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**1 - 5**of**5**### Order-types of models of Peano arithmetic: a short survey

, 2001

"... This paper is a short and slightly selective survey of results on order-types of models of Peano arithmetic. We include few proofs, and concentrate instead on the key problems as we see them and possible ways of responding to the very considerable mathematical di#culties raised ..."

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This paper is a short and slightly selective survey of results on order-types of models of Peano arithmetic. We include few proofs, and concentrate instead on the key problems as we see them and possible ways of responding to the very considerable mathematical di#culties raised

### On the Structure of Initial Segments of Models of Arithmetic

"... Abstract. For any countable nonstandard model M of a sufficiently strong fragment of arithmetic ~ and any nonstandard numbers a, C EM, MF C ~ a, there is a model K of T which agrees with M up to a and such that in K there is a proof of contradiction in T with Godel number ~ 2ac. lntroduction For any ..."

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Abstract. For any countable nonstandard model M of a sufficiently strong fragment of arithmetic ~ and any nonstandard numbers a, C EM, MF C ~ a, there is a model K of T which agrees with M up to a and such that in K there is a proof of contradiction in T with Godel number ~ 2ac. lntroduction For any M a model of arithmetic and a EM, M r a will denote the structure with the universe {iEMIMFi~a}, and with operations inherited from M. Thus + and. are only partial functions in M r a. The general question that we study bere is this: For which functions f(x) the structure M ra uniquely determines the st,ructure Mr f(a)? "Determines " means "up to elementary equivalence " or equivaiently as all Mra are recursively saturated "up to isomorphism", It is easily seen that Mra determines Mrak, for any k<w. Paris and Dimitracopoulos [3J showed that M ra does not determineM r2G. Namely, they proveï that for any countable nonstandard model M of PA and any aEM nonstandard, there is K a model of PA such that aEK, Mra=Kra, but Mr22°$Kr22°. Thus either Mra does not determine Mr2G or Mr2G does not determine M r22O. Later Hájek [3J found ODe L1o formula 4>(x) such that for any aEM as above, there is K, a E K, K r a = Mr a and KF 4>(22°) and MF-,4>(22°) or KF-,4>(22°) and MF 4>(22°). He has also shown that for any a E M as above, there is K such that aEK, Mra=Kra and 224

### Non-Standard Models of Arithmetic: a Philosophical and Historical perspective MSc Thesis (Afstudeerscriptie)

, 2010

"... 1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5 ..."

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1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5

### A NOTE ON STANDARD SYSTEMS AND ULTRAFILTERS

"... Let (M, X) | = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an end-extension N | = T of M such that the subsets of ..."

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Let (M, X) | = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an end-extension N | = T of M such that the subsets of M coded in N are precisely those in X. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.

### A NOTE ON STANDARD SYSTEMS AND ULTRAFILTERS

, 804

"... Abstract. Let (M, X) | = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an end-extension N | = T of M such that the s ..."

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Abstract. Let (M, X) | = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an end-extension N | = T of M such that the subsets of M coded in N are precisely those in X. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T. The standard system of a model M of PA (the first order formulation of Peano arithmetic) is the collection of standard parts of the parameter definable subsets of M, i.e., sets of the form X ∩ ω, where X is a parameter definable set of M, and ω is the set of natural numbers. It turns out that the standard system tells you a lot about the model; for example, any two countable recursively saturated models of the same completion of PA with the same standard system are isomorphic. A natural question to ask is then which collections of subsets of the natural numbers are standard systems. This problem has become known as