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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set theory and real numbers

I'm not a set theorist just done some casual reading so please keep the answer simple... ZFC has a countable model M (provided it's consistent). In this model the real numbers R are countable (from ...
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Is elementhood between transitive sets monotonic under successor?

A set $z$ is transitive if $x\in y\in z$ implies $x\in z$. Given a set $x$, we define the successor of $x$, denoted $x^+$, to be $x\cup\{x\}$. Now, let $x$ and $y$ be transitive sets. If $x\in y$, ...
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+500

Over ZF, does “every Hilbert space have a basis” imply AC?

I know there is a similar result due to Blass [1] that over ZF, "every vector space has a (Hamel) basis" implies AC. Looking around, however, I can't find any results on the question for Hilbert ...
2
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1answer
64 views

How to force p<b?

Two cardinal characteristics (cardinals between $\aleph_1$ and $\mathfrak{c}$ are: $\mathfrak{b}$, the least size of an unbounded family in $\omega^{\omega}$ ordered under eventual domination ...
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1answer
23 views

algebraic poset

I learn domain theory and stack in definition of algebraic poset. Recall $P$ is algebraic if for every $x\in P$,the set of compact element $y$ below $x$ is directed and has $x$ as least upper bound. ...
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53 views

Prove that ordinal multiplication is left distributive

Suppose $\alpha, \beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
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0answers
39 views

Set theory with urelements

I'm looking for a first-order formalization of set theory (not necessarily one-sorted) which makes a distinction between sets and urelements (objects, that are not sets, but can be elements of a set). ...
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2answers
84 views

Does axiom of foundation/regularity protect against Russell-like paradoxes?

In ZF set theory the axiom of regularity (also called axiom of foundation) says that: In all nonempty sets x there is an element y such that x∩y=∅ As I been told that the intention of the axiom ...
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3answers
71 views

Russell's paradox and axiom of separation

I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms). I see that it removes the immediate contradiction ...
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33 views

Ordinal arithmetic and functions

I have two function $G$ and $F$ defined on ordinals and I know that $$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I ...
4
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3answers
180 views

Reference textbook developing NBG set theory

I'm starting Borceux "Handbook of Categorical Algebra". It starts with a brief discussion of the logical foundations of category theory. He describes two approaches: 1.defining universes and 2. With ...
1
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2answers
56 views

set theory proof explanation

The following lemma is taken from the book 'Introduction to Set Theory' by Hrbacek and Jech. chapter $6$ normal form Can anyone explain to me why the first sentence holds ( the existence of ...
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2answers
102 views

Axiomatic Set Theory (ZFC): Intersection

I'm currently reading "Axiomatic Set Theory" by Suppes, and the book gives a proof of the existence of the intersection (which relies on the Axiom of Separation). While I understand the idea of the ...
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1answer
62 views

Axiomatic Set Theory: Why do we need the “Axiom of Union”?

I've been reviewing the axioms of ZFC, and I'm trying to make sense of why the "Axiom of Union" was put in place. While the existence of the intersection (of two) sets seems to be a "Theorem" we can ...
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1answer
43 views

Definition of continuity of ordinal function

In the book Introduction to Set Theory' by Hrbacek and Jech, chapter $6$ Ordinal Numbers, section $6$ Normal Form, I don't understand the definition of continuity of ordinal numbers. Ordinal ...
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137 views

True statements about natural numbers that are undecidable in Peano Arithmetic *assuming consistency of PA only*?

I am looking for statements $P$ of Peano Arithmetic (PA) that have the following properties: They are as concrete and simple as possible. It is provable by finitistic means that neither $P$ nor ...
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1answer
43 views

Dedekind finite set and a special well ordered set

In ZFC, Dedekind finite set and finite set are same things. So I have a set say A(which is equal to N in ZFC) all Dedekind finite set are equivalent to proper subsets of A and A is well ordered set. ...
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2answers
91 views

For a complete truth-set $T$ is a countable transitive model satisfying $T$ unique?

Let $T$ be a maximal (in the sense that either $\phi \in T$ or $\phi \not \in T$ for all $\phi \in \mathcal{L}_\in$) set of sentences consistent with $ZFC$. Question For a countable transitive model ...
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474 views

Is this extension of ZFC known to be outright inconsistent?

Is the following first-order theory known to be outright inconsistent? Adjoin to ZFC a unary function $U$ together with the following axioms. If $\alpha$ is an ordinal, then there exists an ...
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46 views

show that ordinal multiplication is associative

Prove that ordinal multiplication $\alpha \cdot (\beta \cdot \gamma) = (\alpha \cdot \beta) \cdot \gamma$ is associative by using the following facts: $$\beta \cdot 0=0$$ $$\beta \cdot (\alpha ...
4
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4answers
98 views

Summable enumerations of $\Bbb Q$

We say that a set of natural numbers $A$ is summable if $\sum_{n\in A}\frac1n$ is finite. It is not hard to see that $\{A\subseteq\Bbb N\mid A\text{ is summable}\}$ is an ideal on $\Bbb N$: Subsets ...
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1answer
93 views

Questions about a proof using $\omega$-covers and subsets of $\omega$

I've been reading Galvin and Miller's article entitled "$\gamma$-Sets and Other Singular Sets of Real Numbers." Within this article is a Lemma. Supose $X \in [\omega]^\omega$ and $\mathcal{O}$ is ...
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1answer
43 views

Least rank of a definable R

The set of real numbers can be defined in many ways, including by Dedekind cuts and Cauchy sequences. The different definitions will give different ranks for the underlying set. There exists a set S, ...
5
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1answer
101 views

Defining truth predicates in set theory

In this blog post J.D.Hamkins shows that KM set theory can define truth predicate for first-order set theory, which means, I believe, that there is a second-order definition of such predicate and KM ...
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4answers
221 views

Does There Exist an Explicit Formula Describing Every Possible Sequence of Numbers?

This thread was previously titled "Does a Set Require an Explicit Formula to Exist?". I'm reading H. Enderton's Elements of Set Theory and working through understanding the Zermelo-Fraenkel axioms. ...
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1answer
67 views

Absoluteness of $\Sigma_2$ sentences in forcing

Let $M$ be a model of ZFC and $M\models \varphi$ such that $\varphi$ is a $\Sigma_2$ sentence in the language of set theory. Let $M[G]$ some forcing extension of $M$. Is $M[G] \models \varphi$? What ...
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1answer
67 views

Why adding a club of $\aleph_1$ collapses $\aleph_1$ to $\aleph_0$?

Let $\{S_n \mid n < \omega\}$ be a partition of $\aleph_1$ into countably many disjoint stationary subsets. Why adding a club of $\aleph_1$ to each $\aleph_1 \setminus S_n$ collapses $\aleph_1$ to ...
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0answers
90 views

What is set theory about? [closed]

What is the subject set theory about? What knowledge is required? I am thinking about what subjects I will choose, but I am not sure if I should take this one. That is the course content: Brief ...
0
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1answer
43 views

What does the notation $H=\{ a | a^2=e \}$ mean? [closed]

Is it true that the notation $H=\{ a | a^2=e \}$ means $H=\{a,a^2=e\}$?
2
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1answer
61 views

Is this description of “sigma-algebra generated by collection of subsets” right?

Disclaimer: sorry for my poor english and edition. Claim: If $M\subseteq \mathcal{P}(X)$, then $\Sigma(M)=M_3$, where: $\Sigma(M)$ is the sigma-algebra generated by $M$ $M_1=\{A\subseteq X:(A\in ...
2
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1answer
49 views

Bijection between Natural numbers and Infinite Cartesian product of Natural numbers?

Consider a function $f(n): \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N} \times ...$ mapping each number $n$ to the set of exponents to raise each prime number $p$ to in order to obtain $n$. For ...
2
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2answers
55 views

First order formula defining a predicate which asserts that a set is finite.

Is it possible to define a predicate in the first order language of set theory such that $F(x)$ is true iff $x$ is a finite set? I understand that FOL cannot assert that the domain of discourse is ...
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2answers
43 views

What are the ramifications of introducing a universal set this way?

What are the ramifications of introducing a universal set using this axiom? $$\exists x : \forall y (y\neq x \rightarrow y\in x)$$
5
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1answer
74 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
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2answers
59 views

Definability and the Separation and Replacement Axiom Schemata

I am beginning to study Set Theory, so naturally one might begin with the axioms. When reading the schemata of separation and replacement, my initial thought was, "Why would we need separation if we ...
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0answers
35 views

Transfinite Induction (Proof Explanation)

The theorem above is extracted from the book 'Introduction to Set Theory' by Hrbacek and Jech. Questions: $1$)I don't understand the successor case. When $\alpha_2=\beta+1$, why suddenly $W_2$ ...
7
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1answer
91 views

Indiscernible subsequences

Work in a saturated model $\cal U$ of sufficiently large cardinality. Are there assumptions on $\kappa<|\cal U|$ that guarantee that any sequence $\langle a_i:i<\kappa\rangle$ has an ...
1
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1answer
47 views

Cardinal exponentiation formula

Assume GCH and let $k,m$ be infinite cardinals. I would like to show that $k^m = \max \{ k,2^m \}$. We of course have $k=\beth_a$ and $m=\beth_b$ for ordinals $a,b$. If $a$ is a successor ordinal, ...
2
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1answer
128 views

Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...
0
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1answer
45 views

Existence of non measurable set and ZF theory?

Does statement: 'Existence of non measurable set' consistent with ZF theory. or if I throw Axiom of choice from ZFC theory. Can I prove or disprove existence of a countably additive measure function ...
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2answers
133 views

In the surreal numbers, is it fair to say 0.9 repeating is not equal to 1?

I find the surreal numbers very interesting. I have tried my best to work through John Conway's "On Numbers and Games" and self-teach myself from some excellent online resources. I have prepared a ...
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1answer
33 views

How to check which axioms hold for models in set theory?

I started a class in set theory. The professor drew a few diagrams, all of them having big circles on the outside. Inside there are two small circles marked $a$ and $b$. And they have arrows between ...
2
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2answers
95 views

“Partitioning” an uncountable set “equally”

I just observed a simple fact about the real numbers, that if U is an uncountable subset of the set of real numbers, then there exists a real number r, such that both the set V of all elements of U ...
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1answer
33 views

Forcing $M[G] \models CH$

I have seen the proof of transitioning from a model in which CH holds to a model in which CH fails. However, how do we force the other direction? More explicitly, suppose that $M$ is a countable, ...
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36 views

A lattice generated by two particular sublattices of the lattice of binary relations

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions $\bigcup_{X\in S}(X\times Y_X)$ where $S$ is a finite partition of $U$ and $Y_X\in\mathscr{P}U$ for every $X\in S$ (that is $Y$ is a ...
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72 views

Another conjecture about filters and cartesian products

Now I present a similar but different (weaker) question (the difference is a different definition of the set $\Gamma$): Let $U$ be some set. Let $\Gamma$ be the set of all finite unions ...
2
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2answers
105 views

How does one prove $ZF\vDash MC\Rightarrow AC$?

This is somewhat adressed to Andreas Blass, whose papers I have read, in particular I make reference to an old paper of his »Existence of Basis implies the Axiom of Choice« (84). Anyone who happens to ...
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1answer
53 views

Proof that ZF set theory implies Weak König's Lemma

In some of my other questions and in several references one finds the statement that ZF axioms imply WKL I have searched for the proof of this, but I so far cannot find a proof. I am looking ...
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0answers
87 views

Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
4
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1answer
36 views

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$?

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$? The question is motivated by the observation that $\kappa< \kappa^{{\rm cf}\kappa}$ for any ...