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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4
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1answer
83 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
89 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
343 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
224 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
364 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
votes
2answers
746 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
766 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
177 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
262 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
288 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
183 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
102 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 ...
7
votes
1answer
298 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
232 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
146 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
178 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
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4answers
552 views
21
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1answer
842 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
127 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
288 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
334 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
169 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 ...
1
vote
1answer
113 views

A product of 2 Bourbakian posets, ordered by pointwise ordering.

A poset $P$ is called Bourbakian if every order-preserving map $P\rightarrow P$ has a LEAST fixed point. Let $P, Q$ be Bourbakian. Let $P\times Q$ be ordered pointwise, that is $(a,b)\le (c,d)$ if and ...
8
votes
3answers
251 views

Does the category framework permit new logics?

It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't ...
10
votes
1answer
245 views

Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
3
votes
1answer
175 views

Ramsey, Erdős-Rado partitions

This a specific question about Ramsey type colorings. The arrow notation: If $\kappa$, $\lambda$, $\mu$ are cardinals and $n<\omega$, then $$\kappa\rightarrow(\lambda)^n_\mu$$ if for any function ...
1
vote
1answer
100 views

What will happen when Ax.Inf is replaced to its negation in ZFC? [duplicate]

Possible Duplicate: What are the consequences if Axiom of Infinity is negated? In ZFC, if we replace Ax.Inf to such a statement that every set is finite, then does this theory satisfiable? ...
1
vote
0answers
90 views

Ultrapowers by extenders of potential premice

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...
5
votes
1answer
84 views

No $\Delta-$System on a subset of a singular cardinal.

I've been making my way through the new Kunen and I've come across an exercise that I can't work out. The question is this: Let $\kappa$ be a singular cardinal. Show that there is a collection $A$ ...
2
votes
3answers
300 views

Other ways of proving that the set of all countable ordinals is uncountable

I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox. Is there other ways of ...
0
votes
1answer
877 views

Division by two in set theory

Let $A,B$ be two sets such that $2A \cong 2B$ (here $2A := A \coprod A$). Then $A \cong B$. This can be proven without the axiom of choice, which means that one can explicitly construct a bijection $A ...
3
votes
1answer
187 views

Question on the use of a parametric version of Transfinite Recursion Theorem in Introduction to Set Theory 3rd ed. by Hrbacek and Jech

My question concerns a proof given on page 118 in the text Introduction to Set Theory 3rd ed. by Hrbacek and Jech. The authors on page 117 prove a version of the transfinite recursion theorem ...
3
votes
1answer
74 views

What is $\cap_{i\in \emptyset}A_i$?

I tried this: $x\in \cap_{i\in \emptyset}A_i\iff x\in A_i\forall i\in \emptyset$ and the right hand side is vacuously ture--right? So this means it is equivalent to $x$ being an element of ... any ...
5
votes
2answers
330 views

Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
4
votes
2answers
113 views

Relation between inacessible cardinals and CH

I know that the two statements “There exists an inaccessible cardinal” and “Continuum Hypothesis” are both independent of ZFC. Now, are those two statements independent of each other? That older ...
15
votes
2answers
930 views

Where is axiom of regularity actually used?

Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom? This question was to some extent provoked by Dan ...
5
votes
3answers
1k views

The real numbers and the Von Neumann Universe

So I'm going to prefix this question by saying that I probably don't have a great understanding of what I'm asking. We build the cumulative hierarchy as follows: $V_0=\emptyset$ For every $\alpha$, ...
7
votes
3answers
920 views

Ultra Filter and Axiom of Choice

Some person said me: "The fact that Ultra Filters exist is equivalent to the Axiom of choice". Is this correct? I nees some good references about the subject, please help me. Thanks
4
votes
2answers
462 views

Solovay's Model and Choice

Reference; Foundation for analysis without axiom of choice? Please let me know if I'm misunderstanding something and I hope you explain this with relatively easy words. I am eager to learn, but I ...
4
votes
2answers
147 views

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? If so, how? I was reading the Wiki page on the , and it says "Moreover, $\mathbb{R}$ has the same number of elements as the power set of $\mathbb{N}$", but I ...
3
votes
1answer
102 views

Two naive questions about sets

Can every set have a power set ? Does there exist a set A such that there always is a surjection of A onto B , where B is any arbitrary set? (note that positive answers to both the questions lead to ...