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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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 $ ...
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49 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 ...
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107 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 ...
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38 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 " ...
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1answer
58 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 ...
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106 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 ...
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1answer
80 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 ...
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58 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 ...
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79 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 ...
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1answer
89 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 $\psi$ ...
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106 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 ...
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72 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 ...
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59 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". ...
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117 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 ...
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2answers
63 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 ...
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2answers
79 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 $f^{-1}(\alpha)$ ...
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3answers
79 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 ...
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47 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 ...
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1answer
82 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, ...
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$\sf ZF$ — 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; ...
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142 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 ...
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71 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 ...
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100 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 ...
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53 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 ...
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59 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 ...
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70 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 $U$. ...
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64 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. ...
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1answer
83 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 ...
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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 ...
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1answer
56 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 ...
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64 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 $\alpha\in A$, ...
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1answer
98 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 ...
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43 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 ...
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1answer
90 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 $B$ ...
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47 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 $M_n^\#\in \mathscr{M}$?
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151 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, ...
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1answer
82 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 ...
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1answer
89 views

Finitely-additive measure over $\Bbb{N}$

On the set of natural number, we can consider the finitely-additive measure defined as: $$\mu(A) = \lim_{n\to\infty}\frac{\#(A\cap [1,n])}{n}.$$ However, there is a definable (by PA, or some ...
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1answer
90 views

Is there any set theory without something like the Axiom Schema of Separation?

I appreciate any insight to this question, including suggestions for other terms to learn about first. I am self-taught with regards to set theory and not a mathematician, so my question may not be ...
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1answer
325 views

Is there any known uncountable set with an explicit well-order?

There is no known well-order for the reals. Is there a known well-order for any uncountable set? If not, is it known whether or not an axiom stating that only countable sets can be well-ordered is ...
3
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1answer
66 views

Why can fix $W_0 \subseteq \kappa$ and select $W$ such that $W_0\subseteq W \subseteq \kappa$?

Lemma V$.2.19$ (book Kunen) In $M$, let $\mathbb{P}=Fn(\kappa,\omega)$, where $\kappa$ is any cardinal. Let $K$ be $\mathbb{P}$-generic over $M$. Then $M[K]\models \mathfrak{d} \geq \kappa$ Proof: ...
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1answer
98 views

Infinity of every ZF model

Let's define $S(x) = x \cup \{x\}$. Prove that axioms of ZF (semantically) imply that all sets $\emptyset, S(\emptyset), S(S(\emptyset)), \dots$ are pairwise distinct. Prove (without axiom of ...
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2answers
144 views

Equivalence classes of real sequences, an interesting concept of closeness

Consider $\mathbb R^{\mathbb N}$, the set of infinite sequences of reals. Two such sequences are equivalent if and only if they eventualy coincide. That is, if $x_1,x_2,\dots$ is one of the sequences, ...
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1answer
90 views

A conceptual link between trees and Polish spaces

Could somebody explain to me why trees are so relevant for the study of Polish spaces and descriptive set theory? I still do not get the proper connection (...and when I think I got it – see the ...
3
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1answer
59 views

Replacing an ordinal with its cardinality in a partition relation

In The Higher Infinite, Kanamori claims that if $\alpha$ is a cardinal, and $\beta \to (\alpha)^\gamma_\delta$ for some $\beta$, then the least such $\beta$ is a cardinal. I can't seem to think of a ...
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1answer
95 views

versions of diamond,$\Diamond^*_S$

For an infinite cardinal $\lambda$ and stationary subset $S\subseteq\lambda^+$, why does $\Diamond^*_S\Rightarrow \Diamond_S$? We use the notation from page $127$ of Assaf Rinot, Jensen’s diamond ...
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1answer
177 views

Jech's Set Theory logic prerequisites

I have read some of the books suggested in What are the prerequisites to Jech's Set theory text?, so I have some beginning experience with transfinite recursion, ordinals, cardinals, order types, ...
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106 views

How do sets of language used to formulate ZFC axioms escape Russell's paradox?

We formulate sets using ZFC. Though, to write its axioms we already use the notion of sets. For instance, in formulating the Axiom of Extensionality, we write the following concatenation of symbols: ...
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1answer
41 views

Countable union of sets of cardinality $c$ has cardinality $c$

The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$. The proofs given in the ...
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1answer
42 views

Question related to ordinal number without using Axiom of Choice.

Can we proof this result without using Axiom of Choice :- $$A\cap \alpha=\emptyset \,\,\,\, \mbox{and}\, \, \, A\times \alpha \sim A\cup \alpha$$ then there is an $A^{'} \subset A$ such that $\alpha ...