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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1answer
82 views

proof that inner models are transitive

What would be the proof that inner models are transitive? Does it somehow use transitiveness of the model that they are compared to?
0
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1answer
140 views

A formula that defines constructible universe

\begin{multline} \mathrm{Def}(X) := \Bigl\{ \{y \mid y\in X \text{ and } \Phi(y,z_1,\ldots,z_n) \text{ is true in }(X,\in) \} \mid \\ \Phi \text{ is a first order formula and } z_1,\ldots,z_n\in ...
4
votes
3answers
289 views

A set that is not an ordinal

According to what I heard of, an ordinal is constructed by taking an union of {$\alpha$} $\cup$ $\alpha$ where $\alpha$ is a predecessor ordinal. If so, how can there be a set that is not an ...
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2answers
110 views

An inner model and a model that contain same ordinals

In mathematical logic, suppose T is a theory in the language $L = \langle \in \rangle$ of set theory. If $M$ is a model of $L$ describing a set theory and N is a class of M such that $ ...
4
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1answer
96 views

what is $L_{\omega_1} (x)$?

The class of all hereditarily countable sets can be proven to be a set from the axioms of Zermelo–Fraenkel set theory (ZF) without any form of the axiom of choice, and this set is designated . The ...
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1answer
177 views

Why is $\mathbb{R}$ not hereditarily countable set?

Edit: I think I should rather ask a different question to dismantle my confusion. So, why is $\mathbb{R}$ not hereditarily countable set? It seems obvious, but I just can't seem to think of any ...
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2answers
278 views

How does one prove that there are uncountable number of countable ordinals? [duplicate]

Possible Duplicate: Uncountability of countable ordinals How does one prove that there are uncountable number of countable ordinals? Obviously, there are equal to or more than countable ...
7
votes
1answer
228 views

Does every nonempty definable finite set have a definable member?

Does every nonempty definable finite set $S$ have a definable member? EDIT: Here are a few ways to formalize the question, so you can pick your favorite and answer it. Assume whatever large ...
4
votes
1answer
740 views

How to prove that Cantor's normal form can produce all ordinal numbers

How do we prove Cantor's normal form can produce all ordinal numbers? Also, it's a bit difficult to picture $\omega_1$ as a format in Cantor's normal form. Can anyone show how to do this?
2
votes
2answers
229 views

How to prove that $\epsilon_0$ exists?

How do we prove that $\epsilon_0$ exists? The definition of $\epsilon_0$ says that $\omega^{\epsilon_0} = \epsilon_0$. So, how do we know whether such number exists?
6
votes
2answers
576 views

How to define countability of $\omega^{\omega}$ and $\omega_1$? in set theory?

How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two ...
5
votes
1answer
151 views

Generalization of ordinals to well-founded sets?

In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, ...
0
votes
1answer
341 views

Transfinite induction vs transfinite recursion

Let $\mathfrak{A}$ is a well-ordered set. Let $f$ is a function which maps a start segment $\{ i\in\mathfrak{A} \,|\, i<c \}$ of a transfinite sequence of elements of atomic posets into an element ...
4
votes
1answer
540 views

A Dedekind infinite set has a countably infinite proper subset

Set A is Dedekind infinite, i.e. there is a bijective function from A onto some proper subset B of A. Please prove that A has a countably infinite proper subset.
3
votes
1answer
87 views

Undecidability in constructive subsystems of ZFC

The axioms of set theory can be divided into three categories : -The "restrictive" axioms, which impose that set have certain regularity properties (extensionality, foundation). -The "constructive" ...
2
votes
2answers
117 views

Kunen's result and $L(\mathbb{R})$

Recall Kunen's result: If $j:V \to M$ is a nontrivial elementary embedding then $V \ne M$. Assume $AD$ is true in $L(\mathbb{R})$. Let $j : L(\mathbb{R}) \to M$ be a nontrivial elementary embedding. ...
2
votes
2answers
130 views

$\kappa$-complete ultrafilter and bounded subsets of $\kappa$

If $U$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$ then every bounded subset of $\kappa$ has measure $0$. This is because if we had $X \in U$ with $X$ having size less than $\kappa$, ...
0
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5answers
576 views

When to use transfinite induction?

How do we know when we are allowed to use transfinite induction in a proof ? Edit : considering the replies i should say the following Consider an infinite sum of fractions. By induction we can ...
9
votes
1answer
236 views

Atoms necessary for the existence of a generic filter?

I'm reading Some applications of the method of forcing by Todorchevich and Farah and one of the exercises seems to be wrong to me. Definitions We fix some partially ordered set $(P,\preceq)$. A ...
1
vote
1answer
204 views

Lemma required for Cantor-Bernstein-Schroeder Theorem

Let $A$ and $B$ bet sets such that $A \subseteq B$. If there is an injective function $f: B \rightarrow A$, then there is a bijective function $h:B \rightarrow A$. I understand how to prove this ...
4
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0answers
134 views

A question about the proof that forcing extensions don't add ordinals

I've been reading Halbeisen's Combinatorial Set Theory: with a gentle introduction to forcing (well, more like skimming for now...) and I've stumbled into an apparent problem with a proof of the ...
2
votes
1answer
197 views

Limit ordinals vs. Points at infinity

The point at infinity $\infty$ makes $\mathbb{R}$ into a topological circle, i.e. into the real projective space $\mathbb{R}P$: $\infty$ enhances the original structure ($\mathbb{R}$) and makes it ...
17
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3answers
1k views

Why use ZF over NFU?

Forgive me if this question is quite naïve; I've studied axiomatic set theory in the context of ZF, but my knowledge of NF(U) goes little beyond its axioms, what it means for a formula to be ...
15
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3answers
2k views

A nice introduction to forcing

I want to get acquainted with forcing, along with a few friends, and I'm looking for a text to introduce the basic notions (pardon the pun :) ). The point is to study a text (or texts, if they can be ...
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3answers
235 views

Question on a proof from Jech's Set Theory of: If $X$ is a set of cardinals, then $\sup X$ is a cardinal.

The proof: Let $\alpha= \sup X$. If $f$ is a bijective mapping of $\alpha$ onto some $\beta < \alpha$, let $\kappa \in X$ be such that $\beta < \kappa \le \alpha$. Then $|\kappa|=|\{f(\xi): \xi ...
2
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2answers
137 views

A proof for the equality $\aleph_0 card(X) = card(X)$ with $X$ an infinite set?

Good evening, I want to show that all bases of a vector space have the same cardinality, and it needs the following equality : Let $\aleph_0$ be the cardinality of $\mathbb{N}$ and $X$ an infinite ...
4
votes
2answers
833 views

A problem with Cantor's continuum hypothesis

If CH isn't true, this means $$2^{\aleph_0}=\aleph_2\:.$$ Does this imply: $$2^{\aleph_1}=\aleph_3$$ or does the CH hold only for $\aleph_0$ and $\aleph_1$? Thanks
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3answers
681 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
1
vote
1answer
165 views

What should I be able to do with this chapter on Axiomatic Set Theory in order to check if I've learned it decently? [closed]

I've just read a chapter on axiomatic set theory, from Comprehensive Mathematics for Computer Scientists 1. It comes with basic notation on sets and some axioms: Axiom 1 (Axiom of Empty Set) Axiom 2 ...
1
vote
1answer
214 views

Three questions on chapter 7 of Jech's Set Theory

In the proof of Pospisil's Theorem (theorem 7.6) that there are $2^{2^\kappa}$ uniform ultrafilters on $\kappa \geq \omega$, the author writes : Let $\mathcal{A}$ be an independent family of subsets ...
5
votes
1answer
264 views

Cardinality of Reals and Turing Machines

I'm a math hobbyist, so forgive me if what I ask is silly. I just learned that the cardinality of Reals is greater than the Naturals. So, because of that, there can be no turing machines which ...
3
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1answer
376 views

Discussion: Differing definitions for the rank of a set

I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.). $\text{rank}(A)=\min\{\alpha\mid ...
4
votes
2answers
217 views

A weak form of the axiom of choice

I am trying to understand the relationship between two forms of the axiom of choice: If $T=\{X_0,X_1\cdots \}$ is a family of non-empty mutually disjoint finite sets, then $\cup T$ contains at least ...
5
votes
1answer
110 views

Confusion over functors regarding universal objects

Let $\mathcal C$ be a category and suppose it has all finite products. I want to show that there is a functor $- \times - \colon \mathcal C \times \mathcal C \to \mathcal C$ which sends $(A,B)$ to ...
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7answers
3k views

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
5
votes
1answer
291 views

How to exhibit models of set theory

Even though $\mathbb{N}$ cannot be defined by first order means, it can be defined by second order means. Anyway: it can be defined, and there is no doubt, which abstract structure $\mathbb{N}$ ...
3
votes
1answer
253 views

How many weak/strong limit cardinals exist under different assumptions?

I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following ...
6
votes
2answers
265 views

Infinite linear (non-well) orderings

I can rather easily imagine the infinite linear (non-well) orderings $(\mathbb{Z},\leq)$, $(\mathbb{Q},\leq)$, $(\mathbb{R},\leq)$, each one of its own order type. Are there "essentially" other ...
3
votes
1answer
129 views

Existence of sequence of rational numbers in $RCA_0$

In subsystem $RCA_0$ of second order arithmetics, existence of arbitrary functions is not guaranteed since some may not be defined using $\Delta_1^0$ comprehension. In Simpson's book, when real ...
1
vote
1answer
164 views

Do we need the axiom of choice to well-order a set with countably many elements?

Say I index a countably-infinite set $A$ bijectively with the positive integers so that $$A=\{a_1, a_2, a_3,\dots\} $$ The indexing gave an order to the set. Was the choice axiom used?
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votes
2answers
175 views

What is the name of $V_\alpha$?

In the Von Neumann cumulative hierarchy, $V:=\bigcup_\alpha(V_\alpha)$ is called the universe. Is there a name for the individual levels $V_\alpha$? Just as one can say "The closure of $A$ is defined ...
5
votes
3answers
437 views

What does $H(\kappa)$ mean?

As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence: I've heard that if ...
2
votes
2answers
1k views

Finitely but not countably additive set function

Let X be any countable! set and and let F be the cofinite set, i.e., $A \in F $ if A or $A^{c}$ is finite (this is an algebra). Then show that the set function $\mu: F \rightarrow [0,\infty)$ ...
0
votes
1answer
250 views

If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.

I'm having trouble understanding the statement: If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well. I understand the ...
2
votes
4answers
218 views

What is the order type of monotone functions, $f:\mathbb{N}\rightarrow\mathbb{N}$ modulo asymptotic equivalence? What about computable functions?

I was reading the blog "who can name the bigger number" ( http://www.scottaaronson.com/writings/bignumbers.html ), and it made me curious. Let $f,g:\mathbb{N}\rightarrow\mathbb{N}$ be two monotonicly ...
25
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2answers
1k views

Is the fundamental theorem of calculus independent of ZF?

By the fundamental theorem of calculus I mean the following. Theorem: Let $B$ be a Banach space and $f : [a, b] \to B$ be a continuously differentiable function (this means that we can write $f(x + ...
4
votes
3answers
225 views

What is the essential difference between the terminologies “associative” and “semi-group”.

Can only a map $$*:S\times S\rightarrow S$$ be associative? If I look at $$(a* b)* c=a* (b* c),$$ then it seems I have to rule out the more general case $$*:A\times B\rightarrow C.$$ ...
5
votes
2answers
376 views

How does the supremum of the cardinalities of a set of ordinals relate to the supremum of the ordinals?

Suppose we have a set of ordinals such that their supremum is a cardinal (not in the original set). $\sup\{\alpha\}=\kappa$ I'm interested in the supremum of the cardinalities of those ordinals: ...
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vote
3answers
703 views

Is the set of all definable subsets of the natural numbers recursively enumerable?

I asked myself similar questions before, for example "Are the definable real numbers countable"? It seemed to me that the set of all explicitly and unambiguously definable objects is "countable", ...
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3answers
637 views

Are surreal numbers actually well-defined in ZFC?

Thinking about surreal numbers, I've now got doubts that they are actually well-defined in ZFC. Here's my reasoning: The first thing to notice is that the surreal numbers (assuming they are well ...