Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, and the symbols $+$ and $\cdot$ as standing for ordinary addition and multiplication.'' This comment is found in section "Truth definition of the given language'' of article "Metalogic'' in Encyclopedia Britannica online at <http://www.britannica.com/topic/metalogic> 


Is there a source for such a characterisation or another characterisation of the Intended Interpretation in a more traditional publication in a refereed journal or book? 


Note 1. I am asking for a published source for the characterisation provided by the renouned logician Hao Wang, establishing a connection between a standard $\mathbb{N}$ on the one hand and what he seems to take to be the ordinary numbers with "ordinary addition and multiplication", on the other.


Note 2. The historical comments by user Francois Dorais concerning Frege, Peano, and Dedekind are interesting but I think inconclusive so far. It would be nice to get a clarification.


Note 3. Following user @logicute's comment (and also the sources (s)he cited) I will assume that the term *intended interpretation* (henceforth abbreviated II) entails an identification of a mathematical concept and an intuitive (i.e., pre-mathematical) concept. The former is the usual theory of the integers (N) as developed for example by von Neumann in a set-theoretic context. The latter are (the totality of) the familiar numbers that human beings are familiar with before they learn anything about set theory. This II entails an identification of N with the totality of familiar integers. Gabriel responded by giving a page in Rautenberg which stipulates "N is the set of natural numbers." However, the term *natural number* usually refers to an element of the mathematical object namely N, whereas I was referring to "counting numbers" as a synonym for "familiar numbers" as explained above. 


Note 4. Quinon and Zdanowski wrote that intended models could be defined as those that reflect our *intuitions* about natural numbers adequately. See *Quinon, P.; Zdanowski, K. "The Intended Model of Arithmetic. An Argument from Tennenbaum's Theorem." In S Barry Cooper, Thomas F. Kent, Benedikt L\"owe, Andrea Sorbi (Eds.) Computation and Logic in the Real World. Third Conference on Computability in Europe, CiE 2007. Siena, Italy, June 18--23, 2007, 313--317.* This comment on the role of intuition seems close to Wang's comment and is more explicit. Since Wang already made comments about intended interpretations in a paper from the 1950s it is not impossible that a comment like that by Quinon et al may have appeared in some logic textbook somewhere along the way. Hopefully this will turn up eventually.


Note 5. The quote from Dedekind provided by Mauro A. show that Dedekind may have been the first explicitly to propose in writing a connection between "ordinary counting numbers" and a formal system today denoted $\mathbb{N}$. Wang seems to have had Dedekind in mind when he was writing his contribution to the Encyclopedia Britannica. In the intervening decades *somebody* must have mentioned this explicitly in a logic textbook.


Note 6. Souces of this type in ultrafinitists would also be of interest.


Note 7. The best answer seems to be the comment from Kleene: "Since a formal system (usually) results in formalizing portions of existing informal or semiformal mathematics, its symbols, formulas, etc. will have meaning or interpretations in terms of that informal or semiformal mathematics. These meanings together we call the (intended or usual or standard) interpretation or interpretations of the formal system."


Note 8. The matter of the so-called *Intended Interpretation* is dealt with in detail in [this article](https://arxiv.org/abs/1703.00425). 



Looking for a source for Intended Interpretation
Here are quotes from three well-known sources. 


Shoenfield, *Mathematical Logic* (1967), page 23:



> 
> We construct a model of $N$ by taking the universe to be the set of natural numbers and assigning the obvious individuals, functions, and predicates to the nonlogical symbols of $N$. This model is called the *standard model* of $N$, ...
> 
> 
> 


(Emphasis in the original in all quotes.)


Kleene, *Mathematical Logic* (1967), page 200:



> 
> Since a formal system (usually) results in formalizing portions of existing informal or semiformal mathematics, its symbols, formulas, etc. will have meaning or interpretations in terms of that informal or semiformal mathematics. These meanings together we call the (*intended* or *usual* or *standard*) *interpretation* or *interpretations* of the formal system.
> 
> 
> 


Kleene 1967 p. 207:



> 
> A we remarked in § 37, a formal system formalizing a portion of informal mathematics has an "intended" (or "usual" or "standard") interpretation. ... The informal mathematics that we aim to formalize in $N$ is elementary number theory. So for the intended interpretation, the variables range over the natural numbers $\{0, 1, 2, \ldots\}$, i.e. this set is the domain. ... The function symbol $'$ is interpreted as expressing the successor function $+1$, and $0$ ("zero"), $+$ ("plus"), $\cdot$ ("times") and $=$ (equals) have the same meanings as those symbols convey in informal mathematics. 
> 
> 
> 


Here Kleene explicitly speaks of the interpretation as referring to the numbers that were known informally before the axioms of $N$ were laid out. 


Kaye, *Models of Peano Arithmetic* (1991), Chapter 1: "The standard model", p. 10:



> 
> The structure $\mathbb{N}$ (the *standard model*) is the $\mathcal{L}\_A$ structure whose domain is the non-negative integers, $\{0, 1, 2, \ldots\}$ and where the symbols in $\mathcal{L}\_A$ are given their obvious interpretation. 
> 
> 
> 


In contemporary practice, in formal arithmetic, it is normal practice to use the term "natural numbers" and the symbol $\mathbb{N}$ to refer to the *standard* natural numbers, i.e. to identify them with the informal counting numbers (e.g. this is Kaye's convention, and many others'). The need for a distinction between standard and nonstandard models is particularly evident in my own field of Reverse Mathematics; we have a different convention that $\omega$ refers to the standard numbers and $\mathbb{N}$ refers to an arbitrary model at hand (e.g. Simpson's *Subsystems of Second Order Arithmetic*). 


Part of the issue here may be that the meaning of the term "standard model" $\mathbb{N}$ can be interpreted in several ways. From the perspective of a certain kind of realism, it refers to the "actual" counting numbers. From the point of view of a certain kind of formalism, it refers to the natural numbers in whatever metatheory is being used at the moment, so that the "standard numbers" are the ones that are metafinite and "nonstandard models" have numbers that do not correspond to numbers in the metatheory. In any case, the notation $\mathbb{N} = \{0, 1,2, \ldots\}$ is *intended* to convey that $\mathbb{N}$ is identified with the usual counting numbers $0$, $1$, $2$, $\ldots$ from basic arithmetic, whatever we think those are. 


