All Questions
Tagged with real-numbers set-theory
79
questions
1
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2
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96
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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
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 ...
- 329
2
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1
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189
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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
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1
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45
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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
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114
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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
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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
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0
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71
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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
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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
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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
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1
answer
52
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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
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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
92
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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
36
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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
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2
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137
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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
1
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0
answers
90
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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
257
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 ...
- 41
3
votes
1
answer
137
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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
1
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3
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210
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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 ...
- 745
6
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0
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126
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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
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143
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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 ...
- 499
1
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1
answer
144
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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
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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
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0
answers
90
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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
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1
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148
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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 ...
- 605
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votes
1
answer
72
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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
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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
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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-...
- 171
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0
answers
24
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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
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0
answers
101
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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