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 ...

learn more… | top users | synonyms

2
votes
1answer
66 views

Cardinal characteristics

Assuming Continuum hypothesis is not true, How many cardinals $k$ exist which are $\aleph_1 < k < \mathfrak c$? Can I assume that there is a finite number of these cardinals or is there an ...
3
votes
1answer
80 views

Is intuitionistic naive set theory consistent?

I'm asking because the usual argument that a set either belongs in itself or not doesn't apply. I did a quick search and it appears that the logic is also required to be contraction free. If it's ...
1
vote
3answers
62 views

Are dimensions redundant?

I am fairly new to this so apologies for informal terminology. After I discovered what space filling curves are, I came to the conclusion that any point in any number of dimensions can be represented ...
4
votes
2answers
84 views

Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
2
votes
1answer
67 views

Does this argument rely on countable choice?

Consider the following Theorem: Any algebraic field extension $K|F$ of infinite degree contains finite subextensions of arbitrarily high degree. Proof: We'll prove that, for any n, there's a ...
6
votes
4answers
374 views

Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
4
votes
3answers
180 views

Models of set theory

How can one talk of a semantics or model of set theory (lets say ZF or ZFC) when the definition of a structure (and potential model) needs a carrier set in the first place (by its definition)?
3
votes
1answer
66 views

Why are the hypotheses of Zorn's lemma met in this proof about decomposing a Hilbert space into invariant subspaces?

Let $H$ be a separable complex Hilbert space and let $\mathcal{A} \subset B(H)$ be an algebra of bounded linear operators on $H$ which is closed under adjoints. I've just read a very short proof that ...
1
vote
2answers
49 views

$\kappa\cdot\kappa= \kappa,$ for infinite cardinals

I am trying to understand the proof that uses a maximal-lexicographic ordering. For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $<_{cw}$ is ...
1
vote
0answers
71 views

A couple of questions on ordinal numbers

While going over von Neumann's definition of ordinal number I made a couple of conjectures whose veracity I have not been to able to decide yet. I share them here in order to pick up hints ...
1
vote
3answers
67 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
3
votes
2answers
97 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...
-2
votes
2answers
190 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
4
votes
2answers
191 views

How much set theory is necessary for serious logic?

I'm currently studying logic at my university and I have been trying to squeeze in as much set theory on the side as possible. Considering that I am spending quite a lot of time studying set theory ...
0
votes
0answers
20 views

question regarding limit of the a sequence $[x]$ along the a ultrafilter $U$

I've some question regarding limit of the a sequence $[x]$ along the a ultrafilter $U$ Its written let $U$ be an ultrafilter on $N$ where $N=\{1,2,....\}$. Now let $[x]=(x_i \mid i\in N)$ be a ...
3
votes
1answer
47 views

Absolute coequalizers in $\mathbf {Set} $

Let $ A $ be a set and let $ R\subseteq A\times A $ be an equivalence relation on $ A $. Denote by $ p, q $ the projections $ R\longrightarrow A $ on the first and second factor, respectively. The ...
2
votes
1answer
72 views

What cardinals do we actually need?

I recently heard of a great variant of set theory called "Pocket Set Theory" in which the only two cardinalities are that of $\mathbb N$ and that of $\mathbb R$ (and the latter is a proper class). ...
4
votes
1answer
95 views

How many subspace topologies of $\mathbb{R}$?

Say two subsets of $\mathbb{R}$ are equivalent if they are homeomorphic, with the subspace topology. How many equivalence classes are there? It's immediate that there are at least $\beth_0$ (we can ...
2
votes
1answer
55 views

Transfinite Cardinals and Expressive Power

Consider a language with a sufficiently rich lexicon such that, for every (finite and transfinite) cardinal K, it's possible to express the statement that there exist K-many objects. Two general ...
1
vote
1answer
67 views

The Recursion Theorem (Set Theory)

In the book 'Introduction to set theory' by Hrbacek and Jech, there is this theorem stated in the book: Then in the proof, there is this part: I don't understand the induction part. We are trying ...
3
votes
1answer
137 views

A conjecture about filters and finite unions of cartesian products

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions of cartesian products ($X_1 \times Y_1 \cup \dots \cup X_n \times Y_n$) of sets on $U$. Obviously, $\Gamma$ is a a distributive ...
1
vote
1answer
62 views

Is it possible to iterate a function transfinite times?

Let $f:A\rightarrow A$ be a function. We can simply define $f\circ f$, $f\circ f\circ f$, etc., for each given natural number inductively. $f^{(0)}=id_A$ $\forall n\in\omega \qquad f^{(n+1)}=f\circ ...
1
vote
1answer
116 views

Miller's Construction, Partition Principle and Failure of Axiom of Choice

Partition Principle ($PP$) is the following statement: For all sets $a$, $b$ there is an injection $f:a\rightarrow b$ iff there is a surjection $g:b\rightarrow a$ It is known that $ZF\vdash ...
2
votes
0answers
37 views

Is this an equivalent form of Axiom of Choice? [duplicate]

It is known that Axiom of Choice implies the following statement: For each two sets $A$ and $B$, there is a one to one function from $A$ to $B$ iff there is a function from $B$ onto $A$ Is above ...
1
vote
2answers
177 views

How many undecidable statements are there in ZFC?

There are several statements known to be undecidable in ZFC, with the continuum hypothesis probably being the most "popular" one. Is it known how many undecidable statements are there in ZFC? I.e. ...
3
votes
1answer
109 views

Why is the powerset axiom more acceptable than the axiom of choice?

The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that ...
9
votes
0answers
110 views

Does such a first-order theory exist? A question pertaining to transitive models of ZFC.

Assume a proper class of inaccessibles. Does there exist a first-order theory $T$ subject to the following conditions? $T$ admits an infinite model Whenever $M$ is a transitive model of ZFC with $T ...
7
votes
1answer
200 views

Is there “intuition” as to why the Continuum Hypothesis is independent of most large cardinal axioms?

I could not find a question that seemed to answer my specific query, despite lots of material on the Continuum Hypothesis (CH) on MSE and MO. If there is already a question on this, I'd greatly ...
3
votes
0answers
58 views

how many infinities are there? [duplicate]

I'm a past-graduate in mathematics and familiar with the basics of ordinals and cardinals. My question is: how many infinities are there? There are obviously infinitely many, but since we already know ...
3
votes
0answers
66 views

Manifolds of Non-Standard Dimension

Can there exists (non-trivial) manifolds of non-standard dimensions? Certainly, there do exist manifolds of dimension $n$ for any $n \in \mathbb{N}$ (as well as manifolds of countably many ...
3
votes
1answer
49 views

$E$ stationary, $D$ closed and unbounded, then $E \cap D$ stationary.

A subset $S$ of $\omega_1$ is called stationary if the image of every normal function on $\omega_1$ has a non empty intersection with $S$. Let $E$ be a stationary subset of $\omega_1$, and let $S:= ...
5
votes
1answer
110 views

How much maths can we do in NF(U)?

I have recently become interested in non-standard set theories, particularly in NF and NFU and have been reading some things here and there. Of course, I don't know much about it and I'm still trying ...
0
votes
0answers
91 views

Understanding the Banach-Tarski Paradox

How is it possible to prove a paradox? Also, can someone explain the Banach-Tarski paradox in layman's terms (for someone up to calc 3 and ODEs knowledge)?
7
votes
1answer
128 views

All games determined + ZF inconsistent

Let $A$ be a nonempty set, $T\subset A^\mathbb{N}$ a nonempty pruned tree and $X\subset [T]$. The game $G_{A}(T,X)$ is played as follows: Player I and Player II take turns playing $a_{0},a_{1},\dots$ ...
2
votes
1answer
152 views

Can set theory be inverted?

Has anybody investigated or constructed a set theory, call it $T$, which develops in a manner that is the reverse of how the majority of the standard set theories develop? Instead of starting with the ...
2
votes
2answers
86 views

Trouble understanding Jech's version of Easton's theorem

On page 232 of Jech's Set Theory (3rd Edition, 2003), we have the following statement of Easton's theorem. Theorem 15.18 (Easton). Let $M$ be a transitive model of ZFC and assume that the ...
1
vote
1answer
36 views

Definition of an $E$-rudimentary function

For a given set or class $E$, we call $f: V^k \rightarrow V$, where $k < \omega$, $E$-rudimentary, iff it can be generated by the following schemata: $f(x_1,\ldots,x_k) = x_i$ $f(x_1,\ldots,x_k) ...
0
votes
2answers
127 views

Can a set have a subset which doesn't exist?

Is it possible in ZF that given some set $S$, we can informally "describe" a set $P$ such that $P \subseteq S$, and $P$ does not exist (or we can not prove within ZF that P exists)? In other words is ...
4
votes
1answer
72 views

Does the converse of Tychonoff's theorem hinge on the axiom of choice?

Tychonoff's theorem:$\phantom{---}$ If $A$ is a non-empty index set and $X_{\alpha}$ is a non-empty compact topological space for every $\alpha\in A$, then $X\equiv\times_{\alpha\in A} X_{\alpha}$ is ...
3
votes
2answers
77 views

Clarifications on proof of compactness theorem

I've been reading through the following proof of compactness theorem: http://www.princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf One thing that struck me is that this proof seems to ...
3
votes
1answer
41 views

Range of elementary embedding $\pi: V \rightarrow M$ models ZFC?

Let $V$ denote the cumulative hierachy and $M$ be a class together with an elementary embedding $\pi: V \rightarrow M$. As $\pi$ is elementary, we get that $im(\pi)$ models ZFC. But now my textbook ...
1
vote
1answer
43 views

Elementary embeddings and continuity

Let $\pi: V \rightarrow M$ be a non-trivial elementary embedding with critical point $\kappa$, where $M$ is a transitive class. I don't seem to understand a given proof of the following basic ...
0
votes
0answers
39 views

$\left(V_{\pi(\alpha)} \right)^M = \pi \left( V_\alpha \right) $, where $\pi: V \rightarrow M$ is an elementary embedding

Let $\pi: (V;\epsilon) \rightarrow (M;\epsilon \restriction M)$ be an elementary embedding from $V$ into a transitive class $M$. Furthermore, let $V_0 = \emptyset$. $V_{\alpha+1} = \mathcal ...
1
vote
2answers
47 views

Is it possible to prove that $x=\{x\}$ is false in ZF system? [duplicate]

A object is different from the set containing that object seems a basic idea of set theory. That is, for any object $x$, $x≠\{x\}$. But I don't know how to prove it in ZF system (Zermelo-Fraenkel ...
4
votes
0answers
75 views

$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$

I'd like someone to check my proof that $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$. First, let's prove that ...
3
votes
1answer
48 views

$\{\alpha < \kappa \mid cf(\alpha) = \lambda\}$ is not ineffable

We call a subset $X \subseteq \kappa$ of a regular cardinal $\kappa$ ineffable, iff for every family $(A_\alpha \mid \alpha \in X)$ of subsets $A_\alpha \subseteq \alpha$, there is a stationary set $S ...
1
vote
1answer
70 views

Product of Borel $\sigma$-algebras vs Borel $\sigma$-algebra of product

If $X$ and $Y$ are topological spaces with associated Borel $\sigma$-algebras $\mathcal{B}_X$ and $\mathcal{B}_Y$, then the product $\sigma$-algebra $\mathcal{B}_X\otimes \mathcal{B}_Y\subset ...
4
votes
1answer
70 views

Question about cardinals in ZF

In Lévy's basic set theory refers a theorem of Bernstein as exercise problem: Theorem. (Bernstein) Let $\mathfrak{a,b}$ be cardinals. If $\mathfrak{a+a=a+b}$, then $\mathfrak{b\le a}$. I try to ...
3
votes
2answers
243 views

Why exactly is Whitehead's problem undecidable.

I'm trying to get a deeper understanding of Whitehead's problem. It is possible to construct a group of cardinality $\aleph_1$ that satisfies Chase's condition, and is not free. This group is ...
4
votes
0answers
56 views

Cardinality of Countable Support Iteration of Proper Forcing

Suppose $\mathsf{CH}$ holds. Suppose $\mathbb{P}$ is a forcing such that $\mathsf{ZFC} \models \mathbb{P} \text{ is proper and } |\mathbb{P}| = 2^{\aleph_0}$. For instance, Sacks forcing is such an ...