Nonstandard Models of Arithmetic and Set Theory: AMS Special Session Nonstandard Models of Arithmetic and Set Theory, January 15-16, 2003, Baltimore, MarylandAmerican Mathematical Soc., 2004 - 167 This is the proceedings of the AMS special session on nonstandard models of arithmetic and set theory held at the Joint Mathematics Meetings in Baltimore (MD). The volume opens with an essay from Haim Gaifman that probes the concept of non-standardness in mathematics and provides a fascinating mix of historical and philosophical insights into the nature of nonstandard mathematical structures. In particular, Gaifman compares and contrasts the discovery of nonstandard models with other key mathematical innovations, such as the introduction of various number systems, the modern concept of function, and non-Euclidean geometries. Other articles in the book present results related to nonstandard models in arithmetic and set theory, including a survey of known results on the Turing upper bounds of arithmetic sets and functions. The volume is suitable for graduate students and research mathematicians interested in logic, especially model theory. |
Spis treści
| 1 | |
Coding in IΔ0 | 23 |
Automorphisms Mahlo Cardinals and NFU | 37 |
AC Fails in the Natural Analogues of V and L that Model the Stratified Fragment of ZF | 61 |
Working with Nonstandard Models | 71 |
Internally Iterated Ultrapowers | 87 |
On Some Questions of Hrbacek and Di Nasso | 121 |
Turing Upper Bounds of Jump Ideals and Scott Sets | 129 |
Diversity in Substructures | 145 |
Automorphisms of Countable Recursively Saturated Models of Set Theory | 163 |
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algebra arithmetical sets automorphism axioms Borel coding set cofinal completion computes an enumeration consistent construction Corollary countable jump ideal defined definition E-formula element elementary embedding elementary extension elementary submodel elementary substructure enum equivalenced lattice exists family of sets finite formula FRIST Gaifman Hence holds II(A implies induction infinite initial segment interpretation isomorphism iterated ultrapowers Lemma Logic Lt(M Mahlo cardinals Math Mathematics measurable cardinal models of arithmetic models of set n-CPP+ representation n-symmetric natural numbers NFUA non-Euclidean geometry non-standard models nonstandard universe Ord is weakly ordinal partition Peano Arithmetic permutations power set predicate proof of Theorem Proposition prove Rep(T satisfies saturated models Scott set sentences sequence Skolem Solovay standard model stratified structure subalgebras subsets substructure lattice Suppose Turing upper bounds ultrafilter ultrapowers VQ/P weakly compact WSOL