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Addition in nonstandard models of arithmetic

  • R. Phillips (a1)

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In [3] Kemeny made the following conjecture: Suppose *Z is a nonstandard model of the ring of integers Z. Let

and let F be the subgroup of those cosets ā which contain an element of infinite height in *Z. Kemeny then asked if the ring R = {a: ā ∈ F} is also a nonstandard model of Z. If so then Goldbach's conjecture is false because Kemeny also shows in [3] that Goldbach's conjecture fails in R.

The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but we give here a simpler proof based on Mendelson's paper. Suppose R is a nonstandard model of Z. Then each positive number in R is a sum of four squares. Choose a in R so that a is a positive element of R of infinite height in *Z. Then since a is infinite in *Z, a − 1 is positive. Thus , xiR for i = 1, …, 4. Now each xi must be of the form ai + ni, where ai has infinite height in *Z and ni, ∈ Z.

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[1]Gandy, R. O., A note on a paper of Kemeny's, Mathematische Annalen, vol. 136 (1958), p. 466.
[2]Kaplansky, I., Infinite abelian groups, University of Michigan, Ann Arbor, 1954.
[3]Kemeny, J. G., Undecidable problems of elementary number theory, Mathematische Annalen, vol. 135 (1958), pp. 160169.
[4]MacDowell, R. and Specker, E., Modelle der Arithmetik, Infinitistic methods, (Proceedings of a Symposium on Foundations of Mathematics, Warsaw, 1959), Pergamon Press, Oxford; PWN, Warsaw, 1961, pp. 257–263.
[5]Mendelson, E., On nonstandard models for number theory, Essays on the foundations of mathematics, Fraenkel anniversary volume, 1961, pp. 259268.
[6]Phillips, R., On the structure of nonstandard models of arithmetic, Proceedings of the American Mathematical Society, vol. 27 (1971), pp. 359363.
[7]Robinson, A., Nonstandard analysis, Studies in logic and the foundations of mathematics, North-Holland, Amsterdam, 1966.

Addition in nonstandard models of arithmetic

  • R. Phillips (a1)

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