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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61 views

Forcing names, parameters in definitions, and the Iterative Conception of Set

So, I've been trying to learn as much as I can about forcing. I know that a model provides its own (trivial) forcing extension. What I'm curious about is whether there is a way to think of the ...
2
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0answers
64 views

Measurable cardinals are weakly-compact?

First off, this isn't homework, but I'm doing research into large cardinal stuff so I wanna understand these theorems. I'm given this information to work with: a cardinal $\mu$ is measurable if ...
6
votes
1answer
164 views

Description of the Universe $V$ [closed]

For me, the concept "set" seams very ambiguous. This does not satisfy me because sets are used very often in mathematics, and so many questions in mathematics are not definite for me. I want to read ...
2
votes
1answer
88 views

What's the need for the axiom of regularity in ZF?

My understanding is that ZF posits the existence of just one set, the empty set $\emptyset$, and the von Neumann hierarchy is constructed starting from there using the axioms of power set and ...
5
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2answers
57 views

$\alpha$-computable bounded subset of $\alpha$ is in $L_\alpha$

I would like to prove the proposition 1.12b from Chong, Techniques of Admissible Recursion Theory: Let $\alpha$ be an admissible ordinal. A subset $K \subseteq \alpha$ is in $L_\alpha$ ($\alpha$-th ...
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2answers
401 views

Programming and ZFC

Suppose I have a simple program that implements an algorithm (say depth-first search), written in a simple imperative programming language with the standard for loops, recursions, conditional ...
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0answers
58 views

Continuum hypothesis outside of ZFC [duplicate]

My understanding is that it is impossible to prove and disprove the Continuum hypothesis in ZFC. Would it be possible to prove or disprove it in some other axiomatic set theory?
2
votes
3answers
79 views

Is there a set $A$ such that $|\mathbb Z|<|A|<|\mathbb R|$ is undecidable?

CH guarantees that the statement $|\mathbb Z|<|A|<|\mathbb R|$ is false for all $A$, but since $\sf CH$ is undecidable it might still be possible that there exist a set $A$ for which the ...
2
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0answers
41 views

Is there more simple way to get a doughnut property?

The terms and my question appear from the Halbeisen's book Combinatorial set theory with a gentle introduction to forcing. For subsets $a$, $b$ of $\omega$ such that $b-a$ is infinite, define a ...
2
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1answer
87 views

Where am I wrong in this “proof” that the collection of sets is countable?

Looking at sets (in ZF, Set Theory of Kunen) I could not escape from looking at logic as well. A formal language is presented containing basic symbols ($\wedge,\neg,\exists,(,),\in,=,$ and $v_{i}$ ...
4
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2answers
127 views

A structural view to the power set axiom: Is this axiom really justifiable?

The power set axiom in set theory states that the collection of the subsets of a set is a set itself. I wonder if this is a "natural" axiom in the sense that if we consider sets as the simplest ...
3
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0answers
57 views

Is there a “model theoretical” proof of Solovay's Stationary Splitting Theorem?

Consider Solovay's Stationary Splitting Theorem: In $\operatorname{ZFC}$ every stationary subset $S$ of some cardinal $\kappa$ with uncountable cofinality can be partitioned into ...
2
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1answer
49 views

Are every set totally ordered in $\sf{NBG}$ or $\sf{SP}$ or $\sf{TG}$ Set Theory?

I have glanced over this post but it the answer dealt mainly with $\sf{ZFA}$ and $\sf{ZFC}$. There was no mention of $\sf{NBG}$, $\sf{SP}$ or $\sf{TG}$. So my question is, can the same claim be proved ...
4
votes
1answer
160 views

Existence of Hamel basis, choice and regularity

Blass (1984) shows that the existence of Hamel basis for arbitrary vector space over any field implies the axiom of choice. However such implication needs the axiom of regularity. As in Blass' ...
3
votes
1answer
44 views

Inaccessibility of $\omega^V_1$ in $L$ under determinacy

I'm looking for the proof of the following fact: Assume axiom of determinacy. Then $\omega^V_1$ is a strongly inaccessible cardinal in $L$. I have seen this result mentioned in few places (e.g. ...
2
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0answers
29 views

show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$.

Trying to show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$. Here is my attempt: Proof by induction on $\alpha$. For $\alpha = 0$, $|R_{\omega+ 0}| = ...
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2answers
71 views

Difference Between Axiom of choice and axiom of countable choice.

My question is: In particular, does the result that every surjective (continuous or even linear if it matters) function has a pre-inverse depend on the full axiom of choice or just the axiom of ...
0
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1answer
40 views

Defining the pair $(A,B)$ in $\mathsf{ZFC}$ where $A,B$ are classes

In $\mathsf{ZFC}$ a class $C$ is defined by a formula $\phi(x)$: conceptually $C = \{ x : \phi(x) \}$, and if $a$ is in $C$ we mean $\phi(a)$. If $A$ and $B$ are two classes, how does one define the ...
6
votes
1answer
137 views

Detecting incomparability in countable elementary submodel

This might be just an easy exercise in model theory but I can't seem to wrap my head around right now. Let $\theta$ be large enough regular cardinal and $\kappa < \theta$. $(\kappa, \prec)$ is ...
1
vote
2answers
54 views

Is this a valid construction of the natural numbers under ZF?

First, define an equivalence relation, $\sim$, such that two sets, $A$ and $B$ are equivalent, $A\sim B$ if and only if there exists a bijection between them. Then define $$0=[\emptyset]_\sim$$ Where ...
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0answers
33 views

In $ZF$, $AC$ is equivalent to $\forall \alpha (\mathscr P(\alpha)$ can be well-ordered) [duplicate]

This is an exercise from Kunen - An introduction to independence proofs that I have hard time to solve. In $ZF$, $AC$ is equivalent to $\forall \alpha (\mathscr ...
4
votes
1answer
80 views

$\neg \textsf{AC}+ \neg\textsf{CH}$

Is there some interesting\surprising results that have only been proven by assuming $\neg \textsf{AC}$ and $\neg\textsf{CH}$ ? Is there some interesting\surprising results implying both $\neg ...
8
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0answers
83 views

Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to ...
5
votes
1answer
81 views

What is wrong with sets like $\{a,\{a\},\{\{a\}\},\ldots\}$

I know pretty much nothing about set theory beyond first year undergraduate maths, so apologies if this is a stupid question. The axiom of regularity in ZFC as I have understood it would forbid the ...
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0answers
93 views

Functional analysis and $\textsf{CH}$

Does anyone know any result in the field of Functional Analysis which holds only if one assume one of the following : $\textsf{CH}$ $\neg \textsf{CH}$ $\textsf{GCH}$
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6answers
113 views

What is some simple to prove very counter-intuitive result obtained by Choice?

I'm aware of some theorems like the Banach-Tarski's which yield very counter-intuitive results, however, it's proof is far beyond my knowledge, so I'm looking for some result that is easy to prove ...
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vote
1answer
44 views

Statements Equivalent to Axiom of Foundation

Yes, this is homework. We're asked to show that the following 3 statements for the Axiom of Foundation are equivalent: $(1)\quad V = \bigcup_{\alpha} V_{\alpha}$ $(2)\quad \in \text{ is ...
9
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1answer
151 views

Models in set theory and continuum hypothesis

Some days ago I had the opportunity to listen to the talk about model theory and connections with algebra and geometry. I'm not at all specialist in this field so my question probably will be naive ...
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0answers
70 views

extending models of zfc

I'm not sure if this question is coherent. But here goes. $(1)$ Let $M$ be a model of $ZFC$. For which axioms $A$ of $ZFC$ is there a model $M'$, extending $M$ and in which $A$ no longer holds? ...
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0answers
28 views

Statements equivalent to the Continuum Hypothesis [duplicate]

I was wondering if there are any statements that could help forming an opinion on whether the CH is "true" or not. I am thus looking for all statements, preferably of this character or simple ones, ...
4
votes
2answers
84 views

How does one prove transfinite induction in ZFC?

If one is able to use classes, it seems to me that the proof of transfinite induction is a simple extension of the usual proof of induction (and equal to the proof of transfinite induction on sets). ...
4
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1answer
44 views

Discussion on AC and countable unions of countable sets

My question stems from the comments following Asaf Karagila's answer here : Why can't you pick socks using coin flips? At some point there was a discussion about whether or not a countable union ...
6
votes
1answer
84 views

What is a projective object in $\rm Set$?

What property of a set in $\sf{ZF}$ is equivalent to its being a projective object in the category $\rm Set$? Since all sets are projective assuming $\sf AC$ my guess is that it is equivalent to ...
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0answers
36 views

On Cantor's argument [duplicate]

What axioms of set theory are needed for Cantor's diagonalization argument to work and why? What happens if we do away with some of these axioms (for instance Axiom of Choice)?
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0answers
30 views

An example of totally ordered field with cardinality larger than continuum. [duplicate]

Is there any totally ordered field with cardinality larger than the continuum? If such field exist, please give an example (an simple one if possible).
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1answer
66 views

Countably infinite product of countably infinite sets has cardinality of the continuum

How to prove that the countably infinite product of countably infinite sets has cardinality of the continuum? I know that it is uncountable thus the only thing to prove is the existence of a one-one ...
6
votes
1answer
111 views

Cardinality of $\sigma$-algebra generated by an infinite family of sets

Let $\mathcal{F}$ be an infinite family of subset of $X$ of cardinality $\kappa$ (thus $\kappa$ is an infinite cardinal). From the recursive description of generated $\sigma$-algebra, I know that the ...
0
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2answers
66 views

List of limit ordinals [closed]

I am trying to understand ordinals and cardinals. Seeing a list of limit ordinals would be helpful. A small question outside of this, are the limit ordinals related to cardinals? In what way? ...
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0answers
47 views

Untyped KM set theory

I'm writing a particular computational system that manipulates logical propositional statements. After starting with ZF and playing around and scrounging, I've arrived at roughly the NBG set theory ...
1
vote
1answer
67 views

Finite coloring of $[\omega]^{<\omega}$.

It is known that Ramsey theorem does not hold for finite colorings of $[\omega]^{<\omega}$. So I am interested in this "partial" result: First let $S_n = ]n, +\infty[$ be the set of natural ...
8
votes
1answer
114 views

Do the ZF-provable forcing principles differ from the ZFC-provable forcing principles?

In "The Modal Logic of Forcing", Joel David Hamkins and Benedikt Löwe show that the ZFC-provable forcing principles are exactly those of the modal logic S4.2 (interpreting $\Diamond \phi$ as asserting ...
6
votes
1answer
87 views

$M$-amenable ultrafilters on $\kappa$ are $\kappa$-powerset preserving

Let $M$ be a transitive model of $\operatorname{ZFC-}$ and let $$ j \colon M \rightarrow N $$ be elementary with $\operatorname{crit}(j) = \kappa \in \operatorname{wfp}(N)$. Let $U_j$ be defined by ...
2
votes
1answer
84 views

Applications of forcing to Topology

I'm insterested on set theory and general topology: particulary on forcing and compactness. I'm searching for a book which study the interaction between both topics or even the interaction between ...
3
votes
1answer
88 views

Why isn't there a total order of $\cal P(\Bbb R)$?

I have heard that, in some models of ZF, $\cal P(\Bbb R)$ has no total order. How could one prove this? I already know that $\Bbb R$ isn't necessarily well-ordered. I'm guessing that one could reduce ...
4
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0answers
56 views

Constructing ordinals which witness an iteration tree is continuously illfounded

An iteration tree $\mathcal{T} = ( T, \langle M_n, E_n \mid n \in \omega\rangle)$ of length $\omega$ is continuously illfounded iff there are ordinals $\alpha_n$ for $n \in \omega$ such that ...
5
votes
1answer
154 views

Can there be a countable transitive model satisfying the same $MK$ theory as $V$?

A little while ago, I asked whether or not there could be a countable transitive model satisfying the same $ZFC$ theory as $V$ (assuming that we're working within some $V$, or (if you like) that there ...
5
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0answers
139 views

“Clubbiness” of $\Pi^1_n$ sets

I'm sure this is just my google-fu failing me, but: what are sufficient large cardinal axioms to guarantee "Every (boldface) $\Pi^1_n$ set of countable ordinals contains or is disjoint from a club ...
3
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0answers
63 views

Handwriting cardinal numbers [closed]

In books the cardinal numbers are usually written with a gothic font. Is there any convention how to handwrite cardinal numbers, in particular, how to handwrite the continuum ($\mathfrak c$)
4
votes
1answer
55 views

Do Russell's socks form a Dedekind-finite infinite set?

A countable collection of pairwise disjoint two-element sets $(A_n)_{n<\omega}$ is called Russell-sequence if $\prod_{n<\omega} A_n$ is empty. (That is, there is no way to choose one of two ...
6
votes
1answer
159 views

Countable elementary submodels

I'm having some trouble understanding elementary submodels. Let $H_\chi$ be the set of all sets which are hereditarily of cardinality $<\chi$. Let $\textbf{N}=(N,\in)$ be a countable elementary ...