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Abstract
GRIST is an axiomatic framework for nonstandard set theory with many of standardness. The paper establishes a number of general consequences of GRIST, in particular, a very strong form of Transfer principle. 2010 Mathematics Subject Classification 26E35 (primary); 03E70, 03H05 (sec- ondary) This is the last in a series of three articles devoted to GRIST, an axiomatic presentation of nonstandard analysis with many of standardness. The two previous papers, (4) and (5), will be referred to as RST and RST2, respectively. It is shown in RST that GRIST is complete over ZFC: If an extension of GRIST proves a theorem that is not provable in GRIST, then it proves a theorem in the language of ZFC that is not provable in ZFC (see Proposition 6.5). In other words, no additional principles can be added to GRIST while keeping it conservative over ZFC. Yet in mathematical applications it is sometimes awkward to argue directly from the axioms of GRIST. It is convenient to have at one's disposal other principles, provable in GRIST, but tailor-made for certain kinds of applications. A number of such consequences of GRIST is derived in RST, Section 12; see also RST2, Proposition 1.10. For applications of relative set theory see RST2 and (3, 6, 10). This paper focuses on deducing some further useful principles in GRIST. Foremost among them is Strong Stability, perhaps the ultimate generalization of Transfer. Sec- tion 1 begins with a formulation of Strong Stability. Strong Stability is then used to prove that levels represented by elements of a given set are precisely those from a finite union of singletons and closed intervals (in the ordering of levels by inclusion). Section 2 contains the proof of Strong Stability in GRIST. It relies heavily on the development of GRIST in RST. Counterexamples to some natural strengthenings of Strong Stability are also constructed there. Section 3 deals with some variants of Idealization and Choice that are provable in GRIST. It also presents a generalization of Robinson's Lemma due to Andreev.
Highlights
This is the last in a series of three articles devoted to GRIST, an axiomatic presentation of nonstandard analysis with many “levels of standardness.”
It is shown in RST that GRIST is complete over ZFC: If an extension of GRIST proves a theorem that is not provable in GRIST, it proves a theorem in the language of ZFC that is not provable in ZFC [see Proposition 6.5]
No additional principles can be added to GRIST while keeping it conservative over ZFC
Summary
In GRIST, the ordering of levels is dense (and there is a coarsest level). There are other possibilities, and weaker theories (FRIST [2], Peraire’s RIST [9]) agnostic on the details of the ordering of levels. The axioms of GRIST are listed below [see RST2, page 4]. The key principle of GRIST is Transfer, referred to as Stability: For all V ⊆ V and all x1, . The first step towards transgressing this limitation is made in the Local Transfer principle. Local Transfer [RST2, Proposition 1.10 (6)]: For any sets xk+1, . Local Transfer [RST2, Proposition 1.10 (6)]: For any sets xk+1, . . . , xn and any V0 there is V ⊃ V0 such that, for all V0 ⊆ V ⊂ V and all x1, . . . , xk ∈ V0 , P(x1, . . . , xk, xk+1, . . . , xn; V0) ↔ P(x1, . . . , xk, xk+1, . . . , xn; V)
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