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.
11,971 questions
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How applicable are the isomorphism theorems? [closed]
How broadly do the (four?) isomorphism theorems apply? Do they hold only of groups? What do they look like in set theory?
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Construction of a dense linear order with $2^\kappa$ cuts
The question is the following:
Given a cardinal $\kappa$, is there a dense linear order of size $\leq \kappa$ such that there are $2^\kappa$ cuts (a cut is a downward closed subset)?
The question is a ...
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Comparison of Dedekind-finite sets
It is well-known that the trichotomy property of cardinals is equivalent to the axiom of choice, but I wonder whether the trichotomy property of finite cardinals can be proved without invoking the ...
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Defining an integral over surreal numbers
Im not an expert on the surreals but I have noticed that even Conway when writing the 2nd edition of his book mentioned that a natural definition of an integral over surreals is still elusive. So I ...
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Defining ordinal multiplication for infinite families of ordinals without transfinite recursion
In Naive Set Theory p.82-83, Halmos defines ordinal addition by defining the ordinal sum of an infinite family ${A_i}$ of well-ordered sets, indexed by some well ordered set $I$. He then proceeds to ...
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DC in $L(\mathbb{R})$
I assume ${DC}_\mathbb{R}$ and want to show that $\textbf{L}(\mathbb{R})\models \text{DC}$. Let $R\subseteq A^2 \in \textbf{L}(\mathbb{R})$ be a relation such that $R$ has total domain $\text{dom}(R)=...
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What real numbers correspond to second order logic definable Dekekind cuts?
This question spurred from a thought I had: does every (lower) Dedekind cut have a (finite) second order logic formula that defines it?
Fix the usual setting: the domain is $\mathbb{Q}$ with the order ...
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Error or imprecise statement in Mac Lane's "Categories for the Working Mathematician"?
$\DeclareMathOperator{\Nat}{Nat}\DeclareMathOperator{\Map}{Map}\DeclareMathOperator{\Ob}{Ob}\DeclareMathOperator{\Set}{Set}\DeclareMathOperator{\id}{id}$Let the definitions and assumptions in set-...
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Cartesian product interpretation of transfinite numbers up to $\epsilon_0$ as sets of tuples; difficult to compose $-$, $/$
Interpreting expressions of $k\in\mathbb{N}, \omega, $ and $\text{+ - */^}$ as types of tuples
Section 1 - Preamble
The ordinal-like expressions tend to cause confusion, so in this question I define ...
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Does "$A^2$ injects into $A\sqcup\aleph(A)$ for any $A$" imply choice?
This question asks whether "$|A^2|=|A^3|$ for any infinite $A$" imply choice. It is not too difficult to show that this implies that $A^2$ injects into $A\times\aleph(A)$ for any infinite $A$...
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When is the iterated ultrapower innocuous?
Suppose $U$ is an ultrafilter over $I$. When does the isomorphism
$$
\prod_U \Big(\prod_U \mathfrak A_i\Big) \cong \prod_U \mathfrak A_i
$$
occur?
I think I saw somewhere that if $U$ is a uniform $\...
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How small can a set of the second category be?
It is known that a subset $X$ of $\mathbb{R}$ of the second category is always uncountable, but I wonder if $X$ must have the same cardinality as $\mathbb{R}$? (Of course we don't assume that CH ...
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Can Russell's Definition of Numbers be Generalized to Include Infinite Series
According to Bertrand Russell in his Introduction to Mathematical Philosophy, a number is defined as the following: "The number of a class (set) is the class (set) of all those classes (sets) ...
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Relative to what set theory do we do model theory? [duplicate]
In regards to category theory, it's clear to me what the start point is, we collect up all possible spaces have some sort of structure which can be created w.r.t to some fundamentals of mathematics, ...
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Constructing the set of all finite tuples using $F$, showing that it has size $\epsilon_0$, and that it must contain all finite tuples?
Ok, this construction is still a bit wrong and still a bit vague, but I have a better one. I can't delete the question as too many people have commented, but I've learned enough more that I think a ...
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Construction of a normal Suslin Tree according to Jech
In Lemma 9.13 Jech proves the existence of a normal Suslin tree if there exists a Suslin tree.
In the proof he constructs $T_3$ as the set of branching points of $T_2$. I have problems to see why $T_3$...
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Semantically "undefined" terms of non-standard length in FOL with $\omega$-nonstandard metatheory
As preliminary background, the following passage from Foundations of Set Theory by Fraenkel et. al.:
The object-language of a given theory is sometimes an artificial symbolic language...When an ...
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Does $\mathsf{ZF}$ prove that an injection $\mathcal{P}(A+1)\to\mathcal{P}(A)$ implies $A$ being dually Dedekind infinite?
We know that $\mathsf{ZF}$ proves that if we have a surjection $f:X\to Y$, then there is an injection $\mathcal{P}(Y)\to\mathcal{P}(X)$, where $\mathcal{P}(X)$ is the power set of $X$, given by $S\...
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On Dedekind-finite initial segments in nonstandard models of PA under ZF + Axiom of Choice(FALSE).
This is more a question of logic than of mathematics, but it concerns the ZF axiom system, the foundation of modern mathematics.
Let us work within ZF and assume ¬AC (the Axiom of Choice is false). As ...
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Proof that product on surreal numbers preserves the order [duplicate]
I'm trying to understand the proof of the following theorem on Conway's book On numbers and games (page 19 on the second edition):
The proof is done by induction, as usually, but when proving (iii) ...
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Can it be proved that $\mathbb{N}\subseteq e$ if we can prove $0\in e$, $1\in e$, etc. separately?
Suppose for every natural number $n$, $Z_n(x)$ is a first order predicate with a single free variable $x$ that states, informally, that $x = n$. For instance $Z_0(x)$, stating that $x = 0$, i.e. that $...
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What is the size of the class of isomorphism types of groups?
Let Grp be the class of isomorphism types of groups (this can be made sense of using Scott's trick). The class of all ordinals Ord injects into it because every nonempty set has a group structure. ...
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elementary embedding into both $V_\alpha$ and $V_\beta$
It was mentioned in this post that if $V_\alpha\prec V_\beta$ for $\alpha<\beta$ then both are models of ZFC. Now, suppose that there is a countable transitive $M$ and two elementary embeddings $\...
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With ZFC foundations, if all math objects are sets, where do these sets "live"?
Let's say we're doing ordinary mathematics, and we want ZFC to be our foundations, such that all of our mathematical objects are sets.
I have long had the idea in my head that these sets that ...
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Forcing posets that completely embed into $\sigma$-closed posets
For forcing posets $\mathbb{P}$ and $\mathbb{Q}$, recall that $i:\mathbb{P}\to\mathbb{Q}$ is called a complete embedding if $i$ is order-preserving, incompatibility preserving, and for all $q\in\...
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