Sep 21, 2017 · The problem with that definition is that it is circular because the real numbers are constructed from the natural numbers.

37 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, …

Set theoretic construction of the natural numbers Ask Question Asked 14 years, 2 months ago Modified 3 years, 3 months ago

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 …

Jun 20, 2019 · So there's a reason Enderton defines the set of natural numbers as the intersection of all inductive sets rather than as $"\ {0,1,2,\ldots\}"$ or as the set of all sets that can be …

In mathematics, there seem to be a lot of different types of numbers. What exactly are: Real numbers Integers Rational numbers Decimals Complex numbers Natural numbers Cardinals …

Dec 14, 2023 · The usage of the natural numbers and all the properties you know and love came first. The definition involving set theory and emptysets and such came after and was …

Apr 5, 2015 · The natural numbers have different definitions depending on the book; sometimes the natural numbers are just the positive integers $\mathbb N=\mathbb Z^+$, but other times …

Jul 14, 2014 · But this is when our original definition of a number fails to make sense- when we consider complex numbers, which brings me to my main question: what is a rigorous definition …

Nov 3, 2024 · Seeing the comments, it seems the set-theoretic definition of $\mathbb {N}$ is the way to go, but could you give a proof that this set satisfies the induction principle. You could …