At that point, you cannot actually distinguish between the natural numbers starting at $0$ and the natural numbers starting at $1$, except for the (completely arbitrary) name for the initial number. Basically, it is the different definition for addition and multiplication which distinguishes the two choices. $\endgroup$

Sep 21, 2017 · Suppose we did start with some notion of "natural number" which we used to construct a model of the real numbers. Then even in this setting, the quoted definition is still not circular, because it's defining a new notion of "natural number" that will henceforth be used instead of the previous notion of "natural number".

Just based on Ravichandran's assessment of what the natural numbers are, the map that sends each number spoken by Sisyphus to the number spoken by his companion at the same time is obviously an isomorphism between the "Greek natural numbers" and the …

Jan 8, 2015 · I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it. Pros of considering $0$ not to be a natural number: generally speaking $0$ is not natural at all. It is special in so many respects; people naturally start counting from $1$;

Apr 5, 2015 · 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 $\mathbb N=\{0,1,2,\dots\}$.

Dec 4, 2018 · 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...

Dec 8, 2012 · I don't understand why the set of natural numbers constitutes a commutative monoid with addition, but is not considered an Abelian group.

In fact it has the same cardinality as Reals as in some sense we are trying to count all the possible binary numbers in the above argument. If we look at Reals in the binary form the above argument follows to show uncountability and each binary representation of a real number would also represent the subset of a Natural number.

Apr 26, 2019 · What you’ve written just means ‘f is a function mapping natural numbers to integers’, but you haven’t specified what that function actually is. The function that you define after that though is indeed a bijection. This shows, by definition, that the cardinality of these two sets is the same. Your intuition about infinite sets is ...

@PeterWoolfitt Peter you said "The induction argument fails because it shows P(n) is finite for every natural number n, but it does not show that "P(∞)" is finite because ∞ is not a natural number." Right, ∞ is not a natural number, so there is no such thing as P(∞). So how can we use it to reject Induction argument? $\endgroup$