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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6
votes
1answer
109 views

$\{S\} \not\in S$ in ZFC?

The usual argument I see for proving that, assuming S is a set, $S\not\in S$ (in ZFC) is the following: Take $S$. Assume $S$ is a set in ZFC. Define $T=\{S\}$ ; by axiom of pairing, $T$ is a set. ...
2
votes
1answer
49 views

Countable transitive model of ZFC that is well-founded externally

As I am studying set theory, I came to realize that there exists a countable "well-founded" model of ZFC. But I am curious whether countable models can ever be well-founded externally. What would be ...
2
votes
2answers
48 views

Embedding $\omega_1$ into the direct sum/product of $\Bbb R$'s and $\Bbb N$'s

I'm trying to find a way to visualize the first uncountable ordinal $\omega_1$. This is rather difficult, as the visualization tactic that I often use for countable ordinals - namely, the "matchstick" ...
1
vote
2answers
46 views

CH, GCH and exioms of set theory

Is there a set of axioms possible in which CH and GCH are proveable (and we can go on with our lives) ? If so, why don't we use this set of axioms ? (i.e., what goes "wrong" with the rest of ...
2
votes
3answers
128 views

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
2
votes
0answers
26 views

A totally bounded uniformity and certain filters

Conjecture For every totally bounded uniform space $(U;F)$ and filters $\mathcal{X}$, $\mathcal{Y}$ on $U$ such that $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$ ...
4
votes
0answers
51 views

Do I run into set-theoretic problems if I allow Homs in my category to be proper classes?

I speak very little set theory, so please , have patience with potential misconceptions below. Let $C$ be some category. This is, as we all know, given by a class of objects $Ob(C)$ and for each $a,b ...
3
votes
1answer
58 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
4
votes
1answer
199 views

Construction of *ZFC

In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to ...
2
votes
0answers
37 views

Transfert principle of a conservative extension of ZFC

In the following paper, there is a theory called $^*ZFC$ in the language $(^*,\in)$. The *-map is (more or less) defined on the Von Neuman hierarchy $S$ and verifies the following axiom schemata true ...
2
votes
1answer
39 views

Show that a collection is a proper class

I need some help with the following exercise. Let $C$ be a proper class and $B$ a set. Show that $I=C-B$ is a proper class. I've thought to argue by absurd. So suppose that $I$ is a set. $I$ cannot ...
2
votes
1answer
41 views

If $\kappa$ is an inaccessible cardinal then $|V_\kappa|=\kappa$

I am trying to do one of the exercises from Jech chapter 6 as follows: If $\kappa$ is an inaccessible cardinal then $|V_\kappa|=\kappa$ Now clearly we have that $\kappa \leq |V_\kappa|$ as $\kappa ...
0
votes
1answer
28 views

“signature”/“type” of set operations e.g. union

The usual addition function on integers has "signature" or "type" $+: \mathbb Z \times \mathbb Z \to \mathbb Z$. Similarly, one could try and write the "signature" of set union as $\cup: \mathbf{Set} ...
2
votes
1answer
85 views

Inaccessible cardinals

Let $M[G]$ be the full Solovay model, and let HOD be the model of hereditarily ordinal definable sets in $M[G]$. Is it possible for HOD not to have an inaccessible cardinal? Does HOD satisfy GCH?
5
votes
2answers
349 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
3
votes
1answer
49 views

Constructively generating a sigma algebra

We have a collection $\mathcal{C}$ of sets (includes $\Omega)$ and would like to constructively generate the sigma algebra $\sigma(\mathcal{C})$. Would the following process work? Let ...
1
vote
1answer
38 views

Addition of a Sequence of Cardinals

In Keith Devlin's book, "The joy of sets", he says that given a sequence of cardinals $\langle\kappa{_{\alpha}\vert\alpha<\beta}\rangle$, the sum of the cardinals ...
1
vote
0answers
41 views

Subject-level guide for Princeton Companion to Math?

I have the Princeton Companion to Mathematics, which I'm enjoying overall. However, right now it's a lot more useful to me for expanding on topics I'm already somewhat familiar and less useful for ...
0
votes
0answers
36 views

A relation between certain families of filters and filters on a cartesian product of sets

(By filters, I will mean all filters on a set, including the improper filter.) The product $\mathcal{A}\times\mathcal{B}$ of two filters is the filter defined by the base $\{ A\times B \,|\, ...
5
votes
4answers
103 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
3
votes
1answer
68 views

Zorn's Lemma related statement

Consider the following statement: If $X$ is partially ordered set such that every chain in $X$ has un upper bound, then for every $x \in X$ there is a maximal element $m$ in $X$ such that $x \le m$. ...
0
votes
1answer
26 views

Construction by transfinite induction

Théorème 2.1.6 (Construction by transfinite induction). Let $(X;\leq)$ be a well order, and $G$ a law which associates each function $g$ whose domain is a proper initial segment $X$ the element (ie, a ...
2
votes
1answer
35 views

Normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$

I am looking at fine normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$ and I have seen that there are two (seemingly) different formulations of normality, in terms of regressive (sometimes also ...
4
votes
2answers
110 views

The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
0
votes
2answers
53 views

An ordinal that is finite or infite?

Let $\beta $ be an ordinal such that for all $\gamma $ $2^{\aleph_{\gamma}}$ = $\aleph_{\gamma + \beta}$. Does $\beta $ have to be infinity?. Under the continuum hypotesis, is true, let 0= $\gamma $ ...
5
votes
3answers
214 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
1
vote
1answer
59 views

About alephs and beths

If $2^{\aleph_{0}} \ge \aleph_{\omega_1}$, show that $\beth_{\aleph_{\omega}} = 2^{\aleph_{0}}$ , and that $\beth_{\aleph_{\omega_1}} = 2^{\aleph_{1}}$ I don´t know how to start, can you give me a ...
3
votes
1answer
309 views

A question about splitting sets

I've been looking into combinatorics and small cardinals, in particular, the splitting number $\mathfrak{s}$. By definition, a set $X \subseteq \omega$ splits an infinite set $Y \subseteq \omega$ if ...
1
vote
1answer
69 views

Reference Request Scott's Trick

Does anyone know of a reference for Scott's Trick. I can't find it in Set Theory-Jech?
3
votes
0answers
48 views

Absoluteness of Satisfaction Relation for Models of the type $J_\alpha$

Is the satisfaction relation absolute between $J_\alpha \subseteq J_\beta$? That is, given a language $L$, a $L$-structures $M$, a formula $\varphi$, and $x$ which are all in $J_\alpha$, is it true ...
1
vote
1answer
67 views

Axiom of Infinity, existence of empty set circular?

I am reading Jech's "Set Theory". First, he states that the existence of the empty set follows from the axiom of infinity. The empty set is defined, using the separation schema as $$\emptyset = \{ u ...
10
votes
2answers
167 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
3
votes
2answers
78 views

Do the ordinals exist before the universe of sets is constructed?

Should I worry about the following appeIarance of circularity in ZFC set theory? In constructing the universe of sets, you start with the empty set and then keep taking the power set over and over. ...
2
votes
1answer
44 views

Are there ordinals other than the set of natural numbers which satisfy this property?

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ...
2
votes
1answer
45 views

Cofinality assuming GCH

There is this statement that GCH holds iff any pair of regular cardinals $\kappa,\lambda$ such that $\kappa<\lambda$ satisfy that $\lambda^\kappa = \lambda$. Assume we do have two such cardinals. ...
1
vote
1answer
35 views

Describing two-step iteration in terms of complete Boolean algebras.

Suppose $B$ is a complete Boolean and let $\dot{C}\in V^B$ be such that $$\|\dot{C}\text{is a complete Boolean algebra}\|_B=1.$$ Let us consider all $\dot{c}\in V^B$ such that ...
2
votes
1answer
43 views

Is there a (concrete) category of superstructures?

The superstructure $V(X)$ over a set $X$ is usually defined as follows. $V_0(X)$ $V_{i+1}(X) = V_i(X) \cup P(V_i(X))$ $V(X) = ⋃_{i=0}^∞V_i(X)$ This defines a metafunction $V.$ Remark. To make ...
2
votes
1answer
28 views

Identity on singular strong limit cardinals

Let $\lambda$ be a singular strong limit cardinal. Prove that $2^\lambda = \lambda^{\mbox{cf}\lambda}$. It has been a while since I had to prove anything relating to cardinals, and I am not sure ...
4
votes
0answers
48 views

Full Theory of Structure of Set Theory and Generic Extensions

Let $V$ be a model of $ZFC$. Let $M \in V$ be (a possibly not transitive) countable model of large fragments of $ZFC$ (for example countable substructures of large $H_\Theta$). If $M$ models enough of ...
8
votes
0answers
149 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...
1
vote
0answers
61 views

When the continuum hypothesis settles the uniqueness (upto isomorphism) of the Hyper-reals doesn't it mean the hypothesis should be an axiom?

One useful consequence of $ZFC$ is that the real numbers can be shown to be unique upto isomorphism. According to wikipedia: The use of the definite article the in the phrase the hyperreal ...
2
votes
1answer
82 views

the quotient boolean algebra of $P(\kappa)$ over the nonstationary ideal

Let $\kappa$ be a regular cardinal. Then the quotient boolean algebra over the nonstationary ideal, $P(\kappa)/I_{NS}$ is $\kappa^+$-complete. Specifically, any $S \subseteq P(\kappa)/I_{NS}$ of ...
2
votes
2answers
44 views

Is there a commonly accepted notation for k-subsets?

The question says it all. I have once seen the following notation used for $k$-subsets of a set $S$ but I failed to verify that it is commonly used and I was also unable to find any evidence for a ...
0
votes
0answers
33 views

Bounded sequences on free ultrafilter

Let $\mathcal U$ be a free (non-principal) ultrafilter on $\mathbb N$. Two rational sequences $(a_n)$ and $(b_n)$ are called $\mathcal U$-equal if and only if $$\{n\mid a_n=b_n\}\in\mathcal U.$$ ...
6
votes
2answers
159 views

Understanding Zorn's lemma.

A lot of authors assume Zorn's lemma. I am told it is not an obvious mathematical fact, but I am having problems understanding why that is. Zorn's lemma states that if every chain in a partially ...
3
votes
3answers
122 views

Why isn't the inductive set _the_ set of natural numbers?

ZFC's axiom of infinity states: $$\exists x (\varnothing \in x \wedge \forall y \in x (y\cup \left \{y \right \} \in x)) $$ Isn't this set $ x $ really $\mathbb{N}$? It wouldn't be $\mathbb{N}$ if x ...
-1
votes
1answer
69 views

What are morphisms in the category of sets $\mathbf{Set}$?

Do i understand correctly that morphisms in the category of sets $\mathbf{Set}$ are ordered triples $(f, A, B)$ where $f$ is a function $A\to B$? It seems that it is often claimed, even in the ...
3
votes
1answer
88 views

A Question Regarding Forcing in Gödel's Constructible Universe in Infinitary Logics

In his answer to the MathOverflow question Gödel's Constructible Universe in Infinitary Logics, Prof. Hamkins gives a very interesting answer and proof to user46667's second question: (2) What is ...
2
votes
2answers
126 views

Is ZF${}-{}$(Axiom of Infinity) consistent?

Godel's theorem implies that Con(ZF) is not provable in ZF since it contains the axiom of infinity. So is it consistent if the Axiom of infinity is removed?
3
votes
1answer
130 views

Are there classes with different sizes?

Are there classes with different sizes ? I will put a precise statement of my question below: Are there two well formed formulas $P,Q$ each with one free variable such that there is no well formed ...