Questions tagged [set-theory]

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 theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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35 views

Unions of basis sets in the topology induced by the base

The topology generated by a basis is defined as $\tau=\{U\subset X|\forall x\in U, \exists \beta\subset B, x\in \beta\}$. It is also true that all sets in the topology are unions of basis sets. I have ...
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0answers
30 views

Iterated function,Set theory and Collatz conjecture [on hold]

\begin{matrix} \hline & odd& & &even& &\\ \hline 1&2&3&4&5&6\\ \hline n\equiv 0\left ( mod3 \right ) & n\equiv 1\left ( mod3 \right ) & n\equiv 2\...
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1answer
59 views

Example of non well-orderable set in ZF. [duplicate]

Definition: For a set $x$ define its cardinality by $|x|=\min\{\alpha\in On\mid\alpha \approx x\}$. where $On$ is the calss of all ordinals, $\alpha\approx x$ means there is a bijection $f:\alpha\...
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1answer
60 views

Can we conclude this from ZFC?

Assume that sets exist for a given property. Then can we derive from ZFC that there exists a set containing all those (existing) sets?
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1answer
70 views

Isn’t Halmos wrong here?

In his Naive Set Theory, under Section 9, Families, he states the following: Suppose that $\{ X_i\}$ is a family of sets $(i\in I)$ and let $X$ be its Cartesian product. If $J$ is a subset of $I$, ...
2
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1answer
68 views

Exploiting Grothendieck universes

A Grothendieck universe provides an easy-to-understand example for a model of ZFC. Because ZFC, if consistent, cannot prove the existence of any model of itself, the existence of universes needs to be ...
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0answers
24 views

Which expression in TON corresponds to the limit ordinal of Stegert's ordinal collapsing function?

This is probably going to be one of the most difficult questions that has ever been posted on this website. Jan-Carl Stegert defined a very powerful ordinal collapsing function based on Rathjen's OCF ...
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1answer
73 views

Is the top-down approach to universe-juggling mathematical consistent?

Here's an idea I've been sitting on for awhile. I guess it's time to expose it to the flaming sword of expert opinion. Not sure if it will remain standing by the end of this - oh well. Here goes ...
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1answer
41 views

In the world of ZFC + Grothendieck universes, is there a standard model for ZFC?

Consider the set of ZFC axioms extended with the following axiom of universes: For every set $s$, there exists a Grothendieck universe $U$ that contains $s$, i.e. $s \in U$. Is it possible to ...
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2answers
65 views

Can you prove this without axiom of choice?

Let $\{ X_i\} (i\in I)$ be a family such that $X_i\neq\varnothing$ for all $i\in I$. Let $J\subset I$. Let $\prod_{i\in J} X_i=\{ x\in (\bigcup_{i\in J} X_i)^{J} : \forall i(i\in J\to x_i\in X_i\}$ ...
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2answers
101 views

How to prove that the von Neumann universe equals $V$?

Do I understand correctly, that it is possible to prove in NBG set theory that the von Neumann universe, i.e. the union of $$ \begin{align} V_0 &= \varnothing, \\ V_{\alpha+1} &= P(V_{\alpha})...
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2answers
111 views

What properties of a proposed new logic + inference system + set-theory must be checked to make it viable?

Suppose I would like to introduce a new trifecta of logic + inference system + set theory. What are the minimum formal properties of these systems that I must verify to demonstrate the viability of ...
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1answer
22 views

Proving that a minimal set satisfying 'tree' properties is countable.

Let $X$ be a set and let $\mathcal F (X)$ dentoe the finite subsets of $X$ and let there be given a function $$\tag 1 F: X \to \mathcal F (X)$$ Let $K$ be a finite subset of $X$. Definition: A ...
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0answers
41 views

Does this proof of the Schröder–Bernstein theorem work?

$ f : A \mapsto B $ and $ g : B \mapsto A $ with both functions being injective. Prove that there exists a bijective function $ h $ on $A$ onto $B$. Suppose $ \exists A_1 $ such that $ A_1 \...
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3answers
90 views

Why is the class of all sets denoted $V$?

In NBG set theory, why is the class of all sets denoted $V$? $S$ seems to me to be the natural designation.
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0answers
50 views

Infinite Time Turing Machine and Hypertask

Good Day, I would like to ask this question. Infinite Time Turing Machine ordinals are countable ordinals. Also I have read that ITTM either halts or repeats itselve after countably many steps. Does ...
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1answer
39 views

Proving $\|x=y\|\cdot \|\phi(x)\|\le\|\phi(y)\|$ in Boolean valued models

This question relates to the Boolean algebra approach to forcing. Fix a complete Boolean algebra $B$. I'm writing $\|\sigma\|$ for the Boolean value of $\sigma$, where $\sigma$ is a sentence of the ...
3
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1answer
81 views

If $A$ is $\kappa\text{-Suslin}$ then so is $pA$.

So there is this proposition in "The Higher Infinite", that I have a problem with, and I need some help. The prop. Proposition. Suppose that $A \subseteq {^k(^\omega\omega)}$ and that $\kappa \gt ...
6
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1answer
69 views

On the distributive number $\mathfrak h$

The distributive number $\mathfrak h$ is defined as the least cardinal $\kappa$ such that there exists a family of $\kappa$ open dense subsets in the preordered set $([\omega]^\omega,\subset^*)$ whose ...
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1answer
39 views

Does $|A|^{|A|}=2^{|A|}$ hold for any infinite set $A$? [duplicate]

Because I know the cardinality of all the functions$f:[0,1]\to \mathbb{R}$ is $2^{c}$(c is the cardinality of continuum).I wonder whether this holds generally.
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1answer
41 views

proof of existance in the recursion theorem in set theory

consider this theorem of set theory, the recursion theorem. On Wikipedia there is a proof of uniqueness but no proof of existence, I would like to see an existence proof.
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0answers
46 views

Partial ordering with respect to being an initial segment or continuation according to Halmos

There is an exercise in "Naive Set Theory" by Halmos given as (Sec. 17, p. 68): A subset $A$ of a partially ordered set $X$ is cofinal in $X$ in case for each element $x$ of $X$ there exists an ...
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1answer
109 views

Problem 12, Sec. 24 of Munkres' “Topology,” long line cannot be imbedded in reals

I'm doing this exercise in Munkres' "Topology" (~paraphrasing). I have two motivations for reading this book: 1, topology is really pretty, but the main one, 2, is that I am not yet an undergraduate; ...
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1answer
52 views

Predicative separation and the limited principle of omniscience

The axiom schema of predicative separation says that, for a set X and a predicate F containing only bounded quantifiers, there exists a set whose members are exactly those members of X which satisfy F....
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0answers
45 views

Finding a recursive function that codes a relevant tree for a $\Pi^1_1(a)$ subset of $^k(^\omega\omega)$.

So I am studying the chapter "$\Pi^1_1$ Sets and $\Sigma^1_2$ Sets" from "The Higher Infinite" and I got stuck when trying to come up with a definition for the desired function below. Some ...
36
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4answers
4k views

What was the definition of “set” that resulted in Russell's paradox?

Russell's paradox, the set of all sets not containing themselves can be broken down to two statements: A thing that contains all sets that don't contain themselves. This thing/one such thing ...
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0answers
89 views

Set of Ordinal Functions with singular cofinality

Is there a set $A$ of regular cardinals such that the partial order $(\prod A, <)$ has singular cofinality? Here $\prod A$ is the set of all ordinal functions on $A$ with $f(\kappa)<\kappa$ for ...
2
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1answer
49 views

Countable intersection of nested non-empty closed sets in ZF

For all sequences of nested (non-increasing with respect to set-inclusion) non-empty closed sets, their intersection is non-empty. What are the minimum conditions on a topological space for it to ...
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0answers
43 views

Is the codomain of an injective function whose domain is a proper class, a proper class?

In NBG set theory, let $X,Y,F$ be classes, such that $F \subseteq X\times Y$ is an injective function with domain $X$. If $X$ is proper, is $Y$ proper?
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2answers
66 views

ZFC and axiom of power set [duplicate]

It seems that I do not understand thoroughly the axioms of ZFC. I am thinking of "is really the axiom of power set independent from the other axioms, and if it is, how to prove that?". In other ...
3
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1answer
76 views

Infinite time turing machine vs time travelling turing machine

Good day, I would like to ask this question. EDIT: By time-traveling turing machine I mean that if the TM halts then it sends back in time result of computation. The result might be in form of ...
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1answer
47 views

If $S$ is any constant or variable then '$S$ is a set' is a logical proposition?

I have a textbook that claims the following: Set theory involves the introduction of the new phrase is a set and new symbols $\{ ...\mid ... \}$ and $\in$ defined by (Set$1$) If $S$ is any ...
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1answer
34 views

Defining the Order of the Natural Numbers

Is there a way to give the regular partial order of the natural number directly from their definition through the Infinity Axiom? I have only ever seen the partial order of the natural number to be ...
2
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2answers
83 views

Well-ordered Cartesian product in ZF

Can it be proved in ZF that the product $\prod_{\alpha < \kappa} S_\alpha$ is nonempty given$\{ S_\alpha \}_{\alpha < \kappa}$ a family of nonempty set and $\kappa > 0$ is an ordinal? It ...
3
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1answer
67 views

Do the properties of the hyperreal numbers change depending on the ultrafilter used?

One of the more common constructions of the hyperreals is $\mathbb{R}^{*} := \mathbb{R}^{\omega}/\mathcal{U}$ where $\mathcal{U}$ is some ultrafilter containing the filter of sequences in $\mathbb{R}^{...
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1answer
65 views

When is the maximal element guaranteed by Zorn's Lemma unique? [closed]

Zorn's Lemma states that any poset with the property that every chain has an upper has at least one maximal element. Are there necessary or sufficient conditions on the poset for the maximal element ...
3
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2answers
81 views

Some question about the statement of Zorn Lemma

Some textbooks describe the Zorn Lemma as: Every nonempty ordered set S has the maximal element if every totally ordered subset of S has an upper bound in S. Some other books replace the upper bound ...
0
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1answer
49 views

Proof of every measurable cardinal carries a normal measure

I'm reading the proof of Theorem 10.20 in Set Theory by Jech and I don't understand the last argument. The theorem says every measurable cardinal carries a normal measure. The proof goes: Let $U$ be ...
1
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1answer
68 views

Are there any disasters with Projective Determinacy?

It is well-known that the Axiom of Choice entails a number of counter-intuitive results, Banach-Tarski paradox is only one example. In this MSE question, Martin Sleziak asked about similar undesirable ...
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1answer
39 views

characterization of a normal measure

Problem 10.5 from Set Theory, Jech: A measure $U$ on $\kappa$ is normal if and only if the diagonal function $d(\alpha)=\alpha$ is the least function $f$ with the property that for all $\gamma<\...
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0answers
64 views

Is there a name for isomorphism-preserving functions?

Is there a name for isomorphism-preserving functions in the sense that whenever $x,y$ are sets of the same cardinality, so are $f(x)$ and $f(y)$? For instance, working within the NBG set theory, ...
1
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1answer
59 views

Are the $\mathsf{HOD}$s preserved by weakly homogeneous forcings?

We saw the following theorem in class: Let $M$ be a transitive model of $\mathsf{ZFC}$, let $\Bbb P\in M$ be a weakly homogeneous partially ordered set, let $G$ be $\Bbb P$-generic over $M$ and let ...
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1answer
109 views

Forcing with restricted condition: it's definition [closed]

What does it mean to apply the symbol $\Vdash$ to a condition $q$ restricted to $\xi$: $q\upharpoonright \xi\Vdash\ldots$ as used on the page 5 above lemma 2.4 here; is this $\upharpoonright$ there ...
4
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1answer
43 views

Is it true that $(\kappa = 0,1$ or infinite$)$ $\Leftrightarrow (\kappa^\mu = \kappa$ for all $\mu < \kappa)$?

Clearly the statement holds for all cardinals $\kappa$ up to $\aleph_0$ - it is well known that $|\mathbb{N}^n| = |\mathbb{N}|$ for all $n \in \mathbb{N}$. Assuming the generalised continuum ...
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0answers
77 views

Surreal numbers in Set theoretic multiverse

Good day, I would like to ask this question. Proper class of surreal numbers is the largest ordered colection in givent Set theoretic universe. Now If we have a set theoretic multiverse as proposed ...
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0answers
56 views

Can we create all subsets of all sets? [duplicate]

Using only axiom of specification, can we create all possible subsets of any set? For finite sets, it’s true I guess. But for infinite sets, what can we say? If it were true of infinite sets, we’d ...
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0answers
31 views

Why is Axiom of Powers needed? [duplicate]

By axiom of infinity, we have a set. By axiom of specification, we can create all of its subsets. By axiom of pairing and axiom of unions, we can create a set containing all of theses created subsets. ...
3
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2answers
104 views

Given a cardinal $\kappa$, what is the smallest cardinal $\lambda$ for which $2^{\lambda}\geq\kappa$?

What can $\sf{ZF}$ and $\sf{ZFC}$ tell us about the smallest cardinal $\lambda$ for which $2^{\lambda}\geq\kappa$ given a cardinal $\kappa$? In $\sf{ZFC}$, the cardinals are well-ordered, and, hence, ...
7
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2answers
427 views

Peano's successor function [duplicate]

I have been trying to get my head round ZF set theory and Peano's axioms, but I have hit some confusion over Peano's definition of the successor function, or more accurately von Neumann's model. Why ...
1
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1answer
161 views

Proving an equivalence to the Axiom of Choice

A set of sets A is said to be disjointed if $\forall C,D\in A, C\cap D =\varnothing $. Let F be a set of sets; prove that F has a maximal disjointed subset. Prove that this statement is equivalent ...