All Questions
Tagged with real-numbers set-theory
79
questions
1
vote
2
answers
96
views
why is $L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\}$ not the set of all Dedekind cuts?
Let the set $L$ be definded as
$$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$
where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$.
Clearly, every member of $L$ is neither an empty set ...
- 1,266
5
votes
2
answers
228
views
How can a subset of reals not exist?
Let's take a Vitali set in a model of ZFC, then map its elements to the corresponding reals in the Solovay model and consider them as a set. We get a Vitali set in the Solovay model while it shouldn't ...
- 329
2
votes
1
answer
188
views
Is it true but unassertable that there are undefinable real numbers?
I know of Joel David Hamkins's analysis of the so-called "math tea argument", namely that there are undefinable real numbers. Supposedly, he debunked this argument by constructing a ...
- 21.3k
0
votes
1
answer
44
views
Prove: every finite cover $\mathcal U$ of $M\subseteq\mathbb R$ by open intervals contains two sets of disjoint intervals whose union covers $M$
It's not hard to show that if three open intervals in $\mathbb R$ have a non-empty
intersection, then one of the intervals is contained in the union of the other two. The simplest way to show this is ...
- 3,800
0
votes
1
answer
114
views
Shouldn't ℵ₀ be the cardinality of the reals?
If in ZFC any set can be well ordered, and that $\aleph_0$ is the cardinality of every infinite set that can be well ordered, shouldn't $\aleph_0$ be the cardinality of the real numbers?
I know this ...
1
vote
1
answer
159
views
Are the reals a "subset" of the class of ordinals
I am not sure if it's even correct to use subset in this context but I'm sure it gets the point across. I just want to know if the class of ordinals includes non-integer elements like $4.5$, $\pi$, $e$...
- 35
1
vote
0
answers
71
views
Lebesgue measurability
For what $n$ it is (in)consistent that all $\Sigma^1_n/\Pi^1_n$ sets are Lebesgue measurable ?
I remember that there is a result that if all $\Sigma^1_3$ are Lebesgue measurable then
$\omega_1$ is ...
- 3,978
12
votes
2
answers
1k
views
Examples of first-order claims about the reals that are not preserved under forcing
I am looking for an example of a first-order sentence in the signature of the real numbers, $(+,\times, <, 0,1)$, that is true when translated in the language of set theory in the natural way, but ...
- 1,297
4
votes
1
answer
218
views
Continuum many reals with pairwise irrational difference
In "Problems and Theorems in Classical Set Theory" by Péter Komjáth and Vilmos Totik, in the Solutions to Chapter 30, they claim: "It is easy to give continuum many reals with pairwise ...
- 1,647
1
vote
1
answer
52
views
Is the intersection of a uncountable real numbers subset with the complemetary of a countable subset uncountable? [duplicate]
Let be $E,F\subset \Bbb{R}$ two subsets such that $E$ is uncountable and $F^c$ is countable. Is $E\cap F$ uncountable?
I guess it is true, but I am not sure since I don't see a way in order to prove ...
- 733
0
votes
0
answers
86
views
Can $\mathbb{R}$ be written as an uncountable union of disjoint uncountable subsets?
Can $\mathbb{R}$ be written as an uncountable union of disjoint uncountable subsets?
I was thinking of the following: Consider an uncountable proper subfield $F$ of $\mathbb{R}$, then consider $\...
- 67
2
votes
2
answers
102
views
Can a replacement set be uncountable?
I apologize if I mess up my terminology here.
I was reading about solution sets recently and saw that for the formula "x + 1 = 1 + x" the solution set is equivalent to the real numbers.
That ...
- 33
0
votes
0
answers
91
views
Is axiom of choice required to throw away repeated intervals in a constructive argument?
I am looking at this answer to this question:
Let $U \subseteq \mathbb{R}$ be open and let $x \in U$. Either $x$ is rational or irrational. If $x$ is rational, define
\begin{align}I_x = \bigcup\...
- 743
0
votes
1
answer
35
views
Equivalence of Dedekind cuts and Dedekind left sets
I am currently working on the book "Classic Set Theory" by Goldrei. Goldrei is using Dedekind left cuts or left sets, i.e. the subset $L$ of a Dedekind cut. He gives the following definition ...
- 645
1
vote
2
answers
136
views
Example of a complete unbounded dense linearly ordered set that isn't isomorphic to $\mathbb{R}$
I know as a fact that $\mathbb{R}$ is the unique (upto isomorphism) complete linearly ordered field. But if we remove the "field" condition and replace it with "dense unbounded set"...
- 782