Looking for a source for Intended Interpretation

Question

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.

Answer 1

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.

Answer 2

This is difficult to answer for a variety of reasons. If you're looking for a published source for the phrases "intended interpretation" and "standard model" with a suitable definition, many mathematical logic textbooks will do. From a historical perspective, as suggested by the phrase "originally intended", one can say a lot more. However, the key players in this historical development would never use the phrase "intended interpretation" in the modern sense since the phrase entails a distinction that they weren't aware of — that there are other interpretations!

In any case, the key players Peano and Dedekind both made their intent very clear. Especially Dedekind, who wrote essays and letters on the nature of numbers. After some searching, I just found a beautiful digitized copy of his essay Was sind und was sollen die Zahlen? from 1888; you can also find "The Nature and Meaning of Numbers" in his Essays on the Theory of Numbers. In his preface to Arithmetices Principia: Nova Methodo Exposita, Peano states that his system is intended to derive the principles of arithmetic. So both Peano and Dedekind made it clear that their axiomatic system was intended to describe the natural numbers.

The fact that counting numbers satisfy the Peano–Dedekind axioms is now known as Frege's Theorem. Although Frege first proved this using his ill-fated Rule V, it was later observed that that much of Frege's work didn't use the full power of Rule V and thus Frege's derivation could be legitimately called a theorem. (See Richard G. Heck, Jr., The development of arithmetic in Frege’s Grundgesetze der Arithmetik, J. Symbolic Logic 58 (1993), no. 2, 579–601.) So, Frege was the first to check that the Peano–Dedekind axioms did indeed describe the counting numbers.

The modern distinction between "intended interpretation" and "unintended interpretation" was known to Skolem around 1915 as he explains in his early critique of axiomatic set theory ("Some remarks on axiomatic set theory", found in van Heijenoort). However, it is only in the 1930's that he first managed to demonstrate the existence of non-standard models of arithmetic.

Answer 3

Here's a reference: Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, Springer, 2010. On page 42 he defines the operations $+$ and $\cdot$ on $\mathbb N$ as having their "ordinary meaning". On page 62 he says that "interpretation" = "model". Finally, on pages 105-106 he defines $\mathcal N = (\mathbb N, 0, S, +, \cdot)$ and calls $\mathcal N$ the "standard model". ($\mathbb N$ is defined as the set of natural numbers including zero on page xix.)

Answer 4

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.

And also:

• Richard Dedekind, Letter to Keferstein (1890), into: Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic (1967), page 98-on:

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 [1888]). [...]

(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:

• Hao Wang, THE AXIOMATIZATION OF ARITHMETIC (1957), reprinted as Chapter IV of Hao Wang, A Survey of Mathematical Logic (1963), page 67-on.

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 :

• Hao Wang & JBarkley Rosser, Nonstandard models for formal logics (JSL, 1950) (JSTOR), and:

• Thorald Skolem, Peano axioms and models of arithmetic, into: Thoralf Skolem & Georg Kreisel & Abraham Robinson & Hao Wang (editors), Mathematical Interpretation of Formal Systems (1955), page 1-on:

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.

See also:

• Leon Henkin, On Mathematical Induction (1959).

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 $\mathcal A$).

Answer 5

In lieu of answering a question let me sketch why the question is more difficult to answer than it at first appears.

The question asks for a reference, in a journal article or book, to a statement of one of the two forms:

1. The intended interpretation of the axioms of Peano arithmetic is the ordinary counting numbers.

2. The set $\mathbb N$ is meant to represent the ordinary counting numbers.

I guess because ample references exist for the statement that the intended interpretation of Peano arithmetic is the set $\mathbb N$, these two are equivalent.

I think one reason references that make precisely one of these statements, and cannot possible be interpreted as making a different one, in a journal article or book seem to be hard to find, is that mathematicians are uncomfortable making assumption about informal intuitions in a formal context.

How is the author to know what the reader takes to be the ordinary numbers?

Perhaps the reader is a child who believes that there is a largest number.

Or the reader is an ultrafinitist, who believes that only numbers that they can count to in their lifetime exist. (The belief that only these numbers are "ordinary numbers" is quite reasonable. Is $10^{10^{10^{10}}}$ really an ordinary number? But $10^{10^{10^{10}}} \in \mathbb N$.)

Or they may only refer to the numbers by their standard English names and thus not be able to express the concept of 10^66.

There are also ways that their intuitive concept of number could have more numbers than are in $\mathbb N$ - by including fractional numbers, approximate numbers, numbers that depend on some external referent, or multiple names for the same number.

I am sure that if a mathematician wrote such a statement in one of their books they would not face objections from these sorts of people. Why is everyone's ordinary concept of number so similar? The answer is surely that we are taught about numbers in school, and all taught broadly the same things.

This suggests that you should, rather than asking a mathematician to tell you that the set of natural numbers refers to the numbers you learned about in school, go back to your elementary school teacher and demand that they tell you whether, when they taught you about numbers, they meant to teach you how to work with elements of $\mathbb N$. (Of course it is easy to verify that all the properties of numbers that you were taught in school are also properties of $\mathbb N$.) Perhaps this teacher does not know the formal definition of the mathematical set $\mathbb N$, but surely someone on the chain of command at some point does. Should you look for an official memo from your country's Department of Education on the nature of the numbers you were taught about?

Let me close with two positive suggestions:

Rather than in a logical context discussing the axioms of Peano arithmetic, the statement that the set $\mathbb N$ is supposed to correspond to our intuitive notions of number might appear in an introductory work on set theory where that set is defined. I checked one (Naive Set Theory by Paul Halmos) and did not really find the statement with the precision you're looking for.

When you next come up with a novel result on a related topic, you could simply include this statement in the ensuing journal article. In an appropriate context it seems unlikely that a referee will demand its removal. Since such a statement cannot be justified, you will not be required to write a proof, and since it is not an original insight, you will not be stealing credit from the original author.

Answer 6

The idea of "intended interpretation" stems in fact from the ancient times. The Robert Lawlor article discusses in detail the Pytharogas intended interpretation in the collection Homage to Pythagoras edited by Christopher Bamford. I think that any intended interpretation of numbers today is much more immaterial than it was even in the last century. The diversity of interpretations has taken its place. The term "intended interpretation" is obsolete now. The use of the term was and still is a deeply concealed quest or desire for categorical foundations. The best logicians, who had discovered or understood the eternal crash of categoricity, were mainly the regular species of the mathematical community, i.e. practicing realists or platonists as most of the working mathematicians are these days.

Semen Kutateladze 943