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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2answers
35 views

What's the correct form of Axiom of Extensionality?

Different sources report two different forms of the axiom (in which $=$ is considered a primitive notion): ...
3
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4answers
324 views

How to prove that a set exists in ZFC?

This is probably something really trivial, but I don't have any helpful set theory books (nor any library where I could borrow them for that matter) and googling such things as "proving a set exists ...
5
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1answer
62 views

$\aleph_1$ almost sure events that almost never all hold

This recent question sparked my curiosity. Is there a family of events $(E_k)_{k \in I}$ such that$\def\pp{\mathbb{P}}$ $\pp(E_k) = 1$ for any $k \in I$ but $\pp( \bigcap_{k \in I} E_k ) = 0$? Clearly ...
4
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1answer
107 views

What is the *exact* consistency strength of $MK$?

It's well known that the existence of an inaccessible cardinal proves $Con(MK)$. Joel Hamkins has a nice blog post (http://jdh.hamkins.org/km-implies-conzfc/) that explains what you get out of $MK$, ...
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1answer
28 views

Does the relationship between Jonsson Cardinals and Jonsson Algebras require the axiom of choice?

Using Skolem functions, one can see in ZFC that a cardinal $\kappa$ is Jonsson iff there are no Jonsson algebras on $\kappa$. (I.e. every algebra of size $\kappa$ has a proper subalgebra of size ...
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1answer
60 views

Is predicate a set?

A set can be defined by a predicate. Is the predicate itself a set too? A predicate seems to be a sort of relation and since relations are sets (in set theory), it seems that a predicate is a set ...
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1answer
64 views

Proving ZFC is consistent

I've heard from a friend that we can actually prove the consistency of ZFC if we assume at least one inaccessible cardinal exists. How is this carried out, precisely? Googling doesn't help and my ...
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1answer
42 views

show that there's a linear ordering such that $R \subseteq R^*$ [duplicate]

Show that if $R$ is a partial ordering on a set $A$, then there exists a linear ordering $R^*$ on $A$ such that $R \subseteq R^*$
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1answer
40 views

Hausdorff Maximal Principle

"Hausdorff's Maximal Principle" says that any partial order P has a maximal chain (chain = linear suborder). It is equivalent to the axiom of choice. If we restrict Hausdorff's Maximal Principle to ...
1
vote
1answer
27 views

Preserving weakly inaccesible cardinals in generic extensions

I'm trying to see that every weakly inaccessible cardinal $\kappa$ in a ctm $M$ remains weakly inaccesible if we force with a forcing which preserves cofinalities (thus, preserves cardinalities). It's ...
1
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1answer
41 views

Why Can we work with $M$ model countable transitive model of some finite fragment of $\mathrm{ZFC}$ and why is it exist.?

When we say that let $M$ be a countable transitive model of some finite fragment of $\mathrm{ZFC}$. Why Can we work with $M$ model countable transitive model of some finite fragment of ...
2
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1answer
95 views

Regarding the axiom $2^\kappa = 2^{\kappa^+}$ for regular cardinals $\kappa$, and its relationship to a couple of other axioms.

(Take ZFC as background.) The following two statements both follow from GCH: ICF. Injective continuum function. The continuum function (i.e. $\kappa \mapsto 2^\kappa)$ is injective. ...
2
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1answer
38 views

Do the notations for relative constructible universe and for forcing extention coincide?

We have the notations $L[A]$ for the inner-model constructible relative to some $A$, and the notation $M[G]$ for a generic extention of the model $M$. Do they coincide? That is, if we look at the ...
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1answer
33 views

Decidability of the surjectivity of any given function in $\mathsf{ZFC}$

Let $X$ and $Y$ be two non-empty sets and let $f:X\to Y$ be an arbitrary function. Let $\mathsf P$ be the statement that $f$ is surjective. That is, $$\mathsf P\equiv[\forall y\in Y\,\exists x\in ...
0
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1answer
59 views

Forcing exercise from Kunen's book

I'm new in the study of the forcing method and I having some troubbles to solve some of the exercise from Kunen's book (edition 2013): specifically, problem IV 2.46 from page 271. It says the ...
7
votes
1answer
65 views

Does $\operatorname{card}(X) < \operatorname{card}(Y)$ imply $\operatorname{card}(X^2) < \operatorname{card}(Y^2)$ without choice?

I looked to see if this question was already posted, but did not find anything. Please let me know if this is a duplicate. Assume $X, Y$ are infinite sets such that there is an injection $X \to Y$ ...
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0answers
34 views

Stronger Konig's Lemma

Assume T is "the" infinite binary tree: The one generated by branching, starting at the root, indexed by {0,1}. Konig's Lemma (binary case) states that: "There is an infinite (and maximal) branch in ...
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0answers
26 views

Infinite Trees and Axiom of Choice

This is kind of a follow up question to an earlier one. A tree is a partially ordered set T such that: (a) T has a minimum element R, and (b) For each x in T, {y|y less than or equal to x} is well ...
2
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1answer
67 views

Why can't we quantify over propositional functions in the ZFC set theory?

What's the difference between saying if $P(y)$ is some propositional function, then we can create an axiom ${\forall}z{\exists}x:(y{\in}x{\iff}y{\in}z{\land}P(y)$ and saying ...
0
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2answers
61 views

Does Second-Order Comprehension make second-order ZFC inconsistent due to Russell's Paradox?

When we do set theory, we take our first-order variables to range over all sets. But if we take our second-order variables to range over sets of sets in the range of the first-order variables, then ...
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0answers
39 views

How can one formulation of the axiom of extension define set equality and the second one not?

I've seen the claim that there are two ways of writing the axiom of extension. The first one is ${\forall}A{\forall}B({\forall}x(x{\in}A{\iff}x{\in}B)){\iff}A=B$. This one supposedly admits $=$ as a ...
3
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1answer
74 views

an application of Martin's axiom to Lebesgue measure

I am an beginner in set theory and try to solve an exercise in the 2nd chapter in Kunen's book: Assume $MA(\kappa)$, let $A$ be a family of Lebesgue measurable subsets of $\mathbb{R}$,with ...
3
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1answer
39 views

Rigid relations and Choice

A binary relation $R$ on a set $D$ is rigid iff the unique $D → D$ bijection that fixes $R$ is the identity function. Any well-ordering is rigid, so the Well-Ordering Principle has the consequence ...
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2answers
57 views

Ambiguity of the provability of the Continuum Hypothesis

To my understanding, the Continuum Hypothesis cannot be proven true or false under ZFC. But to me this is very ambiguous as to the actual provability of CH. Does this mean that even with infinite ...
1
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1answer
35 views

Number of subsets with the same cardinality

Suppose we have a set $S$ with cardinal number $n$, such that $n+n = n$. Consider the set, T, of all subsets with cardinality $n$. How can I show that the cardinality of $T$ is $2^n$? (Without the ...
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1answer
18 views

Constructing $2^\kappa$ vectors with a certain property

Take an infinite rectangular array with $\kappa$ columns and $2^\kappa$ rows where $\kappa$ is some infinite cardinal. Can you fill it up with at most $\kappa$ different elements in such a way that ...
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2answers
40 views

Proving cardinality of an uncountable sum

I'm trying to prove the following thing: For a family $\mathcal{A}$ of countable sets such that $|\bigcup\mathcal{A}|$ is uncountable and such that $\big|\{A\in\mathcal{A}: x\notin A\}\big| ...
3
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2answers
100 views

Infinite subset contains finite subset of any size

Is it true in ZF that given any infinite set $X$ and any natural number $n$, there is a subset of $X$ with cardinality $n$ (i.e. equinumerous with $n$)? Here, $X$ is defined to be an infinite set if ...
0
votes
1answer
32 views

Proof about the hierarchy of sets: show that $V_{\alpha + 1} = \mathcal P(V_{\alpha})$

Define $$V_0 = \varnothing,$$ $$V_1 = V_0 \cup \mathcal P(V_0),$$ $$V_2 = V_1 \cup \mathcal P(V_1),$$ and so on. In general, $$V_{\alpha + 1} = V_{\alpha} \cup \mathcal P(V_{\alpha}).$$ Show ...
2
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0answers
50 views

Trichotomy of Ordinals. Is $K$ a set?

We want to prove the "Trichotomy of Ordinals": Definiton: An ordinal is a transitive set with elements that are all transitive. Definiton: $\alpha$ and $\beta$ ordinals are comparable if one of the ...
2
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1answer
65 views

Infinite Trees, Infinite Branches, And Choice

https://en.wikipedia.org/wiki/Tree_(set_theory) Defines a tree T order-theoretically as a poset that has a minimum element R, and for each t in T, the set of lower bounds of T is well-ordered. ...
4
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0answers
55 views

Countable Choice And Countable Union Of Countable Sets Being Countable

Quick Question: Does the countable union of countable sets being countable imply the axiom of countable choice?
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1answer
29 views

Does the disjoint union of an indexed family of sets need the axiom of choice?

I have an indexed set $\{O_i:i\in I\}$ (which are orbits of the action of a group on a set), and I would like to define what I know as sum or disjoint union: $$ \sum_{i\in I} \mathscr P(O_i) = ...
3
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1answer
51 views

Proof Of König's Lemma

I am trying, carefully, to prove König's Lemma that an infinite binary tree $T$ has an infinite simple path. Let $R$ denote the root vertex of $T$. If I start at $v_1=R$, there must be a vertex ...
2
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1answer
42 views

Understanding of Zorn's Lemma

Zorn's Lemma. Suppose a partially ordered set $P$ has the property that every chain (i.e. totally ordered subset) has an upper bound in $P$. Then the set $P$ contains at least one maximal element. My ...
2
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1answer
34 views

Decimal expansions and topological connectedness

I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold: One may think of ...
3
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1answer
61 views

Weakly Compact Cardinals are Mahlo Proof

I have a question about a corollary in Jech's set theory text which states: Corollary 17.19. Every Weakly Compact cardinal $ \kappa $ is a Mahlo cardinal, and the set of Mahlo cardinals below $ ...
1
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1answer
39 views

Prove axiom of infinity from ZF$-$Infinity +Zermelo's version of infinity

Assume ZF-infinity and assume that there is a set $X$ such that $\varnothing\in X$ and for all $t\in X$, $\{t\}\in X$. How to deduce axiom of infinity from these new axioms?
5
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1answer
70 views

Continuum hypothesis : Number of connected components of $A^c$ in $\Bbb R$

Let $A \subset \Bbb R$ with a countably infinite number of connected component (for the usual topology). What can be the cardinal of the set of the connected components of $A^c$? With continuum ...
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3answers
1k views

Why can't Axiom of Choice be proven by Rule C

Rule C is appeared in the textbook: Introduction to mathematical logic by Mendelson (Page 81 in the fourth edition). It is said "It is very common in mathematics to reason in the following way. Assume ...
4
votes
1answer
64 views

Quantification over the set(?) of predicates

When learning set theory and logic, one fact that popped up a handful of times was that we did not quantify over predicates. To quote the notes I took in class on the axiom of unrestricted ...
4
votes
1answer
79 views

Understanding sets added by a forcing notion

Consider a coloring $c:[\kappa]^2 \to 2$ ($\kappa$ a regular uncountable cardinal, can be assumed to be $\omega_1$ for simplicity) s.t. the following holds: For every $A \subset ...
2
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1answer
37 views

Exponential of cardinal numbers

there is two wrong statement that I want to find counterexample for them. if $\alpha$ and $\beta$ and $\gamma$ be infinite cardinals then show that these two statements are wrong $\alpha < \beta ...
2
votes
2answers
134 views

Power two of ordinal

if $\omega$ is ordinal number of set $\mathbb{N}$ can I compute $(\omega + 1)^2$ like below $\begin{align*} (\omega + 1)^2 & = \ (\omega + 1).(\omega + 1)\\ & = \ ((\omega + ...
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0answers
68 views

If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U = j_D$, is $U=D$?

Here, $j_U$, $j_D$ are the canonical elementary embeddings induced by the measures $U,D$ on $\kappa$. I actually wish to ask three related questions. Let Ult$_U(V)$, Ult$_D(V)$ be the transitive ...
2
votes
1answer
41 views

Uncountable dense sets of reals without the axiom of choice

In the absence of AC, can there be an uncountable dense set $S\subset\mathbb R$ such that $S\cap(-\infty,a)$ is countable for each real number $a$? (Of course, since $S$ is a countable union of ...
0
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2answers
55 views

Choice function without AC in special case [duplicate]

I read the Jech, Set theory, and saw following proposition. (☆) If S is a finite family of nonempty sets, existence of choice function of S can be proved without axiom of choice. I tried to prove ...
4
votes
3answers
99 views

Minimal model of ZFC without power set axiom

We know that $L$ is the minimal standard model of ZFC. The question is, what is the minimal "standard" model of ZFC$^-$ (meaning ZFC without the Power Set axiom)? This is really two questions: Is ...
2
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1answer
40 views

If every partitioning of $X$ has a choice function, is $X$ necessarily well-orderable?

Working over the ZF axioms, it's clear that if $X$ is a well-orderable set, then every partitioning of $X$ has a choice function, by choosing the minimum of each cell. Question. Does the converse ...
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0answers
41 views

Baire property and perfect set

Be $A\subset X$ whit the Baire property and not meager. Show that $A$ contain a subset perfect nonempty. I try prove that $A$ contain a subset $G_{\delta}$ no-numerable and use the theorem of Cantor ...