For a clear description of what has been later called: the intended interpretation (or standard model), we can see:
- Richard Dedekind, Continuity of irrational numbers (Stetigkeit und irrationale Zahlen, 1872), page 4:
I regard the whole of arithmetic as a necessary, or at least natural,
consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication.
one is, I believe, forced to accept the following facts :
(1) The number sequence $N$ is a system of individuals, or elements, called numbers. This leads to the general consideration of systems as such (§1 of my essay ). [...]
(6) however [...] these facts are still far from being adequate for completely characterizing the nature of the number sequence $N$. All these facts would hold also for every system $S$ that, besides the number sequence $N$, contained a system $T$, of arbitrary additional elements $t$, to which the mapping $\varphi$
could always be extended while remaining similar and satisfying $\varphi(T) = T$. But such a system $S$ is obviously something quite different from our number sequence $N$, and I could so choose it that scarcely a single theorem of arithmetic would be preserved in it. What, then, must we add to the facts above in order to cleanse our system $S$ again of such alien intruders [emphasis added] $t$ as disturb every vestige of order and to restrict it to $N$ ?
Interestingly enough, Dedekind's letter was quoted and (partially) translated by Hao Wang; see:
See page 69:
Dedekind wrote a very interesting
letter (dated 27 February, 1890, addressed to Headmaster Dr.H. Keferstein
of Hamburg) to explain how he arrived at the Peano axioms. [...] The notion of non-standard models (unintended interpretations [emphasis added]) of axioms for positive integers is, for instance, brought out quite clearly in Dedekind's letter.
and page 77:
His [Dedekind] letter supplies a useful clue, when he discusses under (6) the question of excluding undesirable interpretations of the set $N$ for which some ordinary arithmetic theorems would fail to hold. This suggests the following line of argument which may have been followed by Dedekind. The definition
of natural numbers in terms of the chain of $1$ enables us to determine the
abstract character of the set of natural numbers entirely: witness his proof
that any two sets satisfying the definition are isomorphic. If a theorem is
independent of his definition, then there are two possible interpretations of
the definition according to one of which the theorem is true and according
to another the theorem is false. If the definition determines a unique interpretation of the theory, such situations cannot arise. Therefore, by the
uniqueness of interpretation, all theorems about natural numbers must be
derivable. This argument is plausible but not entirely rigorous because, among
other things, the notion of interpretation has not been made sufficiently
explicit to assure that any undecidable theorem will necessarily yield two
different interpretations of the definition.
Dedekind's conclusion that these determine adequately the sequence of
natural numbers is often expressed equivalently by saying that the Peano
axioms are categorical or have no essentially different interpretations. As we
know, the axioms do admit different interpretations such as taking $100$ as
$1$ or taking the square of a number as the successor of a number. But they
are all essentially the same in a technical sense of being isomorphic.
The proof of this is very easy once we concede that the axiom of induction
does assure that the number sequence contains no "alien intruders" beyond the true natural numbers each of which is either the base-element or can be reached from the base-element by a finite number of steps. [...]
and page 79:
In recent years a good deal of research in mathematical logic has been devoted
to the question of unintended interpretations (nonstandard models [emphasis added]) of theories of positive integers. It is therefore of interest to find this question raised in Dedekind's letter. [...] It follows, however, from certain fundamental results of Gödel and Skolem that whenever a language can be effectively set up and proofs can be effectively checked, there are always unintended models of positive integers which satisfy all of the Dedekind-Peano axioms, provided that the properties in the axiom of induction are restricted to ones expressible in the given language.
For earlier occurrences, we can see :
This fact can be expressed by saying that besides the usual number series [emphasis added] other models exist of the number theory given by Peano's axioms.
And page 9:
In these simple cases the definition of non-standrad models [emphasis added] can often be established in a perfectly constructive way.
See, in the same collection, also:
- Hao Wang, On denumerable bases of formal systems, page 57-on; on page 69, regarding the discussion of categorical theries and Lowenheim-Skolem theorem:
there is [...] always some model for each system which is denumerable and therefiore different from the intended model [emphasis added].
and page 71-71:
Vaguely we feel that each formal system is constructed with a unique intended model, which may be called the standard model, in mind. The speaker shares with many the discomfort over the unqualified notion of a standard model. The notion of standard model relative to certain preassigned interpretations of certain
specific notions is easier. [...] In this connection, the situation with number theory is much better than the situation with set theory. The standard interpretation of positive integers can be specified, for example, by emphasizing that every positive integer is either $1$ or obtainable from $1$ by applying the operation of adding $1$ a finite number of times.
Note Wang's contribution to ‘Metalogic’ for Fifteenth edition of Encyclopaedia Britannica is dated 1974.
The term standard model was introduced by Leon Henkin, Completeness in the Theory of Types JSL (1950) (JSTOR) meaning what is today called "normal" model for high-order logic.
See S.C.Kleene, Introduction to metamathematics (1952), page 427-on (discussing Skolem's result of 1933):
[page 429] The axioms of Postulate Group B of our formal number-theoretic system [first-order Peano axioms] admit an interpretation [...] other than the intended one.
As per Carl's answer above, the "mainstream" notion of standard model was codified in the '60s by the leading authors of mathematical logic textbooks, like Kleene and Shoenfield.
We can find it also into :
- Roger Lyndon, Notes on logic (1966), page 61:
consider any set $A$ of axioms for the natural numbers that can be formulated in a language $L$ with the symbol $=$, symbols $+$ and $\times$, and possibly other arithmetical symbols. We assume that the axioms are valid under the standard interpretation, in the domain $N$ of the natural numbers.
- Georg Kreisel & Jean Louis Krivine, Elements of Mathematical Logic: Model Theory (1967), page 43:
The standard realizaion of [a language with equality] $\mathcal L$ is the realization whose domain is $N$, the set of natural numbers, and in which the symbols $0, s, +, \times$, take their natural values in $N$, namely zero, successor, addition and multiplication.
Now consider the following formulas, $\mathcal A$, of $\mathcal L$: [i.e. the f-o Robinson's axioms] the standard realization of $\mathcal L$ is a normal model of $\mathcal A$ (the standard model of $\mathcal A$).