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
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2answers
56 views

Choice function without AC in special case [duplicate]

I read the Jech, Set theory, and saw following proposition. (☆) If S is a finite family of nonempty sets, existence of choice function of S can be proved without axiom of choice. I tried to prove ...
6
votes
0answers
71 views

If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U = j_D$, is $U=D$?

Here, $j_U$, $j_D$ are the canonical elementary embeddings induced by the measures $U,D$ on $\kappa$. I actually wish to ask three related questions. Let Ult$_U(V)$, Ult$_D(V)$ be the transitive ...
2
votes
1answer
43 views

Uncountable dense sets of reals without the axiom of choice

In the absence of AC, can there be an uncountable dense set $S\subset\mathbb R$ such that $S\cap(-\infty,a)$ is countable for each real number $a$? (Of course, since $S$ is a countable union of ...
8
votes
1answer
126 views

Are categories larger than classes?

In the definition of a category on Wikipedia, it is written that a category "consists of" a class of objects and a class of morphisms, as well as binary operations for compositions of morphisms. What ...
0
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0answers
42 views

Baire property and perfect set

Be $A\subset X$ whit the Baire property and not meager. Show that $A$ contain a subset perfect nonempty. I try prove that $A$ contain a subset $G_{\delta}$ no-numerable and use the theorem of Cantor ...
2
votes
1answer
41 views

If every partitioning of $X$ has a choice function, is $X$ necessarily well-orderable?

Working over the ZF axioms, it's clear that if $X$ is a well-orderable set, then every partitioning of $X$ has a choice function, by choosing the minimum of each cell. Question. Does the converse ...
1
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0answers
47 views

Constructing a Borel-measurable function from a semi-analytic one

Consider a function $f: X \rightarrow (0, \infty)$ whose domain $X$ is a standard Borel space. Suppose $f$ is upper semi-analytic, i.e. for every $\lambda \geq 0$ the set $\{x \in X : f(x) > ...
24
votes
6answers
1k views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
0
votes
0answers
106 views

What kind of Properties are Allowed in the Schema of Comprehension?

Studying an introduction to set theory, we were presented with several axioms as the Axiom Schema of Comprehension and the Axiom of Infinity. This last aximom allows the definition of the inductive ...
1
vote
1answer
91 views

What book on Set Theory is best to understand motivation for axiomatization?

I am Master of Science in ICT, and I had always been in loved in math. On University we haven't been doing any Foundational Mathematics, the closest being Automata Theory and mention of Church-Turing ...
0
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0answers
84 views

Gödel's incompleteness theorem applys to ZFC theory

When I assume ZFC's consistency, it is impossible to prove ZFC's consistency in itself from Gödel's incompleteness theorem 2. If ZFC's consistency have done, its proof need to be done in stronger ...
2
votes
3answers
62 views

About definition of model

In Model theory, the definition of a model is a set. Can it be a proper class? ZFC has a model and maybe some models is a proper class. Definition of a model needs to include a proper class. Is it ...
5
votes
1answer
70 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 ...
1
vote
2answers
80 views

Elementary Set Theory, cofinal subset, cofinality, ordinal, totally ordered set problem

Definition 1. Let $\langle u,<\rangle$ be totally ordered set, $v\subset u$. $v$ is cofinal subset of $u$ means that for all $a\in u$, there exist $b\in v$ ($a\le b$). Definition 2. Let ...
0
votes
1answer
62 views

Arithmetic of uncountable ordinal

Assume $\alpha$ is an ordinal such that $\alpha \geq \omega_1$. Is it true then that $\alpha = \omega + \alpha$ with respect to ordinal arithmetic?
0
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0answers
22 views

Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$ [duplicate]

This is a response to the following exchange at Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? [Bill constructs a $\aleph_0$ ...
1
vote
1answer
50 views

Elementary embeddings and measurable cardinals

Given a measurable cardinal $\kappa$ we can consider its associated embedding $j:V\longrightarrow M\cong Ult_U(V)$ where $U$ is a $\kappa-$complete non principal normal ultrafilter on $\kappa$. In ...
1
vote
3answers
91 views

In relatively simple words: What could be a model of $\sf {ZFC}$?

An $\text{interpretation}$ of a theory consists of A domain of discourse $\mathcal U$, usually required to be non-empty. For every constant symbol, an element of $\mathcal U$ as its ...
4
votes
1answer
44 views

Is there a model of set theory with choice but without a universal well-order?

By "universal well-order", I mean a class-function that bijects $V$ with $ORD^V$.
1
vote
1answer
103 views

absoluteness and and transitivity

I'm early in my reading about absoluteness, but one thing has me stuck, so I thought I'd ask. One reason absoluteness seems to matter is that we feel confident that we know what we're talking about ...
5
votes
1answer
118 views

Limits of finite structures - first order logic

Assume that $\mathcal{C}=\{M_i:i\in I\}$ is an infinite collection of different finite $\mathcal{L}$-structures in a first-order language $\mathcal{L}$. The question is: What kind of infinite ...
38
votes
6answers
3k views

Are there any objects which aren't sets?

What is an example of a mathematical object which isn't a set? The only object which is composed of zero objects is the empty set, which is a set by the ZFC axioms. Therefore all such objects are ...
0
votes
1answer
41 views

when $\kappa > 2^{\omega}$,$ 2^{\kappa}$ is not separable w.r.t the discrete topology

When I want to prove the title, the following hint is provided. say: If $D\subset2^{\kappa}$, is countable, there are $\alpha<\beta$ s.t. for all $f\in D$, $f(\alpha)=f(\beta)$. My question is ...
0
votes
1answer
49 views

Given two forcing extensions, is there a common extension?

Working in ZFC. Say $\mathbb V$ is the ground model, and $\mathbb V[G_1]$ and $\mathbb V[G_2]$ are forcing extensions. Is there a forcing extension $\mathbb V[H]$ containing $\mathbb V[G_1]$ and ...
1
vote
1answer
48 views

A set that satisfies the hypothesis of Zorn's Lemma

A set $x$ satisfies the hypothesis of Zorn's Lemma. Let $k \in x$. $\textbf{Prove:}$ There is a $z \in x$ such that $z$ is $S$-maximal in $x$ and $z=k \vee kSz $ $\textbf{Attempt:}$ ...
4
votes
2answers
174 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
0answers
38 views

Strictly increasing function from $\alpha< \aleph_1$ to $\mathbb{R}$ [duplicate]

I know that there is no increasing function $f: \aleph_1 \to \mathbb{R}$, so it seems like for $\alpha < \aleph_1$, there should exists a function $f: \alpha \rightarrow \mathbb{R}$ that is ...
4
votes
4answers
127 views

Why is the infinite set from the axiom of infinity the natural numbers?

Why is the infinite set from the axiom of infinity the natural numbers? Is there any reason such set was chosen? Couldn't the axiom yield a set that looks like $\Bbb R$ for example?
4
votes
1answer
63 views

When can “$j: V \rightarrow M$ is an elementary embedding” be defined in ZF?

This regards elementary embeddings of inner models of set theory. It seems that it is in general "stated" via an axiom schema each member of which states that the class function is elementary with ...
0
votes
1answer
33 views

Limit ordinal in the exponent [duplicate]

How would you prove that $$n^{\gamma} = m^{\gamma}$$ for every limit ordinal $\gamma$ and $n,m$ finite ordinals? It's a rather short solution problem, but I can't construct any slick answer for it. ...
0
votes
1answer
40 views

Given $\Bbb N$ can you reach every infinite cardinal by performing succesive power set operations?

Suppose $X$ is a set with $|X|=\mathcal K\geq \aleph_0$. Does it always exist an $n$ such that $|\mathcal P (\mathcal P(\cdots\mathcal P(\mathcal (P (\Bbb N ))\cdots)|\geq\mathcal K$ (where the ...
2
votes
1answer
51 views

Prove that $1^\alpha + 2^\alpha = 3^\alpha $ if $ \alpha $ is a limit ordinal

I am trying to prove the following statement: Suppose $ \alpha $ is a limit ordinal. Then $ 1 ^\alpha + 2^\alpha = 3^\alpha$. I'm not sure how to grasp this. Obviously induction won't work, since ...
10
votes
2answers
177 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 ...
2
votes
1answer
73 views

some basic questions about V=L

I am reading through Robert Wolf's A Tour Through Mathematical Logic, which is excellent, but very quick (for a self-studying beginner, like me, at least). I wanted to follow up on four points. (1) ...
1
vote
0answers
62 views

Consequences of the Fact Every Uncountable Set of Reals Has Perfect Set Property

I have recently been working through the consequences assuming the Axiom of Determinacy has and came across the fact that this axiom implies that every uncountable set of reals has the perfect set ...
3
votes
1answer
134 views

How are halting oracles related to set theory?

By the Curry Howard isomorphism, constructive type theory and computation are intimately related to mathematical logic and proofs. Moreover, type theory gives us a nice framework for describing ...
2
votes
1answer
62 views

An ordinal number which satisfies $\omega^{\alpha} = \alpha$

Is there an ordinal such that $\omega^{\alpha} = \alpha$? It seems to me there should be, but I can't explicitly point it out. I know it is possible to prove $\alpha\leq\omega^{\alpha}$, but the ...
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, ...
3
votes
1answer
71 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 ...
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 ...
1
vote
3answers
81 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 ...
2
votes
1answer
59 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
1answer
23 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
88 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
573 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
3answers
2k views

The existence of the empty set is an axiom of ZFC or not?

I found in the Wolfram MathWorld page of the Axiom of the Empty Set that this is one of the Zermelo-Fraenkel Axioms, however on the page about these ZFC Axioms I read that it is an axiom that can be ...
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
85 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 ...
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 ...