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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. $\endgroup$
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".
$\begingroup$ +1 I'm glad you posted this. My answer was prompted by your early comment re: isomorphism! I agree we can prove something in first-order ZFC about isomorphism, e.g. that a smallest infinite ordinal exists, but that this doesn't "fully capture the natural numbers."
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 $1$;
The natural numbers are defined by the Peano axioms, as in the answer of Isaac. You can also view the natural numbers as the cardinalities of finite sets, which implies that zero is a natural number. Now the other number domains arise because mathematicians want to give values for certain operations which otherwise are only defined partially.
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 $\mathbb N=\{0,1,2,\dots\}$.
(using the inheritance property what you have said was true). In R, N should not be open since no neighborhood of maximal distance r around any natural number should have only natural numbers in it (i.e. take r = 0.5) $\endgroup$ –
Sometimes a natural number. But it's up to the author to define the meaning, which if you look carefully was probably done in the contexts you are asking about (without references). $\endgroup$ – hardmath
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
$\begingroup$ It's not countable, as provable by diagonal argument, but the set of all FINITE subsets, and even ordered sequences, of natural numbers, or even integers or rational numbers, is, which I first realized by using extended definitions of prime factorization as ordered sequences of exponents to first however-many primes, though there ...