Combinatorial isols and the arithmetic of Dekker semirings. (English) Zbl 1005.03042
In a recent paper [“Hereditarily odd-even and combinatorial isols”, Pac. J. Math. 206, No. 1, 9-24 (2002)], J. Barback introduced the notion of combinatorial isol, a generalization of completely torre isol. He showed that an isol \(X\) is combinatorial iff its associated Dekker semiring \(D(X)\) is linearly ordered. In the present paper, McLaughlin extends Barback’s work to show that \(X\) is combinatorial iff every \(\Pi_2\) sentence (in a suitable language) that holds for the natural numbers must also hold for \(D(X)\). McLaughlin also supplies an existence proof for combinatorial isols that avoids the completely-torre route.
Reviewer: Leon Harkleroad (Wilton)
MSC:
| 03D50 | Recursive equivalence types of sets and structures, isols |
| 03F30 | First-order arithmetic and fragments |
References:
| [1] | Barback, Math. Logic Quarterly 44 pp 229– (1998) · Zbl 0903.03027 · doi:10.1002/malq.19980440209 |
| [2] | Hereditarily odd-even and combinatorial isols. Pacific J. Math. (to appear). · Zbl 1050.03030 |
| [3] | Personal communication. April 1999. |
| [4] | Dekker, Math. Zeitschrift 83 pp 345– (1964) · Zbl 0122.01002 · doi:10.1007/BF01111167 |
| [5] | Hirschfeld, Israel J. Math. 20 pp 111– (1975) · Zbl 0311.02050 · doi:10.1007/BF02757881 |
| [6] | and , Forcing, arithmetic, division rings. Lecture Notes in Mathematics 454, Springer-Verlag, Berlin-Heidelberg-New York 1982. |
| [7] | Regressive Sets and the Theory of Isols. Lecture Notes in Pure and Applied Mathematics 66, Marcel Dekker, New York 1982. · Zbl 0484.03025 |
| [8] | Mc Laughlin, Proc. Amer. Math. Soc. 97 pp 495– (1986) |
| [9] | Mc Laughlin, Math. Logic Quarterly 39 pp 431– (1993) · Zbl 0805.03051 · doi:10.1002/malq.19930390146 |
| [10] | Nerode, Annals Math. 73 pp 362– (1961) · Zbl 0101.01203 · doi:10.2307/1970338 |
| [11] | Nerode, Annals Math. 84 pp 421– (1966) · Zbl 0158.25104 · doi:10.2307/1970455 |
| [12] | Mathematical Logic. Addison-Wesley, Reading (Mass.) 1967. |
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