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.
11,974 questions
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Definition of hyperprojective sets
I've seen two definitions of hyperprojective sets:
sets that are both inductive and co-inductive (cf. p.315 of Moschovakis's book);
sets that belong to the smallest $\sigma$-algebra that contains ...
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Does definition of empty set use universal specification
In chapter 3 of Analysis I by Terence Tao, the following definition of empty set is given:
(Empty set). There exists a set $\phi$, known as the empty set, which contains no elements, i.e., for every ...
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How to properly understand $\text{Con}(\Gamma)$ expressing consistency
I've read in Kunen's Set Theory that given two theories (i.e. Axioms) $\Gamma$ and $\Lambda$, we have $\Lambda \lhd \Gamma$ if and only if $\Gamma \vdash \text{Con}(\Lambda)$, so that $\Gamma$ is ...
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Can countably many giraffes guess the color of their own scarf correctly if each may be mistaken about the color of finitely many scarves?
This question is based on the question Is it possible to formulate the axiom of choice as the existence of a survival strategy? (MathOverflow).
Consider the following "computable giraffes, lion &...
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Is the "cardinality of the dimension of a vector space with basis" well-defined? [duplicate]
Or equivalently say, suppose $V$ is a vector space, any two bases of $V$ have the same cardinality?
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Shouldn't all possible sets be covered by ZFC axioms? [closed]
Are there models of ZFC where Continuum Hypothesis (CH) fails? The answer should be yes as if CH were true in all models then it should be provable from ZFC axiom but we know that it is independent of ...
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Measurable cardinal strongly inaccessible without AC
Let $\kappa$ be a measurable cardinal. I want to show that it is still inaccessible in ZF. Using ultrapower and Los I can show that it's inaccessible in ZFC, but that doesn't seem to work here $\dots$ ...
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PSP and $\mathbb{R}\times\omega_1$ vs $\mathbb{R} \sqcup \omega_1$
Working in ZF+PSP, every uncountable subset of $\mathbb{R}$ contains an homeomorphic copy of the Cantor space. I want to show that $|\mathbb{R} \sqcup \omega_1|<|\mathbb{R}\times\omega_1|$. Since ...
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A naive question on actual numbers which live in an infinite aleph set [closed]
An amateur query:
Just as the transcendental number $\pi$ is an actual concrete example of
a number living in the infinite set $\aleph_1$, Is it possible to construct
an actual "number" ...
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Set-theoretical conventions in Pedersen's Analysis Now
In chapter 1 of Gert Pedersen's Analysis Now (specifically the exercises), when dealing with "collections" of proper (equivalence) classes, one avoids standard set-theoretical difficulties ...
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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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Does the Axiom Schema of Separation in any sense "provide" only countably many definitions of subsets of $\mathbb{N}$?
Section 6 (pg. 24-25) of this paper contains the following excerpt (bold for emphasis added by me):
The axiom of Separation could also be called the axiom of Definable Subsets. A subset $T$ of $S$ is ...
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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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Is "factor" the appropriate term when dealing with intersections and unions?
In
$$A = \displaystyle \prod_{i \in \mathcal I} A_i,$$
we call $A$ the product of the factors $A_i$.
What do we call the "factors" in
$$B = \bigcap_{i \in \mathcal I} A_i$$
and
$$C = \...
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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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