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logic - What are natural numbers? - Mathematics Stack Exchange
What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational numb...
Is $0$ a natural number? - Mathematics Stack Exchange
From the Wikipedia article: In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers $\ {1, 2, 3, \dots\}$ according to the traditional definition; or the set of non-negative integers $\ {0, 1, 2,\dots\}$ according to a definition first appearing in the nineteenth century.
Is the sum of all natural numbers $-\frac {1} {12}$? [duplicate]
Is the sum of all natural numbers $-\frac {1} {12}$? [duplicate] Ask Question Asked 12 years, 3 months ago Modified 11 years, 6 months ago
What is a natural number? - Mathematics Stack Exchange
That being said, this "inductive set" definition of the natural numbers comes from a book on analysis, not axiomatic set theory. The goal of the book is to do real analysis (i.e. to study limits, differentiation, integration, etc. in the context of real functions of real nubmers). If you had to start from $\mathsf {ZFC}$ and build up your number systems from there, you could never get to the ...
discrete mathematics - What is the difference between natural numbers ...
Alternatively, if you were asked to define the natural numbers, you could do so axiomatically. With this method, you begin by declaring an element is a natural number, and it really doesn't matter what this element is called, be it $0$ or $1$ or $\Delta$ - this is the idea I'm trying to convey.
About Math notation: the set of the first $n$ natural numbers
You can define $ [n]$ however you want, so it can be true. Despite that, it is a common notation for the set $\ {k\in \mathbb {N} : k\leq n\}$, yes. This notation is used more often on Elementary Set Theory and Discrete Mathematics. Unfortunately analysts don't use it much. I've never seen it being used in Abstract Algebra or Linear Algebra either.
Is it correct to say that the natural numbers are a proper subset of ...
3 Yes. Integers are the essentially the natural numbers and their opposites, plus zero. Since $\Bbb Z$ contains one or more element not found in $\Bbb N$ (namely $0$ and the negative numbers) and all elements of $\Bbb N$ are found in $\Bbb Z$, then $\Bbb N$ is a proper subset of $\Bbb Z$.
What is the motivation for sequences to be defined on natural numbers?
A "sequence" is just a fancy word for a function whose domain is the natural numbers. Mathematicians introduce such fancy words when they meet a concept that appears over and over again and they want to reduce the number of characters needed to describe it. As it happens, functions whose domain is the set of natural numbers appear EXTREMELY often, so this compression scheme turns out to be ...
real analysis - Proving unboundedness of the natural numbers via the ...
Here I think is the crux of the issue: once you've embedded the natural numbers into the real numbers, there could be a real number that's larger than all of the natural numbers.
Produce an explicit bijection between rationals and naturals
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, ...
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