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The positive integers are $\mathbb Z^+=\{1,2,3,\dots\}$, and it's always like that. The natural numbers have different definitions depending on the book, sometimes the natural numbers is just the postivite integers $\mathbb N=\mathbb Z^+$, but other times the natural numbers are actually the non-negative numbers $\mathbb N=\{0,1,2,\dots\}$.
In a "realist" perspective the natural numbers are the counting numbers, something that we can perceive through our rational faculties. In that view the natural numbers have an existence independent of any axiomatization of their properties. A "formalist" perspective cannot promise much, if anything, about a definite system of natural numbers.
the 1 1. st number is 1 1. ; in making limits, 0 0. plays a role which is symmetric to ∞ ∞. , and the latter is not a natural number. Pros of considering 0 a natural number: the starting point for set theory is the emptyset, which can be used to represent 0 0. in the construction of natural numbers; the number n n.
The most well-known construction of the natural numbers, found in texts such as those by Enderton or Hrbacek and Jech, begins as follows: An inductive set is any set X such that ∅ ∈ X, and for all sets a, if a ∈ X then S(a) ∈ X. Here, S(a) = a ∪ {a} denotes the successor of a. Then the Axiom of Infinity states:
My friend showed me this youtube video in which the speakers present a line of reasoning as to why $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$ My reasoning, however, tells me that the previous
Closed 2 years ago. According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that 0 is in the set and for every real number x in the set, x + 1 is also in the set and a natural number is a real number that every inductive set contains. The problem with that definition is that it is ...
I would say that in the natural language the correspondence between cardinal numbers and ordinal numbers is off by one, thus distinguishing two sets of natural numbers, one starting from 0 and one starting from 1st. The 1st of January was day number $0$ of the new year. And zeroth has no meaning in the natural language...
ZFC does have a notion of the "set of natural numbers" and can be definitionally extended to include such a set, as seen above. However, A is potentially wrong in that A assumes that the set , defined in ZFC, has anything to do with the "actual natural numbers". If ZFC were inconsistent, then ZFC would prove statements like " " (interpreted ...
The set N N is the natural numbers. But usually in the context of number theory when one writes n n, one means a particular natural number. For example, n n could be 5 5 or 12 12 or 1 1 or 1, 234, 453, 564, 234, 134, 179, 200 1, 234, 453, 564, 234, 134, 179, 200. But they would usually not use it to denote any integer and certainly not a real ...
Within set theory, having the natural numbers $\mathbb{N}$ built as the minimal inductive set with the corresponding additive and multiplicative operations defined, integers $\mathbb{Z}$ can be set as equivalence classes of parallel diagonals of $\mathbb{N}\times\mathbb{N}$, which contain a copy of the natural numbers.