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Question 1

0votes

0answers

24views

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 ...

VizDracViz

162

asked3 hours ago

Question 2

1vote

1answer

87views

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 ...

Link L

947

asked2 days ago

Question 3

7votes

1answer

227views

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 &...

Elia Immanuel Auer

4,960

askedNov 26 at 8:47

Question 4

0votes

0answers

86views

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?

Yiming Zhang

11

askedNov 25 at 15:05

Question 5

1vote

1answer

128views

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 ...

Ismail Khan

21

askedNov 25 at 6:35

Question 6

9votes

1answer

281views

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 $\dots$ ...

L. R.

205

askedNov 25 at 1:09

Question 7

4votes

2answers

101views

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 ...

Sigma Femb.

546

askedNov 23 at 1:27

Question 8

-5votes

0answers

68views

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 ofa number living in the infinite set $\aleph_1$, Is it possible to constructan actual "number" ...

Gene Partlow

1

askedNov 22 at 23:41

Question 9

2votes

1answer

155views

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 ...

user1349439

45

askedNov 21 at 13:17

Question 10

5votes

2answers

159views

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 ...

Matteo Bisi

115

askedNov 20 at 11:20

Question 11

2votes

2answers

262views

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 ...

NikS

2,303

askedNov 20 at 8:28

Question 12

3votes

1answer

202views

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 ...

apprenant

808

askedNov 19 at 13:27

Question 13

4votes

0answers

91views

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 = \...

Markus Klyver

2,582

askedNov 17 at 19:28

Question 14

4votes

0answers

159views

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 ...

Leonid

1,737

askedNov 17 at 13:31

Question 15

6votes

3answers

723views

Does truth not exist in ZFC, or is it merely not definable?

I am having a lot of trouble with the concept of Tarski's undefinability theorem as it relates to set theory.Tarski's undefinability theorem says that there is no formula $Tr$ on the natural numbers ...

William Oliver

1,262

askedNov 16 at 15:41

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Questions tagged [set-theory]

Does definition of empty set use universal specification

How to properly understand $\text{Con}(\Gamma)$ expressing consistency

Can countably many giraffes guess the color of their own scarf correctly if each may be mistaken about the color of finitely many scarves?

Is the "cardinality of the dimension of a vector space with basis" well-defined? [duplicate]

Shouldn't all possible sets be covered by ZFC axioms? [closed]

Measurable cardinal strongly inaccessible without AC

PSP and $\mathbb{R}\times\omega_1$ vs $\mathbb{R} \sqcup \omega_1$

A naive question on actual numbers which live in an infinite aleph set [closed]

Set-theoretical conventions in Pedersen's Analysis Now

Construction of a dense linear order with $2^\kappa$ cuts

Does the Axiom Schema of Separation in any sense "provide" only countably many definitions of subsets of $\mathbb{N}$?

Comparison of Dedekind-finite sets

Is "factor" the appropriate term when dealing with intersections and unions?

Defining an integral over surreal numbers

Does truth not exist in ZFC, or is it merely not definable?

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