All Questions
Tagged with real-numbers set-theory
79
questions
1
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2
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138
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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"...
- 4,307
1
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3
answers
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 ...
- 65.8k
1
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0
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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
259
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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
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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
6
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0
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127
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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
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1
answer
146
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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 ...
- 249k
1
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1
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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
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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 ...
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-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 ...
- 19
-2
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1
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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
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0
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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
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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
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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}...
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