All Questions
Tagged with real-numbers set-theory
79
questions
2
votes
1
answer
192
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 ...
- 208
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 ...
- 43.8k
5
votes
2
answers
229
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 ...
- 397k
0
votes
1
answer
46
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 ...
- 79k
0
votes
1
answer
115
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 ...
- 44.6k
1
vote
1
answer
163
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 $4.5$, $\pi$, $e$...
- 344
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 ...
- 249k
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 ...
- 249k
1
vote
1
answer
53
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 ...
- 18k
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 ...
- 24.3k
4
votes
1
answer
369
views
Maximal model for ℝ?
I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...
- 2,796
0
votes
0
answers
93
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\...
- 47.4k
0
votes
1
answer
36
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 ...
- 24.3k
1
vote
2
answers
138
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"...
- 4,307
1
vote
3
answers
210
views
Do we draw a distinction between a number as an element of the reals, and an element of the naturals?
I see in some explanations of attempts to formalize numbers such as Von Neumann's ordinals like in this rather philosophical question that we can draw a distinction between a real number '1' and a ...
- 65.8k
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
3
votes
1
answer
259
views
Does $\pi$ have countably or uncountably many decimal digits?
I think I know the answer - countably many, and intuitively it does make sense i.e. it wouldn't make sense that a number has uncountably many decimal digits (is that even possible).
However, I've been ...
- 449
3
votes
1
answer
137
views
Is there a link between uncountable sets and infinite information?
There are only countably many things you can express with a finite number of words. This implies that any uncountable set has to contain uncountably many elements which you cannot define by any finite ...
- 149
6
votes
0
answers
127
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
3
votes
1
answer
146
views
What are the real numbers?
I know "What are real numbers" has probably been asked before, and the answer would be "the unique complete ordered field" BUT, isn't there some subtlety going on here? In the ...
- 249k
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
5
votes
1
answer
258
views
Contradiction of axioms of real numbers
I am just starting out in real analysis, so please bare with me. My questions concerns three specific properties of the real numbers, at least as far as i understand them. Those are:
The natural ...
- 71
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
-5
votes
1
answer
148
views
The set of irrationals numbers is countable?
I tried to prove this using statement using the difference of sets
$\mathbb{R}-\mathbb{Q}$ and the fact that $\mathbb{R}$ is not countable and $\mathbb{Q}$ is countable.
In general, is it possible to ...
- 19
-2
votes
1
answer
72
views
Is it possible (in principle and in meaningful way) to describe any subset of n-dimensional real Euclidean space?
Let us start with some background and motivation. My main question is very simple and it is available few paragraphs further and it is written in bold.
My problem is based from the emerging theory of ...
- 1,323
0
votes
0
answers
37
views
Bijection from $\mathcal{P} (\mathbb{R})$ to the set of functions from $\mathbb{R}$ to $\mathbb{R}$ [duplicate]
I’m a bit confused as to how we get the bijection between a powerset of a set to the set of functions from that self to itself
I can see the obvious bijection from the powerset to the set ${[0,1]}^{R}$...
- 427
1
vote
1
answer
70
views
Can we uniquely define for arbitrary, real-valued, finite sequence $X$, infinitely many pairs (real-valued $f(X)$, rank order of elements of $f(X)$)?
For an arbitrary sequence $X$ of $n$ distinct real numbers, can we uniquely and exhaustively define a set of infinitely many pairs of the form: $[f_{j},$ order$(f_{j}(x))]$, where $f_{j}$ is a real-...
- 39.5k
0
votes
0
answers
24
views
How to construct an increasing $\aleph_1$ sequence of real numbers. [duplicate]
We have $\aleph_1\leq |\mathbb{R}|$. Do we know if there exists an increasing $\aleph_1$ sequence of real numbers? (That is, a set $\{a_\theta\in\mathbb{R}:\theta<\omega_1\}$ such that $a_{\theta_1}...
- 782
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
0
votes
0
answers
31
views
Set representation of (real) numbers [duplicate]
Using the Von Neumann representation we can represent the non-negative whole numbers using the empty set, e.g. $1$ as {$\emptyset$}.
How do we represent with this notation numbers like $\sqrt2, -1, \...
- 43
3
votes
1
answer
116
views
In ZF, is it possible that there is no cardinal such that Reals injects into?
Working in ZF, is it possible that there is no cardinal number such that $\mathbb{R}$ can inject into? For if there exists a cardinal number $\kappa$ such that $\mathbb{R}$ injects into $\kappa$, then ...
- 249k
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
9
votes
4
answers
2k
views
Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.
Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
- 11.5k
3
votes
1
answer
216
views
Can you prove that $\Bbb R$ is uncountable using the Lebesgue measure?
I have been studying measure theory from the ground up, and am quite excited by the seeming power it holds. I thought of this last evening, and I wish to ask if the following proof of uncountability ...
- 10.4k
11
votes
1
answer
2k
views
The well ordering principle
Here is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered.
In the book, it says that the chief ...
- 3,047
4
votes
3
answers
450
views
Possible interpretation of real numbers as functions? [duplicate]
At the end of the day, a real number can be viewed simply as a function over the integers —> the naturals which tells you the digit as that ten’s place (assuming base ten)? You could augment this ...
- 4,655
1
vote
1
answer
144
views
Defining real numbers to exclude incomputable numbers
The real numbers are normally constructed via Dedekind cuts or similar approaches, which result in incomputable numbers: numbers that no finite algorithm can produce to arbitrary precision. This is ...
- 249k
1
vote
2
answers
167
views
Continuum family of continuum subsets of $\mathbb R$ which are not pairwise order isomorphic
I need to construct a continuum family of pairwise order inequivalent subsets of $\mathbb R$, such that cardinality of intersection of every constructed subset and every nontrivial interval is also ...
- 620k
11
votes
1
answer
1k
views
More than the real numbers: hyperreals, superreals, surreals ...?
I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ''...
- 5,125
0
votes
1
answer
63
views
Exist strictly increasing $f \colon \omega _1 \mapsto \mathbb{R}$ [duplicate]
Does there exist a strictly increasing injective function $f \colon \omega _1 \mapsto \mathbb{R}$, where $\omega _1$ denotes the first uncountable ordinal?
- 397k
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
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
2
votes
1
answer
105
views
Where is the copy of $\mathbb{N}$ in the constructible hierarchy relative to a real closed field?
Let $X$ be a real closed field. Let us define a constructible hierarchy relative to $X$ is defined as follows. (This is slightly nonstandard terminology.). Let $L_0(X)=X$. For any ordinal $\beta$, ...
- 249k
-2
votes
2
answers
34
views
Countability of the set [closed]
Let $f$ be differentiable function from $\mathbb{R}$ to $\mathbb{R}$. Consider the set
$$A_y=\{x \in \mathbb{R} : f(x)=y \}$$
I want to know whether $A_y$ is countable for each $y\in \mathbb{R}$. I ...
CommunityBot
- 1
3
votes
2
answers
604
views
Why does axiom of choice not imply the set of real numbers is countable?
The axiom of choice implies all sets can be well ordered. If that is true, you can well order the set of real numbers and the set of the integers. Now, why can one not just pair the set of real ...
- 133k
0
votes
1
answer
610
views
How is the Continuum Hypothesis equivalent to the existence of a well-ordering on $\Bbb R$ whose bounded initial segments are countable?
There exists an well-ordering $(<)$ on $\Bbb R$ such that the set $\{x \in \Bbb R\mid x < y \}$ is countable for every $y \in \Bbb R.$
How to prove that the above statement is equivalent to ...
- 397k
3
votes
1
answer
188
views
A closed countable set with Cantor-Bendixson rank of $\omega +1$
I'm looking to find a closed countable set that has a Cantor-Bendixson Rank of $\omega +1$.
I know that $\{0\}\cup\{\frac{1}{x+1}|x\in\omega\}$ has a Cantor-Bendixson Rank of $2$ because we take out
...
- 79.6k
0
votes
1
answer
150
views
Cuts and continuous sets
I'm reading Kuratowski's "Set theory", and here is a question from the Chapter 6.
Consider a linearly ordered set $A$ and its cuts, i.e., such pairs $\langle X,Y\rangle$ of $A$'s subsets that $X=Y^-$ ...
- 1,773