Tagged Questions

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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0
votes
1answer
54 views

How to deal with probability problems in proper class sample spaces?

Here my focus is mainly on $Ord$ and questions such as: An ordinal is chosen by random. What is the probability of the event that it is a cardinal number? A coin is tossed proper class many ...
0
votes
2answers
48 views

Is the set of all cardinals smaller then a strongly inaccessible cardinal closed?

Given a strongly inaccessible cardinal $k$ (i.e. $k$ is regular uncountable and for each $\lambda < k$, $2^\lambda < k$), is the set of all cardinals smaller then $k$ closed or open? Mahlo ...
0
votes
0answers
45 views

Can we unify every pair of inner models of ZFC by a same hierarchy?

Definition: Fix a ground model $V$ of ZFC. Let $F:V\rightarrow V$ be a definable class function (we call it an operator). The hierarchy $W^F$ corresponding to $F$ is defined as follows: ...
12
votes
1answer
210 views

Infinite prisoners with hats — is choice really needed?

The problem is this (recently asked about here): A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat line up single file line. Each prisoner faces ...
3
votes
0answers
32 views

Proving existence of $\overline{\Bbb Q_p}$ without AC

The proof that every field has an algebraic closure is known to require at least a weak form of AC, the boolean prime ideal theorem. But I recall reading somewhere that for concrete, sufficiently ...
7
votes
5answers
350 views

When can ZFC be said to be “born”?

The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I understood from it is that ZFC appeared after 1922. In what book or paper was ZFC first explicitly ...
4
votes
0answers
46 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} ...
2
votes
1answer
32 views

Well founded relations.

I'm reading a proof in Jech Set theory and I cannot understand a line. Why is it the case that the replacement axiom guarantees the existence of $\theta$ such that $P_\theta = P_{\theta + 1}$? Last ...
-1
votes
1answer
28 views

A Question Regarding Representing $\mathscr P$($\omega$) as a Digraph and CH

It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy ...
0
votes
2answers
67 views

Adding Substructures by Forcing

Consider a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $M$. Let $V$ be a model of ZFC (or ZF) the general question is that what would happen to classes, $Sub(M):=\{N~;~N~\text{is ...
11
votes
2answers
198 views

What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
1
vote
1answer
39 views

invariant well-ordering on binary words

Let $X$ be the set of all infinite to the right binary words, $X^0$ -- the monoid of all finite binary words. There is a natural action of $X^0$ on $X$, namely the concatenation. Does there exists an ...
1
vote
1answer
38 views

Equivalence Relation on a Class

I'm trying to understand Scott's proof of the incompatibility of axiom of constructibility and the existence of a measurable cardinal. I'm stuck in the use of Łoś's Theorem in the universe. Jech's ...
1
vote
1answer
24 views

Does a stationary set of a regular cardinal $k$ always contain a segment $(\alpha,k)$ for some $\alpha<k$

As a continuation to this question. Suppose that $A$ is a stationary set of a regular ordinal $k=[0,k)$. Does this equivalent to saying that $A$ contains a subset of the form $(\alpha,k)$ for some ...
3
votes
3answers
73 views

Axiom of infinity and empty set

The axiom of infinity is formulated as $$\exists S ( \varnothing \in S \wedge (\forall x \in S) x \cup \{x\} \in S)$$ Can someone explain why the use of $\varnothing$ in the axiom of infinity makes ...
1
vote
1answer
46 views

Fixed Point Equivalences of Axiom of Choice

Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular ...
4
votes
1answer
90 views

How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?

In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the ...
2
votes
0answers
35 views

Is the following notion equivalent to subtle cardinals?

Let $\kappa$ be a regular, uncountable cardinal. We call $\kappa$ $\dagger$ if for every sequence $(A_\alpha \colon \alpha < \kappa)$, $A_\alpha \subseteq \alpha$ and every $\xi_0 < \kappa$ ...
1
vote
0answers
32 views

Looking for example of particular type of proper subgroups of $\big(\mathcal P(A), \Delta \big)$ [closed]

i) Let $A$ be a set with more than two elements , give example of a proper subgroup $H$ of $\big(\mathcal P(A), \Delta \big)$ such that $H$ has more than two elements ii) Let $A$ be an infinite set ...
5
votes
1answer
66 views

If $\mathbb{P}$ is a separative poset that doesn't add $\theta$-sequences then every intersection of $\theta$ dense open sets is dense in $\mathbb{P}$

I am looking for a hint (not a solution) to exercise IV.7.28 of Kunen's Set Theory book (2013). Recall that a poset $\mathbb{P}$ is separative if for every $p,q\in \mathbb{P}$, $p\nleq q$ implies ...
0
votes
1answer
61 views

How much conservative ZF+AC and ZF+DC are over ZF?

A logical theory $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if the language of $T_2$ extends the language of $T_1$; every theorem of $T_1$ is a theorem of $T_2$; and any ...
2
votes
1answer
54 views

Zermelo–Fraenkel Set Theory

So I'll try keeping this real short and simple. Assume that language $L$ is defined as $\{ x\in \{0,1\}^* \}$ (finite binary strings) such that $x$ encodes a proof in ZFC that 4 is a prime number. I ...
3
votes
1answer
58 views

Suslin Tree implies a Suslin Line

A Suslin line is defined as a non-empty, complete, dense, linear order without endpoints and in which every collection of disjoint intervals is countable. A Suslin tree on the other hand, is an ...
3
votes
1answer
29 views

Should a stationary set of an ordinal contain all it's limit poins?

Should a stationary set of an ordinal contain all it's limit poins? By a stationary set I refer to Jech's definition: If $k$ is a cardinal then a set $S \subset K$ is called stationary if $S \cap C ...
4
votes
1answer
60 views

Proving $V=V_{\sf On}$ in $\sf Z+Reg$

Define the function $$V_0=\emptyset,\qquad V_{\alpha+1}={\cal P}(V_\alpha),\qquad V_{\delta}=\bigcup_{\beta<\delta}V_\beta.$$ Since we are working in $\sf Z$ (i.e. $\sf ZF$ without the axiom of ...
2
votes
2answers
58 views

A forcing that is $\omega_1$-closed and $\omega_2$-c.c.

I am reading an article (on second order characterizability) which at some point in a proof states that by forcing with $\mathbb P=\{f:\alpha\to\{0,1\},\alpha\in\omega_1\}$ we do not add subsets to ...
2
votes
2answers
114 views

Recursion on a class instead of a set

According to Wikipedia, the recursion theorem states the following: Let $X$ be a set, and $f:X\to X$ a function. For any $x\in X$, there exists a unique function $g:\omega\to X$ such that $g(0)=x$ ...
2
votes
1answer
26 views

to show that there is no injection from a finite successor of finite ordinal to itself

im trying to prove this proposition let $\alpha \in \omega$ so i want to show that there is no injection (not necessarily keeps order) from $\alpha +1$ to $\alpha$ without using cardinality or the ...
3
votes
0answers
104 views

Which infinity do we mean by $\infty$ in the symbol $x\rightarrow\infty$? [closed]

In ordinary mathematics we use the "limits" frequently. In principle the notion of "limit" is closely related to the notion of "infinity". Intuitively when we are calculating a "limit" we begin from ...
1
vote
0answers
45 views

Is there any inconsistent large cardinal axiom which its inconsistency proof is essentially different from proof of Kunen inconsisteny theorem?

There is a long list of large cardinal axioms. Most of them deemed to be consistent with ZFC but there are also some axioms like existence of Reinhardt or $\omega$-huge cardinals which are natural ...
1
vote
0answers
37 views

Intuitive diagrams for models of non-well founded set theory

Based on our intuition about von-Neuman's rank, there is a standard view to describe a model of ZFC as a large V-shape world. When we remove the Axiom of Foundation (AF) from ZFC and replace it with ...
2
votes
0answers
73 views

How large is an uncountable regular cardinal which is closed under arbitrary fast operators?

Let $Card$ be the proper class of all cardinals, define an infinite set of operators like $\otimes_{n}:(Card\setminus \omega)\times (Card\setminus\{0\})\longrightarrow Card$ which are defined for each ...
4
votes
1answer
49 views

Constructing a function whose domain is $\omega$ using successor operation recursively

Let $x$ be a set. Does there exist a functional relation $f:\omega\to \bf{V}$ which has the following property? \begin{eqnarray*} f(0)&=&x\\ f(1)&=&S(x)=x\cup\{x\},\\ ...
1
vote
2answers
50 views

Showing $\prod_{n < \omega} n = 2^{\aleph_0}$ [duplicate]

I have to show that $\prod_{n < \omega} n = 2^{\aleph_0}$. I'm having trouble getting started. I know that $2^{\aleph_0}$ is the set of binary sequences, or the space of functions from $\mathbb{N}$ ...
3
votes
1answer
68 views

$\kappa$ ineffable $\Rightarrow$ $\kappa$ tree-property

Let $\kappa$ be an uncountable, regular cardinal. We call $\kappa$ ineffable iff for every sequence $(A_\xi \colon \xi < \kappa)$ of subsets $A_\xi \subseteq \xi$ there is a stationary subset $S ...
2
votes
1answer
36 views

For any $a\in S_\Omega$, either it has an immediate predecessor or there is an increasing sequence in $S_\Omega$ whose l.u.b. is $a$. [duplicate]

Specifically I would like to show: Let $S_\Omega$ be the minimal uncountable well-ordered set. For any $a\in S_\Omega$, either $a$ has an immediate predecessor in $S_\Omega$, or there exists an ...
14
votes
1answer
903 views

What is the “opposite” of the Axiom of Choice?

One might think that, trivially, the "opposite" of AC is $\neg$AC. However, thinking about it differently, I'm not sure this is intuitively the case. AC says that every set has a choice function. ...
3
votes
1answer
51 views

Topological form of Martin's Axiom

I'm currently studying consequences of Martin's Axiom: Martin's Axiom (MA): Suppose that $\left\langle P, \leq \right\rangle$ is a ccc partially ordered set and $\{D_\alpha\}_{\alpha < \lambda}$ ...
0
votes
1answer
68 views

Is there a set theory that handles collections of proper classes?

Is there a class of all classes? If not, can I define the “banana” so we have the “banana” of all classes? Is there a banana of all bananas? If I define an apple of all bananas, is there an apple of ...
-1
votes
1answer
43 views

Cardinality of the set of all monoids with countably many elements

How can I prove assuming the continuum hypothesis, that the cardinality of the set of all monoids with countably many elements has cardinality the same as that of the power set of the real numbers. ...
2
votes
1answer
59 views

Doubt about the proof of $V_\omega\models\mathsf{Separation}$

In Kunen's 'set theory', he introduce following theorem: Suppose that for each formula $\phi(x,z,\vec{w})$ with no variable besides the displayed ones free,$$\forall z, \vec{w}\in M:\{x\in z:\phi ...
3
votes
3answers
154 views

The Free Set Lemma

The statement of the lemma is as follows: if $$f: \omega_1 \rightarrow \{x\ :\ x\ \textrm{is finite}\}$$ then there is an uncountable $S \subseteq \omega_1$ such that for all distinct $\alpha,\ \beta ...
6
votes
1answer
75 views

Regarding functions on $\omega_1$

I've been trying to prove a property that apparently all functions $g: \omega_1 \rightarrow \omega_1$ have, where $\omega_1$ is the least uncountable ordinal. For $\alpha \in \omega_1$, define ...
1
vote
1answer
61 views

A Question Regarding the Origin of the Axiom of Symmetry

It is my understanding that Chris Freiling's "Axiom of Symmetry" is based on a counterexample to CH given by Sierpinski in his book "Hypothese de continu". Since I neither read nor speak French, I ...
2
votes
0answers
81 views

Formula Complexity of $\models_n$

I want to show $\models_0$ is $\Sigma_1$, and $\forall n \geq 1, \models_n$ is $\Sigma_n$. So for the base case, $\models_0 \ulcorner \phi \urcorner$ is true iff $\ulcorner \phi \urcorner \in ...
3
votes
0answers
46 views

Is there any filter space characterization for strong cardinals?

The theory of filter spaces is introduced by Apter, Diprisco, Henle & Zwicker, in their joint paper: Arthur Apter, Carlos Di Prisco, James Henle, and William Zwicker, Filter spaces: towards a ...
2
votes
1answer
113 views

How to define mathematical objects from scratch?

Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language ...
17
votes
2answers
543 views

Was there anybody before Cantor who conjectured existence of infinities of different sizes?

Georg Cantor is formally known as the first one who discovered existence of infinities of different sizes. But the history of thinking about the concept of "infinity" in maths and philosophy goes back ...
1
vote
1answer
65 views

A question from Kunen's book: chapter VII (H9), about diamond principle

Assume $(\mathbb{P}$ is c.c.c.$)^M$ and $\Diamond$ holds in $M[G]$. Show that $\Diamond$ holds in $M$. Hint: It is sufficient to verify $\Diamond^-$ in $M$. Should I try to create a ...
1
vote
1answer
20 views

Initial segments of well-ordered sets are isomorphic

I want to prove that if $(X,\prec)$ and $(Y,<)$ are well-ordered sets that X must be isomorphic to an initial segment of $Y$ or vice versa. I am trying to do this by defining the function: ...