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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2
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1answer
627 views

Countable infinity and the axiom of choice

Many of the states of affairs about infinite cardinalities and their size, depend on the axiom of choice. What would a comprehensive list be of the properties of the cardinality of the natural numbers ...
5
votes
1answer
345 views

Set theory like first-order theory of ordered pairs

The widely-accepted Kuratowski definition of an ordered pair is $(a,b):=\{\{a\},\{a,b\}\}$. This definition makes perfect sense in the context of first-order set theory. However, how would a ...
3
votes
1answer
614 views

Ordinal non-commutative addition example

I know this is a newbie question, so please bare with me :) I'd like to prove within ZF axiomatic set theory that the addition of two ordinals is not commutative. In particular, I'd like to prove ...
3
votes
1answer
140 views

Question about cofinality of an ordinal

Let $\alpha$ an ordinal and $\langle\alpha_\xi\rangle$ a cofinal sequence of elements of $\alpha$. The length, $\gamma$, of this sequence is at least $\operatorname{cf}\alpha$ but can be equal to any ...
4
votes
1answer
479 views

Counterexamples to the continuum hypothesis

Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably ...
1
vote
1answer
91 views

Equivalent definition of a closed set.

I try to prove the equivalence between "C is closed in an ordinal $\alpha$" and "each strictly increasing sequence of elements of C of length $< cf\alpha$ converge in C". For $\Rightarrow$, it's ...
7
votes
2answers
441 views

A question about the cardinality of the set of all the bijections from $M$ to itself

$M$ is a set.We denote the set of all the bijections from $M$ to itself by $K$.Is the cardinality of $K$ larger than the cardinality of $M$?
13
votes
1answer
651 views

Set of All Groups

In my undergraduate Group Theory class, while discussing the impossibility of equivalence relations on groups, my professor said that the set of all groups does not exist. Are there any group ...
6
votes
1answer
335 views

Set theorist as a physicist or physicist as a set theorist?

I'm majoring physics, but really interested in mathematics. I liked physics since it was really beautiful to have an analysis on a nature with mathematical tool. However, the more i study, the more ...
1
vote
1answer
107 views

How can a non-dense, well-ordered set like a long ray be uncountable?

How can a non-dense, well-ordered set like a long ray be uncountable? If it's a set of an uncountable number of [0,1) line segments laid end to end, shouldn't there be a bijective function between the ...
1
vote
2answers
213 views

Inhabited versus nonempty sets - disasters without excluded middle?

The attached screenshot, from Goldblatt's Topoi, shows at a glance the distinction between the constructive concept of inhabited set versus that of the classical nonempty set. These definitions ...
2
votes
2answers
81 views

Reaching elements with finite applications of the successor relation in well-ordered sets

Given an well-ordered class $(A,\leq)$ we want to proof that every element $a \in A$ can be reached by applying the successor relation on a the smallest element of $A$ or on an limes element of $A$ ...
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0answers
118 views

Questions about $\sigma$-algebra, algebra and topology

I know the definitions of $\sigma$-algebra, algebra and topology, but why countable/finite union, as in $\sigma$-algebra/algebra, and finite intersection, arbitrary union as in topology? What inspire ...
2
votes
2answers
355 views

Hall's theorem vs Axiom of Choice?

From Wikipedia Let $S$ be a family of finite sets, where the family may contain an infinite number of sets and the individual sets may be repeated multiple times. A transversal for $S$ is ...
9
votes
4answers
290 views

Hyperreal field extension

In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension? Is it ...
0
votes
2answers
118 views

Injective function from set of all functions $f: \mathbb{R} \to \mathbb{R}$ to $\mathcal{P}(\mathbb{R})$

I'm looking for an injective function from the set $A$ of all functions $f: \mathbb{R} \to \mathbb{R}$ to $\mathcal{P}(\mathbb{R})$. Any hints? I think the other direction is easy: An injective ...
7
votes
1answer
599 views

Can an infinite cardinal number be a sum of two smaller cardinal number?

Let $\kappa$ be an infinite cardinal number. My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$? If AC holds, then the answer is definitely ...
2
votes
1answer
173 views

Is there a “more powerful” form of set theory that would enable this?

I was wondering about this. In Conway's "On Numbers and Games", he discusses the "surreal numbers", and in one point mentions that they are full of "gaps". That the surreal number line is riddled with ...
6
votes
1answer
223 views

Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
4
votes
1answer
84 views

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular? I would appreciate very much an answer
2
votes
1answer
90 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
3
votes
1answer
118 views

An infinite cardinal agrees with all its well-orders on sets of full size.

Suppose $\kappa$ is an infinite von Neumann cardinal (well ordered by $\in$), and take ${<}$ a well-order on $\kappa$. Does there necessarily exists a subset $X\subset\kappa$ of full size (in ...
7
votes
4answers
346 views

What can I do with proper classes?

There are standard tricks, constructions and techniques in ZFC when working with proper classes; for instance one can form the cartesian product of a pair of classes without difficulty, or more ...
5
votes
1answer
226 views

Do we really need the recursion theorem if we deal only with specific recursively defined functions?

How is it possible to define in a totally rigorous (i.e. from the axioms) was the functions $$h:\mathbb{N}\rightarrow \mathbb{N}, \ n\mapsto 1\cdot\ldots \cdot n$$ or $$ g:\mathbb{N}\rightarrow ...
8
votes
2answers
367 views

Which sets are present in every model of ZF?

As in the title: the existence of which sets is implied by the axioms of $\mathsf{ZF}$? For example one such set would be the empty set whose existence is demanded by the Axiom of the Empty Set. But ...
0
votes
2answers
3k views

Definition of denumerable (countable) set

When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there exists such a bijection or do we mean that we have one and are talking about a pair ...
0
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2answers
758 views

Defining “minimum” using logic and set-theoretic operations.

Express the notion of a minimum of a set of number (where numbers are defined via sets). That is, define a relation Min(S,x) using logic and set-theoretic operations such that it is true whenever x ...
0
votes
1answer
62 views

Set with lower bound but without an infimum w.r.t. $\subseteq$

I'm looking for a set $M$ which is partially ordered by $\subseteq$. $M$ should have a lower bound but no infimum. Is that possible? A lower bound is an element $x \in N$ with $M \subset N$ such that ...
2
votes
1answer
264 views

The relation between order isomorphism and homeomorphism

Let $X$ be a set and let $<_1,<_2$ be order relations on $X$. Let $T_1,T_2$ be the topologies induced on $X$ respectively. If $(X,T_1)$ is homeomorphic to $(X,T_2)$, does that imply that ...
2
votes
4answers
790 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...
1
vote
3answers
181 views

Question about models, cardinalities and collapsing

I have a (seeming) contradiction and I can't seem to figure out the (obvious) mistake in my reasoning. The following (related to my previous question): (1) $\omega$, defined to be the least infinite ...
10
votes
2answers
265 views

Why does $\omega$ have the same cardinality in every (transitive) model of ZF?

As in the title: Why does $\omega$ have the same cardinality in every (transitive) model of ZF? I've been thinking about this for some time now. Can someone show me how to show this by showing me a ...
3
votes
1answer
184 views

Self-similarity in ultrafilters over N

First, some notation: Set variables, $X, Y$, range over sets of natural numbers, $\mathbb{N}={1,2,3,..}$. Square brackets represent sets of natural numbers based on a formula. ...
3
votes
1answer
293 views

Number of non-isomorphic subgroups of $p$-adic integers.

What is the cardinality of the set of non-isomorphic subgroups of $p$-adic integers $\mathbb Z_p$ for a given $p$? The obvious upper bound is $2^{2^{\aleph_0}}$. But are there $2^{2^{\aleph_0}}$ ...
7
votes
1answer
185 views

Given a model of ZF where $ \mathbb{R} $ is the countable union of countable sets, does every subset of $ \mathbb{R} $ have measure zero?

The question basically says it all. It is a well-known result that there exists a model $ \mathcal{M} $ of ZF with the property that $ \mathbb{R}^{\mathcal{M}} $ (here, $ \mathbb{R}^{\mathcal{M}} $ is ...
0
votes
2answers
103 views

What is a notation for the minimal ordinal of $\mathbb{R}$?

What is a notation for the minimal ordinal of $\mathbb{R}$? I know that $\beth_1$ and $\mathfrak{c}$ designate the cardinality of $\mathbb{R}$, and that $\Omega$ denotes the smallest uncountable ...
4
votes
1answer
117 views

Is this function constructed using AC necessarily discontinuous everywhere?

Assume AC. Let $x_\alpha$ be a well-ordering of $\mathbb{R}$. For all $\alpha < \mathfrak{c}$, let $F(x_\alpha) = x_{\alpha+1}$. Can it be proven that $F$ is discontinuous everywhere?
1
vote
2answers
136 views

Give an example of a field of order $\beth_2$

I'd like an example of a field of order $\beth_2$ (that is, the cardinality of the power set of the continuum). I'd prefer more explicit constructions, if possible. This is just out of curiosity, as I ...
8
votes
1answer
302 views

Existence of non-trivial ultrafilter closed under countable intersection

Under what conditions on $\Omega$ does there exist $\mathcal{F} \subset \mathcal{P}(\Omega)$ such that $\mathcal{F}$ is a non-trivial ultrafilter and, for every sequence $(F_{i})_{i \in N}$ of ...
2
votes
1answer
233 views

Better representaion of natural numbers as sets?

Natural numbers can be represented as $0=\emptyset$ $1=\{\emptyset\}$ $2=\{\{\emptyset\}\}$ $...$ or as $0=\emptyset$ $1=\{0\}=0\cup\{0\}$ $2=\{0,1\}=1\cup\{1\}$ $...$ What are the names of ...
2
votes
1answer
147 views

Prüfer groups are countable

I have read that for any prime number $p$ the Prüfer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
2
votes
1answer
182 views

Is the set $S$ of functions $S = \{ f : \mathbb R \to \mathbb R \}$ from the reals to the reals a totally ordered set?

Is the set $S = \{ f : \mathbb R \to \mathbb R \}$ of functions $f$ from the reals to the reals a totally ordered set ? I think the question is clear. Note that there is a bijection to $g(x)$ gives ...
4
votes
4answers
560 views
22
votes
1answer
847 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
0
votes
1answer
128 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
0
votes
2answers
168 views

Why is the class of all sets a stage?

I want to prove that the class of all sets $\mathbb{S}=\{x \mid x=x \}$ is a stage (p. 15) (and then that it is a limit thus that it is the successor of another stage). One way to do it is to proof ...
3
votes
2answers
292 views

Is there a categorification of (infinite) ordinal arithmetic?

Background for the curious reader: An ordinal $\beta$ is a transitive set in the sense that $\alpha\in\beta$ implies $\alpha\subset\beta$. Any ordinal is naturally well-ordered under $\in$ (so any ...
4
votes
1answer
203 views

Why do we need a pullback for the definition or classification of subobjects?

Regarding the subobject classifier construction, why do we need the pullback? Monos from $U$ to $X$ are called subobjects, but I see that there might be injections which just have elements of the X ...
7
votes
1answer
335 views

Existence of non-atomic probability measure for given measure zero sets

Let $\Omega$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $\Omega$. Let $N$ be a collection of measurable subsets of $\Sigma$. Question: What conditions on $\Sigma$ and $N$ guarantee ...
5
votes
1answer
171 views

What is the definition of the well-founded part of a model of set theory?

I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...