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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0answers
4 views

how to show a function is bijection

I have taken two numbers $p$ and $r$ where $p,r\in \{0,1,\ldots,4i + 1\}$ where $i\geq 1$ and $q\in \{0,1,\ldots,n-1\}$. Let $X$ contains all elements obtained by cartesian product of $p$ and $q$. I ...
0
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3answers
38 views

Equivalence class for the relation $R$ (work shown)

For an integer $n\in \mathbb{N}$ define $P(n) = \{p : p \mid n \text{, where $p$ is a prime} \}$. For example $P(12)=\{2,3\} $ and $P(1)=\emptyset$. Question: Consider the relation $R$ on ...
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2answers
120 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [on hold]

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 ...
33
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11answers
8k views

Why is “the set of all sets” a paradox?

I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is ...
2
votes
2answers
35 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
0answers
49 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
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0answers
17 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 ...
2
votes
1answer
29 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
58 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). ...
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0answers
26 views

Let $A$ be a partially ordered set in which every totally ordered subset has a lower bound, then $A$ has a minimal element. [duplicate]

Let $A$ be a partially ordered set in which every totally ordered subset has a lower bound, then $A$ has a minimal element. This is the opposite version of Zorn's lemma. However, since I can't use ...
3
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1answer
78 views

A conjecture about filters and finite unions of cartesian products

Let $U$ be some set. Let $\Gamma$ be the set of all finite joins 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 ...
4
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1answer
71 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
53 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
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1answer
54 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 ...
1
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1answer
93 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 ...
1
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1answer
51 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 ...
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0answers
50 views

Inner Models of ZFC which satisfy V=L [closed]

Let $M$ be an inner model of ZFC which satisfies the Axiom of Constructibilty ($V=L$). What is known about general form of such a model? Should it be similar to Godel's constructible universe ($L$) in ...
2
votes
1answer
145 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
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0answers
36 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 ...
7
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0answers
85 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 ...
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2answers
161 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. ...
2
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1answer
91 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 ...
14
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2answers
218 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
7
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1answer
158 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 ...
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6answers
901 views

Proof that aleph null is the smallest transfinite number?

The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page ...
3
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0answers
59 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
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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 ...
5
votes
1answer
103 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 ...
3
votes
1answer
44 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:= ...
7
votes
1answer
123 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 $a_{0},a_{1},\dots$ ...
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0answers
75 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)?
3
votes
2answers
212 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 ...
1
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1answer
30 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
117 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 ...
2
votes
2answers
77 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 ...
7
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2answers
159 views

Is $\mathrm{ZFC}^E$ outright inconsistent?

From $\mathrm{ZFC},$ define a new theory $\mathrm{ZFC}^E$ by adjoining a constant symbol $E$ together with axioms to the effect that: $E$ is countable and transitive $(E,\in)$ is an elementarily ...
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0answers
92 views

Absoluteness of $\forall x (x=x)$

Is there any kind of (set-theoretic) absoluteness result for the formula $\forall x (x=x)$? And what about for $\exists x (x=x)$? I know $x=x$ is absolute given that it's $\Delta_0$. Also, if I'm ...
4
votes
1answer
51 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
64 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 ...
16
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5answers
336 views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
3
votes
1answer
35 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
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1answer
40 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
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0answers
36 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
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2answers
44 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
72 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
45 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
60 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
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1answer
65 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 ...
4
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0answers
48 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 ...
1
vote
1answer
80 views

Is infinitary Levy hierarchy well-defined?

The well-known Levy hierarchy of formulas consist of two $\omega$-sequences of sets of formulas of different complexity $\langle\langle \Sigma_n:n\in \omega\rangle,\langle \Pi_n:n\in ...