## Nonstandard arithmetic and reverse mathematics.(English)Zbl 1101.03040

Reverse mathematics is based on a hierarchy of fragments of second-order arithmetic. Keisler develops an alternative approach in which the key role is played by the separation of integers into the (standard) natural numbers and hyperintegers. This is done in a hybrid second-order/two-sorted language $$L_2\cup {^*\!L}_1$$. Keisler shows that each of the basic theories $$\text{ WKL}_0$$, $$\text{ACA}_0$$, $$\text{ATR}_0$$, and $$\Pi^1_1$$-$$\text{CA}_0$$ (which are formulated in the language of second-order arithmetic $$L_2$$) has a natural counterpart in $$L_2\cup {^*\!L}_1$$. The language $${^*\!L}_1$$ has all the symbols of the first-order arithmetic and variables of two sorts: $$N$$ and $$^*\!N$$. The universe of sort $$N$$ is a subset of the universe of sort $$^*\!N$$ and variables and terms of sort $$N$$ are allowed in argument places of sort $${^*\!N}$$. The role of the basic theory $$\text{ I}\Sigma_1$$ is played by $$^*\Sigma \text{PA}$$ which includes the basic axioms of $$\text{I}\Sigma_1$$ with variables of sort $$^*\!N$$, an internal induction axiom for a special class of bounded formulas, and two special axioms. One of the axioms says that $$N$$ is a proper initial segment of $$^*\!N$$, and the other expresses a property of coded sequences all of whose terms with standard indices are standard (the Finiteness Axiom). This theory is denoted by $$^*\Sigma \text{PA}$$. The stronger axiom systems describe properties of structures of the form $$(M,\,^*\!N)$$, where $$M=(N,P)$$ is an $$L_2$$ structure and $$(N,\,^*\!N)$$ is an $$^*\!L_1$$ structure. Then, $$^*\text{WKL}_0$$ is a theory in the language $$L_2\cup {^*\!L}_1$$ defined as $$^*\Sigma + \text{ STP}$$, where STP is the Standard Part Principle declaring that $$P$$ is the standard system of $$^*\!N$$ relative to $$N$$. It is shown that $$^*\text{WKL}_0$$ implies $$\text{WKL}_0$$ and that $$^*\text{WKL}_0$$ is conservative with respect to $$\text{WKL}_0$$. An important ingredient of the proof is the theorem of K. Tanaka [Ann. Pure Appl. Logic 84, 41–49 (1997; Zbl 0871.03044)], on self-embeddings of countable models of $$\text{WKL}_0$$ which are not $$\omega$$-models. The equivalent of $$\text{RCA}_0$$ is obtained by weakening STP in $$^*\text{WKL}_0$$. The equivalents of $$\text{ACA}_0$$ and $$\Pi^1_1$$-$$\text{CA}_0$$ are obtained by adding suitable comprehension schemes to $$^*\text{WKL}_0$$; $$^*\text{ATR}_0$$ involves a $$\Sigma^*_1$$-separation scheme. It is also shown that $$^*\Pi^1_1$$-$$\text{CA}_0$$ plus the First-Order Transfer Principle (FOT) is conservative with respect to $$\Pi^1_1$$-$$\text{CA}_0$$ proving a conjecture of C. W. Henson, M. Kaufmann and H. J. Keisler [J. Symb. Log. 49, 1039–1058 (1984; Zbl 0587.03048)].

### MSC:

 03F35 Second- and higher-order arithmetic and fragments 03H15 Nonstandard models of arithmetic

### Keywords:

reverse mathematics; second-order arithmetic; hyperintegers

### Citations:

Zbl 0587.03048; Zbl 0871.03044
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### References:

  Logic in Tehran 26 (2006)  Model theory (1990)  DOI: 10.1016/S0168-0072(95)00058-5 · Zbl 0871.03044  DOI: 10.2307/2274260 · Zbl 0587.03048  Infinitistic methods pp 257– (1961)  DOI: 10.2307/2274061 · Zbl 0624.03051  Subsystems of second order arithmetic (1999) · Zbl 0909.03048
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