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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3
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1answer
51 views

Proof Of König's Lemma

I am trying, carefully, to prove König's Lemma that an infinite binary tree $T$ has an infinite simple path. Let $R$ denote the root vertex of $T$. If I start at $v_1=R$, there must be a vertex ...
2
votes
1answer
44 views

Understanding of Zorn's Lemma

Zorn's Lemma. Suppose a partially ordered set $P$ has the property that every chain (i.e. totally ordered subset) has an upper bound in $P$. Then the set $P$ contains at least one maximal element. My ...
2
votes
1answer
35 views

Decimal expansions and topological connectedness

I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold: One may think of ...
3
votes
1answer
61 views

Weakly Compact Cardinals are Mahlo Proof

I have a question about a corollary in Jech's set theory text which states: Corollary 17.19. Every Weakly Compact cardinal $ \kappa $ is a Mahlo cardinal, and the set of Mahlo cardinals below $ ...
1
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1answer
40 views

Prove axiom of infinity from ZF$-$Infinity +Zermelo's version of infinity

Assume ZF-infinity and assume that there is a set $X$ such that $\varnothing\in X$ and for all $t\in X$, $\{t\}\in X$. How to deduce axiom of infinity from these new axioms?
5
votes
1answer
70 views

Continuum hypothesis : Number of connected components of $A^c$ in $\Bbb R$

Let $A \subset \Bbb R$ with a countably infinite number of connected component (for the usual topology). What can be the cardinal of the set of the connected components of $A^c$? With continuum ...
4
votes
3answers
1k views

Why can't Axiom of Choice be proven by Rule C

Rule C is appeared in the textbook: Introduction to mathematical logic by Mendelson (Page 81 in the fourth edition). It is said "It is very common in mathematics to reason in the following way. Assume ...
4
votes
1answer
64 views

Quantification over the set(?) of predicates

When learning set theory and logic, one fact that popped up a handful of times was that we did not quantify over predicates. To quote the notes I took in class on the axiom of unrestricted ...
4
votes
1answer
82 views

Understanding sets added by a forcing notion

Consider a coloring $c:[\kappa]^2 \to 2$ ($\kappa$ a regular uncountable cardinal, can be assumed to be $\omega_1$ for simplicity) s.t. the following holds: For every $A \subset ...
2
votes
1answer
38 views

Exponential of cardinal numbers

there is two wrong statement that I want to find counterexample for them. if $\alpha$ and $\beta$ and $\gamma$ be infinite cardinals then show that these two statements are wrong $\alpha < \beta ...
2
votes
2answers
134 views

Power two of ordinal

if $\omega$ is ordinal number of set $\mathbb{N}$ can I compute $(\omega + 1)^2$ like below $\begin{align*} (\omega + 1)^2 & = \ (\omega + 1).(\omega + 1)\\ & = \ ((\omega + ...
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
41 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 ...
0
votes
2answers
55 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 ...
4
votes
3answers
99 views

Minimal model of ZFC without power set axiom

We know that $L$ is the minimal standard model of ZFC. The question is, what is the minimal "standard" model of ZFC$^-$ (meaning ZFC without the Power Set axiom)? This is really two questions: Is ...
2
votes
1answer
40 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 ...
0
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0answers
41 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 ...
8
votes
1answer
123 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 ...
1
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0answers
45 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) > ...
2
votes
3answers
58 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 ...
0
votes
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
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0answers
82 views

How can I understand about ZFC and Gödel's Completeness theorem [closed]

English 1 ZFC could be formulated as First order logic. 2 Gödel's Completeness theorem is a theorem within ZFC. 3 I think a lot of books about set theory is implicitly assuming Gödel's ...
1
vote
1answer
88 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
votes
0answers
102 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 ...
0
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1answer
60 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?
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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
48 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 ...
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
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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 ...
1
vote
2answers
79 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 ...
1
vote
1answer
102 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
108 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 ...
0
votes
1answer
40 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
44 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
47 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 $ $\textbf{Attempt:}$ ...
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 ...
1
vote
0answers
36 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 ...
0
votes
1answer
30 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. ...
4
votes
1answer
62 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
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
49 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 ...
2
votes
1answer
70 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 ...
4
votes
4answers
122 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?
3
votes
1answer
133 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
59 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, ...
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 ...