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In his book set theory and the continuum hypothesis, Cohen writes:

Let us state Peano's axioms in the usual form:

  1. each integer has a unique successor

  2. there is an integer $0$ - which is not the successor of any integer

  3. two distinct integers cannot have the same successor

  4. if $M$ is a set of integers such that $0$ is in $M$, and such that if an integer $X$ is in $M$ then its successor is in $M$, then every integer is in $M$.

Checking wikipedia, an integer is a whole number $\in\mathbb Z:=\{\dots, -2, -1, 0, 1, 2, \dots\}$, but Cohen seems to use the word integer to just denote the nonnegative integers (i.e. $x\in\mathbb Z$ with $x\geq 0$). Thus my question:

Do some authors mean by integer the same as natural number, although some authors refer to the whole set $\mathbb Z$ as integers?

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Yes, unfortunately in logic sometimes the word "integer" is used to refer to "natural number."

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  • $\begingroup$ Why? D: I do really want to know. Is that again something like equal up to units? $\endgroup$ – user251257 Dec 2 '16 at 18:57
  • $\begingroup$ @user251257 I'm honestly not sure. It's a terrible ambiguity, but there you have it. $\endgroup$ – Noah Schweber Dec 2 '16 at 19:00
  • $\begingroup$ Huh! I have never encountered this ambiguity, but I guess it's good to be aware of it. I guess one could really trip up on this. $\endgroup$ – Wojowu Dec 2 '16 at 19:24
  • $\begingroup$ @Wojowu Yeah, it's pretty rare, but I've seen it on occasion - usually in older texts. $\endgroup$ – Noah Schweber Dec 2 '16 at 21:03
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The set of Natural Numbers in and of itself is ambiguous.

Some define $\mathbb{N}$ as the set of positive integers (A.K.A. {1, 2, 3, 4, ...}).

Some define $\mathbb{N}$ as the set of non-negative integers (A.K.A. {0, 1, 2, 3, 4, ...}).

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  • $\begingroup$ That's not the the point of my question. It's a completely different issue. $\endgroup$ – user384011 Dec 2 '16 at 22:14
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Depending on the context it is often convenient to narrow or widen the denotation of common terms. You met one example of the former (narrowing). An example of the latter (widening) occurs in algebraic number theory, where "integer" means an algebraic integer (in the ambient number field), and elements of $\,\Bbb Z\,$ are referred to as rational integers.

Another example is "irrational". In elementary contexts this is restricted to real numbers, but in more advanced contexts it is extended to other fields, e.g. see the thread Is $\,i\,$ irrational?

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