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
26 views

Every II-finite set is III-finite

I need some help proving that if a set $X$ is II-finite then it is III-finite, i.e. if every non-empty family of subsets of $X$ which is linearly ordered by inclusion has a maximal element under ...
4
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0answers
26 views

$\phi(\mathbb{P}\ast \mathbb{\dot{Q}})$ has property $\mu$-linked and $\mu$-centered.

Let $\mathbb{P}$ be a poset and $\mathbb{\dot{Q}}$ be a $\mathbb{P}$-name of a poset. If $\phi(\mathbb{P})$ and $\Vdash_{\mathbb{P}} \phi(\mathbb{\dot{Q}})$, then $\phi(\mathbb{P}\ast ...
1
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1answer
30 views

forcing over a dense set.

I want to show that given a countable transitive model $M$ a notion of forcing $\mathbb P$ and a generic filter $G$, then forcing over a dense set of $\mathbb P$, $D$ is just as forcing over $\mathbb ...
0
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1answer
29 views

Function on a well-ordered set

Let $(W,<)$ be a well ordered set. Let $f : W\rightarrow W$ be a function such that $u < v$ implies $f(u) < f(v)$. Show that $\forall w \in W, w \leq f(w)$. I was thinking to consider $T=\{x ...
3
votes
1answer
39 views

A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
2
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1answer
43 views

The Amoeba poset has property Knaster.?

The Amoeba poset has property Knaster.? Amoeba poset $\mathbb{P}=\{p \subseteq \mathbb{R}: p$ $\text{is open} \wedge \mu(p)< \epsilon\}$ for $\epsilon>0$. Any suggestion. Thanks
4
votes
1answer
79 views

No Borel well-order of the reals?

I'm told there is no Borel well-order of the reals (in ZFC). I'm told, in fact, that this is because of Borel determinacy. However, this is usually a vague handwave of the form (a) take the usual ...
11
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4answers
190 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
3
votes
2answers
124 views

Do we know if the consistency of $ \mathsf{ZFC} $ is sufficient to prove that $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) \to \mathsf{SM} $?

This is basically a sequel to this earlier post. There, I asked: If $ \mathsf{ZFC} $ is consistent, then is $ \mathsf{ZFC} \nvdash \neg \text{Con}(\mathsf{ZFC}) $, i.e., $ \neg ...
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1answer
35 views

uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be ...
1
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1answer
52 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
2
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0answers
49 views

Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, ...
1
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1answer
92 views

Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.

Gödel’s Second Incompleteness Theorem says that if $ \mathsf{ZFC} $ is consistent, then $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) $, i.e., $ \text{Con}(\mathsf{ZFC}) $ is not provable in $ ...
1
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0answers
36 views

Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
0
votes
1answer
75 views

Breaking the AC barrier using Kelley-Morse set theory

In her blog post Variants of Kelley-Morse set theory, Prof. Gitman proves that every model of the the common version of Kelley-Morse set theory ($\mathsf{KM}$) is a model of the Wikipedia version of ...
0
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1answer
32 views

Every set with more than point admits a permutation with no fixed point and the Axiom of Choice [duplicate]

Assuming axiom of choice , for any set $S$ with more than one point , there exist a bijection $f:S \to S$ such that $f(s) \ne s , \forall s \in S$ . Is the converse true , i.e. Does the statement " ...
2
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1answer
47 views

How to define “is a structure” by a first-order formula of ZFC

So, I've been trying to define a series of notions by first-order formulas in ZFC, and I'd like to know if I'm getting this right. In particular, I'd like to know if my formula for defining a ...
7
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0answers
92 views

Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
2
votes
1answer
53 views

Consequences of the negation of the Axiom of Dependent Choice

It seems to me that a proper reason to include The Axiom of Choice as a foundational axiom of set theory should be based on the observation that the negation of The Axiom of Choice has absurd ...
-2
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0answers
26 views

Cardinal numbers greater than $\omega$ , ZF [duplicate]

Prove in $ZF$ that for every cardinal number exists a greater cardinal number. I managed to prove this fact in $ZFC$. But without using axiom of choice I can't well-order P(A), and can't build ...
1
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0answers
54 views

Pathologies involving non-generic filters

This is a question regarding exercise (IV.2.46) in Kunen's Set Theory (2011); it reads: "Assume that $M$ is a ctm for $ZFC$, and let $\mathbb{P}$ $=$ Fn$(\omega,2)$. Then there is a filter $G$ on ...
7
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0answers
71 views

Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics": Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
4
votes
1answer
87 views

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ ...
2
votes
1answer
76 views

Why does a nontrivial $V \to V$ have a critical point?

Let $V$ denote the von Neumann universe, and let $j: V \to V$ be a nontrivial (non-identity) elementary embedding. The critical point is the smallest $\kappa$ such that $j(\kappa) > \kappa$. The ...
0
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1answer
68 views

Is the cofinality function monotonic?

Is the cofinality function $\operatorname{cf}$ monotonic? I.e., if $\lambda \le \kappa$ for cardinals $\lambda$ and $\kappa$, does it then follow that $\operatorname{cf}(\lambda) \le ...
1
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2answers
51 views

Sets of “Isolated” Cardinals

Let $C\neq\emptyset$ be a set of infinite cardinals with the property that NO member of $C$ occurs as the supremum of strictly smaller members of $C$. So the cardinals in $C$ are sort of "isolated". ...
4
votes
3answers
100 views

Can I deduce ZFC Standard from “ZFC Dedekind”?

ZFC Standard: Infinity, Extensionality, Specification, Pairing, Union, Replacement, Power Set, Choice and Regularity. ZFC Dedekind: Infinity replaced bij Dedekind Inifinity, other 8 axioms the same as ...
1
vote
2answers
55 views

Increasing sequence of open sets in a separable metric space.

Suppose X is a separable metric space and ($U_α$ : α < γ) is an increasing sequence of open sets (i.e. $U_α$ ⊆ $U_β$ for α < β). Show that there is a countable $γ_0$ such that $U_α$ = $U_β$ for ...
1
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2answers
68 views

Fodor's Lemma for clubs

Fodor's (or Pressing Down) Lemma states that for every stationary subset $S$ of a regular cardinal and every regressive function $f:S\to \mathrm{Ord}$, there is an $\alpha$ such that $f^{-1}(\alpha)$ ...
1
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3answers
72 views

Decidability of the cardinality of a set given that the Continuum Hypothesis is independent from ZFC

I'm a Total Amateur (TM), please forgive me if this question makes no sense. The Continuum Hypothesis states that there are no sets with cardinality strictly between that of the integers and the ...
4
votes
1answer
44 views

Cardinality of sets of reals without choice

Assuming just ZF (no axiom of choice): Does $\aleph_n\leq|\mathbb{R}|$ for all $n<\omega$ imply $\aleph_\omega\leq|\mathbb{R}|$? (with $\kappa\leq|\mathbb{R}|$ meaning that there is a set of reals ...
2
votes
1answer
62 views

Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, ...
6
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0answers
85 views

$\sf ZF$ — Sets that can be proven to exist

There are only countably many formal proofs in $\sf ZF$. Thus, there are only countably many sets that can be proven to exist in $\sf ZF$. This collection of sets seems to satisfy $\sf ZF$'s axioms; ...
3
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0answers
114 views

How should one understand the foundation of set theory?

I have read the answer of Carl Mummert for the question on how to avoid circularity. I would like to ask further as I want to study models of set theory. As I understand, with say assuming the ...
0
votes
1answer
67 views

Number of “small” subsets to a “large” set

For the following we assume the axiom of choice. Let $X$ be a set of cardinality $l$ for some infinite cardinal number $l$, and let $p(X)$ be the number of subsets of $X$ that have cardinality ...
3
votes
1answer
90 views

What is wrong with this proof that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$?

I'm reading books on set theory and I came up with the following 'proof' that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$. What is wrong with it? I really cannot tell! (By ...
2
votes
1answer
49 views

Why this set is finite $E_n=\{l \in \omega: \exists{j<n}[r_j \Vdash \check{l} \leq \dot{h}(n)]\}$?

Lemma IV (4.9) (Kunen book). Let $M$ be a countable transitive model of $ZFC$ and fix $\mathbb{Q}\in M$ such that $(|\mathbb{Q}| \leq \aleph_{0})^{M}$. Let $G$ be $\mathbb{Q}$-generic over $M$. Then ...
0
votes
1answer
53 views

Viewing a $\kappa$-tree as a set of functions.

Trees are defined as posets $(T,<)$ such that for all $x \in T$ the set of predecessors of $x$ is well ordered by $<$. A $\kappa$-tree has height $\kappa$ and every level $T_{\alpha}$ has ...
0
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1answer
67 views

An elementary proof about filters

I my book draft I have proved a theorem which is equivalent to the following. My proof uses ultrafilters, Galois connections, and the cofinite filter. Let $S$ be a set of filters on some set $U$. ...
1
vote
1answer
55 views

Failures of categoricity not related to size?

The paradigmatic cases I know of theories failing to be categorical are cases that seem tied to the language's inability to distinguish between different infinite cardinalities of the theories domain. ...
2
votes
1answer
66 views

What axioms are needed in proofs of the independence of the continuum hypothesis?

My understanding is that the proofs that CH and not-CH are consistent with ZFC are both about ZFC and in ZFC. Is it possible to do these proofs about ZFC but in a weaker axiomatic system? (It is also ...
1
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0answers
51 views

Is there a notion of a club set in a partial order?

Is there a notion of a club set in a general partial order? I know the term club for an ordinal but, what does it mean that $A$ is unbounded in a general tree of a general well order? (for example ...
2
votes
1answer
53 views

Forcing reference

Who first proved that, over ZF, the statement (1) The reals are well-orderable is strictly stronger than the statement (2) Every real-indexed family of nonempty sets of reals has a choice ...
0
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0answers
52 views

Cardinality of countable subsets of the continuum

Assume the following result: If $A$ is an index set with $\#A\leq\#\mathbb R$ and $\{X_{\alpha}\}_{\alpha\in A}$ is a family of sets such that $\#X_{\alpha}\leq\#\mathbb R$ for each $\alpha\in A$, ...
1
vote
1answer
84 views

Can we avoid an axiomatic theory of sets by never formulating paradoxes?

We know that ZFC was formulated to avoid some paradoxes inherent to Cantor's naive set theory, such as Russell's paradox, which inquires about the truth of the existence of the set of all sets. The ...
2
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0answers
40 views

Forcing $M[G] \models \text{cov(meager)}\geq \kappa$

Let $M$ a countable transitive model $ZFC$. In $M$, let $\mathbb{P}=Fn(\kappa,2)$, where $\kappa$ is any cardinal. Let $G$ be $\mathbb{P}$-generic over $M$. Then $M[G] \models \text{cov(meager)}\geq ...
3
votes
1answer
70 views

Why in Teichmüller-Tukey lemma finitness is essential?

First we will state a Teichmüller-Tukey Lemma: Let $A$ be a set and $\phi$ be a property defined on all finite subset of $A$. Assume that $B$ is a subset of $A$ such that each finite subset of $B$ ...
2
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0answers
46 views

Models and its iterates

Let $1\leq n<\omega$ and suppose that $M_n^\#$ exists. If $\mathscr{M}$ is an iterate of $M_n^\#$, why cannot happen that $M_n^\#\in \mathscr{M}$?
3
votes
2answers
113 views

equivalence between formal and informal proof

I'm reading Cohen's book on the independence of the continuum hypothesis, and I see that all the proofs that he gives when he's defining the basic notions of set theory (ordinals, cardinals, ...
3
votes
1answer
75 views

Help with intuition on Cardinal Arithmetic Problems

It happens a lot to me that when I find an intuitive model (picture) of a mathematical entity, the proofs left as exercises in books are very easy to solve. For example when dealing with filters and ...