All Questions
Tagged with real-numbers set-theory
14
questions with no upvoted or accepted answers
6
votes
0
answers
126
views
A 'measure' on $\mathcal{P}(\mathbb{R})$
Question: Is there function $\mu : \mathcal{P}(\mathbb{R}) \to [0, \infty]$ with the following properties:
$\mu$ is countably additive. (on disjoint sets)
$\mu((a, b])) = b-a$, i.e., it extends the ...
- 3,623
5
votes
0
answers
138
views
Where is the first gap in the constructible hierarchy relative to a real closed field?
This is a follow-up to my question here. Let $X= (A,+,*)$ be a real closed field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*\}$. For any ...
- 10.3k
4
votes
0
answers
125
views
Existence of two subsets of $\mathbb R$ with a certain property, undecidable in $ZF$?
Inspired by Question # 2420627 "Prove that there is a bijection $f:\mathbb R\times \mathbb R\to \mathbb R$ in the form of $f(x,y)=a(x)+b(y)$" and the answer and comments by Thomas Andrews.
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- 58.4k
3
votes
0
answers
74
views
How high in the constructible hierarchy do you need to go to see Dedekind-incompleteness?
This is a follow-up to my questions here and here. Let $X= (A,+,*,<)$ be an ordered field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*,&...
- 10.3k
2
votes
0
answers
265
views
About Bernstein sets
Remember that a subset $X\subseteq \mathbb{R}$ is a $G_{\delta}$-set if $X$ is a countable intersection of open sets in $\mathbb{R}$. For example closed subsets of $\mathbb{R}$ are $G_{\delta}$-sets. ...
- 645
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
1
vote
0
answers
90
views
What is the framework in which we can talk about the procedure of Richard's paradox rigorously?
It seems there are two variants of Richard's paradox: one pertaining to natural language and one pertaining to first-order logic. I will focus on the latter.
Now as pointed out in this post, there are ...
- 2,461
1
vote
0
answers
90
views
Kolmogorov's construction of real numbers cardinality of functions that represent real numbers
Hi i am reading about lesser know construction of real numbers by Kolmogorov. In his construction real numbers are defined as a set $\Phi$ of functions $\alpha: \mathbb{N} \rightarrow \mathbb{N}$ that ...
- 11
1
vote
0
answers
101
views
Skolem's Paradox and undefinable reals
I'm trying to understand Skolem's paradox, and also some related ideas about definable numbers. I'm pretty new to learning about model theory. I'll lay out what I think I'm understanding, and maybe ...
- 293
1
vote
0
answers
140
views
Brief overview of the foundations of math?
I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don'...
- 23.8k
1
vote
1
answer
144
views
Sets with Unique Subset Summing to Every Real
Do there exists sets of reals such that every real has a unique subset that sums to it. Formally, do there exists sets $S\subset\mathbb{R}$ such that every $r\in\mathbb{R}$ has a unique (up to ...
- 2,346
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
0
votes
0
answers
92
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
0
answers
105
views
Can every perfect set that is not a closed interval, or $(-\infty,a]$, or $[b,\infty)$ be written as a union of these types of intervals?
I have been reading the book "Introduction to Set Theory" by Jech and Hrbacek and have come to the following exercise in the chapter on sets of real numbers :
Every perfect set is either an ...
- 595