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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1answer
64 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
87 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
116 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
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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
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1answer
93 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
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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
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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
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1answer
56 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
75 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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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
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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
56 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
41 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
73 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
114 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
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1answer
76 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 ...
3
votes
1answer
80 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 ...
3
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1answer
66 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 ...
6
votes
1answer
306 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 ...
2
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1answer
62 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: ...
0
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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 ...
3
votes
2answers
124 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, ...
4
votes
1answer
89 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
votes
1answer
50 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 ...
0
votes
1answer
86 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 ...
4
votes
1answer
130 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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3answers
89 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: ...
2
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1answer
32 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 ...
2
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1answer
40 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 ...
1
vote
1answer
57 views

The Definition of Definition by Recursion

The following is presented as the Transfinite Recursion on well-founded relations in Kenneth Kunen's book. Assume that $R$ is set like and well founded on $A$ and ...
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0answers
21 views

Resources for studying different set theories [duplicate]

Ok so lately I have been fascinated about general structure of mathematics and I have read some books on set theory. I have gone through the Endertons introductory book on set theory which operates ...
1
vote
1answer
143 views

Construction of a model of Peano Arithmetic

I'm studying the axioms of Zermelo-Frankel Set Theory at the moment. I already know the following six axioms: The axiom of empty set The axiom of extensionality The axiom of pairing The axiom of ...
2
votes
2answers
83 views

In ZFC, which axioms of set are not required to class?

In Set Theory , Thomas Jech says Classes Although we work in ZFC which, unlike alternative axiomatic set theories, has only one type of object, namely sets, we introduce the informal ...
4
votes
1answer
61 views

Does $\lambda^2 \leq \kappa^2 \Rightarrow \lambda \leq \kappa$ imply the axiom of choice?

I'm aware that the statement "for all cardinals $\kappa$, $\kappa^2 = \kappa$" is equivalent to the axiom of choice (I believe this was proved by Tarski). More generally, does anyone know if the ...
2
votes
1answer
92 views

Existence of an uncountable set of sequence

I'm back with a question really close to this one : Does it always exist an infinite subset of sequences that satisfy this property? Now I'm asking myself : does it exist an uncountable set that is ...
0
votes
1answer
65 views

Enderton's Elements of Set Theory Scott's Trick Exercise (page 207 problem 31)

31. Define kard $A$ to be the collection of all sets $B$ such that (i) $A$ is equinumerous to $B$, and (ii) nothing of rank less than rank $B$ is equinumerous to $B$. (a) Show that kard $A$ is ...
2
votes
1answer
35 views

Generalization of binomial coefficients

Let $X$ be a set. Write $S(X)$ for the set of all bijections $X\longrightarrow X$. One can easily see that $S(X\sqcup\{\operatorname{pt}\})\cong (X\sqcup\{\operatorname{pt}\})\times S(X)$, where ...
5
votes
1answer
75 views

There is no infinite sequence $x_1 \ni x_2 \ni x_3 \ni …$

We take the "usual" axioms of Zermelo-Fraenkel set theory (axiom of extensionality, axiom of the unordered pair, axiom of the sum set, axiom of the power set, axiom of the empty set, axiom of choice ...
1
vote
0answers
62 views

A universally valid second-order sentence only if CH holds.

I'm looking for a second-order sentence that is universally valid only if CH holds. I'm thinking a surjection between all not enumerable sets onto $\mathbb{R}$ but I don't know how to write it. Thank ...
0
votes
1answer
26 views

Given a collection of functions $f_i$ with the same domain, how to replace with values (w/o axiom replacement)

I know from a collection of ordered pairs we can project onto the first coordinate. I'm interested if there's a way (without using the axiom of replacement) to "project" a collection of functions onto ...
2
votes
1answer
65 views

Quasi-disjoint subsets of an infinite cardinal

Let $\kappa$ be an infinite cardinal and let $S$ be a collection of subsets of $\kappa$ such that for $s\neq t\in S$ we have $|s\cap t| \leq 1$. Is it possible that $|S|>\kappa$?
7
votes
3answers
99 views

What is a $P$-name in forcing theory

I am having troubles understanding what is a $P$-name is forcing theory and what is the purpose of this term in the forcing tecnique. Is there any simple way to explain this term. If there was I ...
2
votes
2answers
62 views

Halmos' Naive Set Theory Cardinal Arithmetic Exercise

On page 95 of Halmos' Naive Set Theory, we get the exercise If $\{a_i\}$ and $\{b_i\}$ are families of cardinal numbers such that $a_i< b_i$, then $$\sum_i a_i<\prod_ib_i$$ I know that we ...
8
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0answers
108 views

Can anyone explain what is the intuition behind the following definition of $p \Vdash^* \phi $?

Can anyone explain what is the intuition behind the following definition? I know that the sign $p \Vdash \phi(x_1,...,x_n)$ somehow suppose to tell me that for any generic filter which contains ...
1
vote
1answer
41 views

If $S$ is an Infinite Set, and $f(S)\subseteq{G}$ is a Set that Generates a Group $G$, then the Cardinality of $G$ is $\le$ Cardinality of $S$?

As the header says, if we map an infinite set $S$ into $G$ by $f$, such that $f(S)$ generates $G$, then is $|G|\le|S|$? I know that this should be true in the case where $S$ does not include into $G$. ...
6
votes
1answer
95 views

how to collapse $\omega_2$ to a smaller cardinal

Let $M$ be a model of ZFC and take the forcing notion $P(\omega,\omega_2)$ where: $P(\omega,\omega_2)=\{p|p \space is \space a \space function \space and \space \exists n \space s.t. (dom(p)=n) \space ...
3
votes
1answer
50 views

Function $f\in M[G]$, $f:\kappa\to M$ is in the ground model implies $\kappa^+$-Baire

Let $M$ be countable transitive model of ZFC, $P\in M$ be poset and $\kappa$ be a cardinal in $M$. In addition, for every $P$-generic filter $G$ over $M$, if a function from $\kappa$ to $M$ is in ...