}$-vector space. My question is about models where $\mathbb{R}$ does have a basis, but ...
0
votes
1answer
41 views

Ordinals recursed

Consider any limit ordinal $\alpha$. It seems intuitively obvious to me that for any such ordinal there is $f:ORD \to ORD$ (with $ORD$ being the class of ordinals) with $f(1)=\alpha$ and $f$ ...