All Questions
Tagged with natural-numbers elementary-set-theory
152
questions
1
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1
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55
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Finiteness, finite sets and representing its elements.
A set $S$ is called finite if there exists a bijection from $S$ to $\{1,...,n\}$ for one $n \in \mathbf{N}$. It is then common to write its elements as $s_1,...,s_n$. I now wonder, why this is ...
- 389
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3
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258
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Can the natural numbers contain an element that is not representable by a number?
I read the following document: https://www.math.wustl.edu/~freiwald/310peanof.pdf . In this document, the author wants to formalize that natural numbers, that are informally thought of as a collection ...
- 389
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votes
2
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100
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Directly proof $S$ is countable, where $S$ is set of function from $\{0, 1\}$ to $\mathbb{N}$
Suppose $S=\{f_1,f_2,f_3,f_4,f_5,........\}$ where $f_i$ is a function $f:\{0, 1\}\to\mathbb{N}.$ I have to prove $S$ is countable.Then need to prove direct one-to-one correspondence between $S$ and $\...
user976798
2
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1
answer
106
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Why Cantor diagonalization theorem is failed to prove $S$ is countable, Where $S$ is set of finite subset of $\mathbb{N}$?
I have an given set $S$ where $S=$ set of finite subsets of $\mathbb{N}.$ We need to prove $S$ is countably infinite.
My approach: I need to prove there is one-to-one correspondence between $S$ and ${\...
user976798
-1
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1
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173
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Why bijection between $\mathbb{N^2}$ and $\mathbb{N}$ is not possible directly?
To understand bijection between $\mathbb{N^2}$ and
$\mathbb{N}$ I found this pdf on internet. But have couple of confusion.
N.B: Here we take $0 \in \mathbb{N}.$
Confusion:1 Why directly proof of ...
- 1
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0
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209
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Are odd natural numbers an inductive set?
The definition of inductive set my textbook gave is:
A set $T$ that is a subset of the integers is an inductive set
provided that for each integer $k$, if $k$ is an element in the set
$T$, then $k+1$ ...
- 53
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103
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Why is the principle of induction for natural numbers not "self-evident"? [duplicate]
The principle of induction can be stated, in first-order logic, as follows. Let $S\subseteq\mathbb N$, and suppose that
$0\in S$.
$\forall n:n\in S\to n+1\in S$.
Then, $S=\mathbb N$. Now, suppose ...
- 20.7k
-1
votes
1
answer
135
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$X$ is infinite thus we have an injection from $\mathbb{N}$ to X [duplicate]
Hey guys I'm trying to prove the following:
$X=\emptyset \lor$ There is a surjection $g: \mathbb{N} \rightarrow X \implies X$ is finite $\lor$ there is a Bijection from $\mathbb{N}$ to X
I did case ...
- 385
2
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0
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128
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Statement of Well-ordering principle
The statement of well ordering principle appears in different mode - on subsets of natural numbers, or well-ordering of every (non-empty) set. For the question below, I am considering it w.r.t. non-...
- 3,047
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515
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Adding a Fixed Value to Each Element in a Set (How to Denote)
To denote a set such as, for example, the set of every natural number that is 3 greater than a multiple of 5, would $5\mathbb{N}+3$ be generally understood as $\{8,13,18,23,28,33,\dots\}$? If not, how ...
- 11
3
votes
3
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187
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Infinite natural numbers?
Only using the successor function $\nu$ and the other axioms, how do we guarantee that the "next" generated number is different from all the "previous" numbers (I am using ...
- 325
3
votes
1
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364
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Showing that the natural numbers are totally ordered with respect to set membership
Working with the usual set theoretic construction of the natural numbers, denoted $\omega$ for now.
I am trying to show that $\omega$ is totally ordered with respect to set membership, that is, $n<...
- 3,612
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85
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In ZFC, do we use the set $\mathbb{N}$ in the definition of $\mathbb{N}$ recursively?
In ZFC set theory, we define the set of the natural numbers as follows: By the axiom of infinity, an inductive set exists. Let I be an inductive set. Then, $\mathbb{N}$ is defined as $\{ x\in I |\...
- 375
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118
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A Doubt about Well Ordering Principle and Principle of Mathematical Induction
I have had this lingering doubt in my mind for a very long time: One of the standard constructions of N starts by assuming the 5 Peano Axioms, proving that every non-zero is a successor and s(n) is ...
- 21
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33
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Defining $A \in \mathcal{P}(\Bbb N \times\Bbb N)$ such that it is not any member of a countable subset $M \subseteq \mathcal{P}(\Bbb N \times \Bbb N)$
$
\newcommand{\N}{\mathbb{N}}
\newcommand{\P}{\mathcal{P}(\N \times \N)}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\i}{^{(i)}}
\newcommand{\x}{^{[x]}}
\newcommand{\y}{^{[y]}}
\newcommand{\...
- 45.9k