Relative arithmetic

Mathematical Logic Quarterly 56 (6):564-572 (2010)
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In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y -standard’. Thus, objects are not standard in an absolute sense, but standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity



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Sam Sanders
Ruhr-Universität Bochum

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Nonstandard arithmetic and reverse mathematics.H. Jerome Keisler - 2006 - Bulletin of Symbolic Logic 12 (1):100-125.
Transfer and a Supremum Principle for ERNA.Chris Impens & Sam Sanders - 2008 - Journal of Symbolic Logic 73 (2):689 - 710.
Internal Set Theory: A New Approach to Nonstandard Analysis.Edward Nelson - 1977 - Journal of Symbolic Logic 48 (4):1203-1204.
More infinity for a better finitism.Sam Sanders - 2010 - Annals of Pure and Applied Logic 161 (12):1525-1540.

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