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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."

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.

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.

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."

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.

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.