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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4
votes
1answer
53 views

Order type of the set of all the limit elements in $\omega^{\omega}$

What do you think is the order type of $\{\alpha\in\omega^{\omega}:\alpha\ \text{is a limit ordinal number}\}$? My first thought was $\omega$, but when I visualize the above mentioned set, ...
0
votes
1answer
45 views

Are there any collections in the NBG set theory that are neither classes nor sets?

Just as proper classes in ZFC are defined as collections which don't fulfill the ZFC set axioms, are there any objects which don't fulfill not only the set, but also the NBG class axioms? How are ...
3
votes
1answer
69 views

Erdös cardinals in $L$

I've readed in Jech's book that the existence of the $\omega-$Erdös cardinal $\kappa(\omega)$ (that is the minimumm cardinal $\kappa$ for which $\kappa\rightarrow (\omega)^{<\omega}$) it is ...
2
votes
1answer
52 views

Equivalence to Martin's Axiom

I know that MA implies $2^\kappa = 2^{\aleph_{0}}$ for each cardinal $\kappa <2^{\aleph_{0}}$. Is the converse true? I mean, does $2^\kappa = 2^{\aleph_{0}}$ for every cardinal $\kappa ...
0
votes
1answer
32 views

How to solve the problem of $(a,a)$ in the Kuratowski formalisation of ordered pairs? [closed]

$(a,a)={\{{\{a\}},{\{a,a}\}}\}={\{\{{a}\},{\{a}\}}\}={\{\{a}\}\}$ Is this any problem in Kuratowski formalisation? If yes, how to solve it?
1
vote
3answers
79 views

What's the difference between ${\{a}\}$ and $a$?

${\{a}\}={\{a,{\emptyset}}\}$ ∧ $a={\{a,{\emptyset}}\}$${\implies}{\{a}\}=a$ How is the above wrong? And if it's actually right, how do we solve the problem with the ZFC axiom of foundation asked ...
1
vote
1answer
22 views

Confusion About Pointclasses

I am doing some work learning about the Axiom of Determinacy and its consequences. This has led me to learning about the properties of the Baire space, $\omega^\omega$. I have recently come across the ...
5
votes
0answers
74 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
18
votes
2answers
567 views

Does ZFC decide every question about finitely generated groups?

In ZFC, we can easily say when a triple $\mathscr{G}=\left\langle G,\cdot,1 \right\rangle $ is a group. Furthermore, we can say when a group is finitely generated: First define a "canonical" finitely ...
10
votes
2answers
171 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 ...
5
votes
2answers
344 views

Seeking a new, more natural definition of the cartesian product of sets

In "standard" set theory usually we have the definition $(a,b) := \{ \{ a \} , \{a,b\}\}$, see for example wikipedia for other similar ones. Then if we set for two sets $A,B$ $$ A\times B := \{ (a,b) ...
3
votes
1answer
83 views

Ordering of Large Cardinal Axioms

One of the unexplained phenomena about large cardinal axioms is that they tend to be in a linear order (in terms of consistency strength). So it brings to mind other natural questions about this ...
4
votes
2answers
173 views

Constructing sets of certain measure from classes of bijections on the continuum

Suppose that for each $\alpha < 2^\omega$, $f_\alpha:2^\omega \rightarrow 2^\omega$ is a bijection. I want to know whether it's always possible to construct an $X\subseteq Y\subseteq 2^\omega$ ...
1
vote
1answer
35 views

Functions on the continuum with an unboundedness property

Is it possible to construct bijections, $f_\alpha:2^{\aleph_0}\to 2^{\aleph_0}$, for each $\alpha<2^{\aleph_0}$ such that for each $\beta,\gamma$ there is an $\alpha$ such that ...
1
vote
2answers
49 views

Do the relative consistency results involving the axiom of choice use completeness of FOL?

The proof of the Completeness Theorem for first order logic uses the axiom choice and is a central result in model theory. Chang and Keisler claim that model theory has important applications to set ...
4
votes
0answers
36 views

GCH in $L[A]$ when $A\subset \omega_1$

I'm trying to prove the GCH in the constructible universe $L[A]$ when $A$ is a subset of the ordinal $\omega_1$. For this purpouse, first I want to see that if we take any elementary substructe ...
12
votes
0answers
199 views

Spanning the reals with a small set - choicelessly

Working in ZF (so, no choice): is it possible that there is a set of reals $X$ such that $\vert X\vert<\mathbb{R}$, but $X$ generates $\mathbb{R}$ as a subgroup under addition? This seems ...
1
vote
0answers
36 views

For a set $A$, property P and cardinal $k$, when is the statement “$|A|<k$” equivalent to “Every function $A \to A$ satisfies P?” [closed]

Question in title. We know that a set $A$ being finite is equivalent to every injection from $A$ to itself being a bijection. Are there other cardinals $k$ such that every function from $A$ to itself ...
1
vote
1answer
42 views

Partial order is intersection of linear extensions also if poset is infinite?

I cannot understand the statement "every partial order is the intersection of its linear extensions. (See e.g. Davey and Priestley [DP]" here (page 6). The book Introduction to Lattices and Order on ...
3
votes
1answer
41 views

Ordinal arithmetic inequality

I have 3 ordinals $\alpha,\beta,\gamma$, $\gamma\neq 0$. Is this implication true? $$\alpha<\beta\implies\gamma\cdot\alpha<\gamma\cdot\beta$$ I have reason to believie it is not, namely ...
1
vote
1answer
48 views

Existence of coinitial and cofinal sequences on uncontable and totally ordered set

Let be $\Omega$ an uncountable set and $\leq$ a linearly ordered relation on $\Omega$. Than can we say that exists a co-initial sequence and cofinal sequence in $\Omega$? Where we say: a subset $B$ ...
1
vote
2answers
58 views

Standard Complete Model of ZFC and Reflection

I am reading Levy's Axiom Schemata of Strong Infinity in Axiomatic Set Theory. In theorem 6, which says that this Reflection principle $N_0$ is equivalent to Replacement + Infinity in $ZF$ (in $S$ ...
4
votes
1answer
57 views

What's the least class of ordinals closed under successor and the limits of omega-sequences? [duplicate]

What is the smallest class $S$ of ordinals that contains $0$, closed under successor and the limits of omega-sequences, i.e., $$ \forall i \in \omega [\alpha_i \in S] \Rightarrow \sup_{i} \alpha_i \in ...
0
votes
0answers
38 views

How do I prove the pairing axiom by the other ZFC axioms? [duplicate]

To prove the pairing axiom I thought about using the replacement axiom but I do not really know how. Can someone show me how it is done?
1
vote
1answer
39 views

Troubles with Kanamori's Theorem 7.8

In Kanamori's Book The Higher Infinite is proved in Theorem 7.8. (implication (b) $\rightarrow$ (a) ) that every inaccesible cardinal with the tree property is a weakly compact cardinal by means of ...
6
votes
1answer
65 views

Is it true that $V$ and $H_{\omega_1}$ agree on the truth value of $\Sigma_1$ sentences?

I want to see if the following result is correct or not: Let $\varphi(x)$ be a formula in $\mathcal{L}_{\mathrm{ZF}}$ with only bounded quantifiers such that $\exists x\,\varphi(x)$ holds in $V$, ...
2
votes
0answers
67 views

Large cardinals: Every weakly compact cardinal has the extension property.

I'm starting to study the basic theory of Large cardinals with Jech's book Set Theory. I'm having struggles understanding the details of the proof that if a cardinal $\kappa$ has the extension ...
1
vote
2answers
53 views

Set Theory - Prove that there is a function $f:\omega\to X$ such that $f(n)<f(n+)$ when $X$ has no maximal element.

Suppose that $X$ is non-empty strictly ordered set, and that $X$ has no maximal element in a sense that for every $x\in X$ there exist $y\in X$ such that $x<y$. By using AC, show that there ...
2
votes
2answers
65 views

Hausdorff Maximal Principle and Axiom of Choice

I need to show that Hausdorff Maximal Principle is equivalent to the Axiom of Choice. Suggested is to use Tukeys Lemma. So far I have that Hausdorff Maximal Principle states that whenever < is a ...
3
votes
1answer
60 views

Diamond in Subtle Cardinals

In Jensen's manuscript on combinatorial principles, he defines the notion of a subtle cardinal: Definition. A regular cardinal $\kappa$ is subtle if for all $C$ club, $(A_\alpha)_{\alpha\in C}$ a ...
1
vote
0answers
84 views

References about positivism

To found the current set theory, it has been necessary to remove some paradoxes like the well-known Russel paradox. It was thus necessary to clarify why things like $$\{x : x \notin x\}$$ can't ...
2
votes
3answers
140 views

Well-ordering on closed subsets of R

I don't know how to approach the proof of the following statement: If $A$ is a family of closed subsets of $\mathbb{R}$ well-ordered with respect to inclusion, then $A$ is countable. Thank you in ...
4
votes
1answer
49 views

Equivalence of forcing notions from dense embedding between them

In general I want to prove that $\mathbb{P}=\left(P,\leq\right)$ and $\mathbb{Q}=\left(Q,\leq\right)$ are forcing notions and there is a dense embedding $h:P\longrightarrow Q$ , then ...
1
vote
0answers
42 views

Which ZF axioms are satisfied in $V_{\omega+\omega}$?

Promoted to a separate question from $\mathbb R$ vs. $\omega+\omega$. Show that all axioms of ZF except the scheme of Replacement hold in $V_{\omega+\omega}$. Extension: Holds for all subsets ...
4
votes
0answers
40 views

Determinacy and uniformization

It's known that $PD$ implies projective uniformization. Assuming $AD$, is there an analogous theorem that holds for all subsets of the plane (where the uniformizing functions are reasonably definable ...
3
votes
3answers
122 views

$\mathbb R$ vs. $\omega+\omega$

Show that there is a subset of $\mathbb R$ which is order-isomorphic to $\omega+\omega$. Show that all axioms of ZF except the scheme of Replacement hold in $V_{\omega+\omega}$. How ...
1
vote
1answer
38 views

Extending well-founded relations to well-orderings

Consider the following exercise: Let $r$ be a binary relation on a set $a$. Show that $r$ is well-founded iff there exists a function $h:a\to \alpha$ for some ordinal $\alpha$, suc hthat $(x,y)\in ...
2
votes
1answer
62 views

Set Theory: Tree Property

Why does the tree property hold for regular cardinals but not singular cardinals? (I.e. There exists a tree of height $\kappa$ with countable levels and no cofinal branch for $\kappa$ a singular ...
2
votes
1answer
44 views

Is the dominating number $\frak d$ regular?

For a poset $(P, \leq)$ we say a subset $C\subseteq P$ is cofinal if for all $p\in P$ there is $c\in C$ such that $p\leq c$. We set $$\text{cf}(P,\leq) = \min\{|C|: C\subseteq P \text{ and } C \text{ ...
5
votes
1answer
68 views

Does forcing preserve the least undefinable ordinal from a model of ZFC?

Let $M$ be a transitive model of ZFC. For convenience let assume that $M$ is countable. Now let us consider the least undefinable ordinal $\vartheta_M$ which is not definable from elements in $M$ and ...
4
votes
1answer
90 views

A possible alternative to the Axioms of Pair, Union, Infinity and Replacement

In this question we assume that all formulae are in the language of $\sf ZFC$ and that $\sf ZFC$ is consistent. Recall that we say that a formula $\varphi(x,y)$ represents a set-like class relation ...
2
votes
0answers
56 views

Linear order embeddable in reals and the power set of naturals

I'm asked to prove that for $\langle X,\preceq\rangle$ linear order $$ \langle X,\preceq\rangle \ \text{embeddable in}\ \langle P(\mathbb{N}),\subseteq\rangle \iff \langle X,\preceq\rangle \ ...
2
votes
1answer
46 views

Elements of the power set of $\textit{On}$ are elements of $\textit{On}$?

I don't know the truth value of this statement so this is just my own attempt. I am attempting to prove this in NBG set theory. Let $X \in \textit{P(On)}$ By the definition provided in my notes $X ...
1
vote
1answer
31 views

Measure supported on finite subsets of $\mathbb N$ that is biased towards large numbers

Let $\Omega$ be the family of all finite subsets of $\mathbb N$, and let $A_n\subset \Omega$ denote those sets which contain $n$. Does there exist a probability measure $\mu$ on $\Omega$ such that ...
1
vote
0answers
54 views

Is there a model of ZFC in which every real number is definable? [duplicate]

https://en.wikipedia.org/wiki/Definable_real_number Wikipedia defines a "definable real number" as one definable by a parameter-free formula in the language of set theory. The article says, "The set ...
1
vote
0answers
45 views

Are there any online sources/books that could help me further study set theory?

I would like to study set theory more intuitively. Searching the internet for this will only provide study for the basics of set theory (unions, intersections, etc.). There are topics I would like to ...
2
votes
1answer
68 views

Partitioning an infinite set into two equinumerous subsets

Let $X$ be an infinite set. Then there exists a partition $\lbrace A, B \rbrace$ of $X$ such that $A, B, X$ are equinumerous. Can you prove it in $\mathrm{ZFC}$? Can you prove it in ...
1
vote
0answers
52 views

Application of Cantor-Bernstein Theorem? [duplicate]

This is a problem in " Basic set theory "- S.Shen: prove that if $ [0,1]=A \cup B$ then either $A$ or $B$ has the cardinality of the continuum. (Does this follow from the Cantor-Bernstein theorem?)
7
votes
1answer
132 views

$|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.

$A$ and $B$ are sets. $|A \cup B| = \mathfrak{c}$, prove that $A$ or $B$ has cardinality $\mathfrak{c}$. This is an exercise problem from my textbook. It's easy if I assume CH to be true. But how can ...
2
votes
1answer
51 views

Elementary subsctructure of $L_{\omega_1}$

Let us consider $A$ the family of all elements of $L_{\omega_1}$ definable without parameters. I was trying to prove that assuming $V=L$ then $A$ is an elementary substructure of $L_{\omega_1}$ by ...