Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
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Proof related to natural numbers
We had allowed two operations of division and subtraction on the set of natural numbers they could lead to fractional numbers or negative numbers, both not defined in the set of natural numbers. ...
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$\omega^\omega$ correspondence with $\mathbb R$
How does the natural continuous bijection between $\omega^\omega$ and $\mathbb R$ look like? I.e. why elements of $\omega^\omega$ are called reals?
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is the divisibility of two ditinct primes an independent event?
In a post of April I rasied a question of "The meaning of the Euler Formula for Zeta?"
anon brought an absolutely beautiful explanation, with the first part:
"Heuristically, if $p$ and $q$ are ...
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How to prove that $N\setminus A$ is finite? [closed]
$A \subseteq \mathcal{R}(N)$ and given that (by inductive definition):
$N ∈ S$.
If $a \in R$, then $R \setminus \{a\} \in A$.
I need to prove that for each $A \in S$, $N\setminus A$ is finite. How ...
user567507
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Does the average primeness of natural numbers tend to zero?
Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
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How does Peano Postulates construct Natural numbers only?
I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook.
Axiom 1.2.1 (Peano Postulates). There exists a set $\Bbb N$ with an ...
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Every natural number is covered by consecutive numbers that sum to a prime power.
Conjecture. For every natural number $n \in \Bbb{N}$, there exists a finite set of consecutive numbers $C\subset \Bbb{N}$ containing $n$ such that $\sum\limits_{c\in C} c$ is a prime power.
A list of ...
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List of powers of Natural Numbers
Greatings,
Some time ago a friend of mine showed me this astonishing algorithm and recently i tried to find some information about it on the internet but couldn't find anything... Please help.
...
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Consistency of Peano axioms (Hilbert's second problem)?
(Putting aside for the moment that Wikipedia might not be the best source of knowledge.)
I just came across this Wikipedia paragraph on the Peano-Axioms:
The vast majority of contemporary ...
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Commutativity of multiplication in $\mathbb{N}$
I'm trying to prove that $a\cdot b=b\cdot a$ when $a$ and $b$ are two natural numbers.
In the rest of this question I'm using $a'$ for the successor of $a$.
Addition is defined as:
$a+0=a$
$a+b&...
user112679
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Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?
We have the followings:
$\sin(\frac{\pi}{1})=\frac{\sqrt{0}}{\sqrt{1}}$
$\sin(\frac{\pi}{2})=\frac{\sqrt{1}}{\sqrt{1}}$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{\sqrt{4}}$
$\sin(\frac{\pi}{4})=\frac{\...
user108850
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How to construct natural numbers by set theory?
Definition 1: For any set $a$ , its successor $a^+=a\cup \{a\}$.
Informally , we want to construct natural numbers such that :
$0=\emptyset,1=\emptyset^+,2=\emptyset^{++},3=\emptyset^{+++}$,... ...
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Is it necessary to use the axiom of Regularity to prove the successor function being injective?
Basically the problem is that given an inductive set $X$ we can define the successor function on $X$ such that $S:X\longrightarrow X$ and for all $x\in X$, $S(x)=x\cup \{x\}$. So, one of Peano axioms ...
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Is there a specialized formula for Lagrangian interpolation on equispaced points?
If we know $f(0),f(1),f(2),\cdots f(n)$, is there a specialized version of the Lagrangian interpolation formula and a shortcut to compute the coefficients ?
(Stability is not a concern.)
user65203
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Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?
One of the nicer consequences of compactness is that there is no statement in first-order logic whose content "$\leq$ is a well-order". So we can show that there are countable structure $(M,\leq)$ ...
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