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A NOTE ON THE JOINT EMBEDDING PROPERTY IN FRAGMENTS OF ARITHMETIC
"... It is known that full Peano Arithmetic does not have the joint embedding property (JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We prove, using some conservation results about fragments of arithmetic, that if T is a theory consistent with PA and T\ I ..."
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It is known that full Peano Arithmetic does not have the joint embedding property (JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We prove, using some conservation results about fragments of arithmetic, that if T is a theory consistent with PA and T\ IE ^ (bounded existential parameterfree induction), then any two models of PA which jointly embed in a model of T also jointly embed in an elementary extension of one of them. In particular, any fragment of PA extending IE [ fails to have JEP. 1.
Arithmetic and the Incompleteness Theorems
, 2000
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A FEW REMARKS ON nINFINITE FORCING COMPANIONS
"... Abstract. We show that the basic properties of Robinson’s infinite forcing companions are naturally transmitted to the so called ninfinite forcing companions and start with the examination of mutual relations of ninfinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the art ..."
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Abstract. We show that the basic properties of Robinson’s infinite forcing companions are naturally transmitted to the so called ninfinite forcing companions and start with the examination of mutual relations of ninfinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the article L is a first order language. In general discussions mostly it is irrelevant whether it is with equality or not; however, in some cases, for instance when it comes to finite models, the supposition of the existence of the equality relation could be of significance – see 2.6. For a theory T of the language L, µ(T) will be the slass of all its models (as usual, by a theory we assume a consistent deductively closed set of sentences – thus, T ϕ means ϕ ∈ T). By Σnformula we mean any formula equivalent to a formula in prenex normal form whose prenex consists of n blocks of quantifiers, the first one is the block of existential quantifiers (Πnformulas are defined analoguosly). The models (of the language L) will be denote by A,B..., while their domains will be A,B,.... For a model A, Diagn(A) is the set of all Σn, Πnsenteneces of the language L(A) (the simple expansion of the language L obtained by adding a new set of constants which is in one to one correspendence with domain A) which hold in A. In particular, for n = 0, Diag0(A) is not the diagram of A in the sense in which it is used in model theory, but this difference is of no importance for the text (the same situation we had when we were dealing with the generalization of finite forcing). As usual, we will not distinguish an element a from A and to it the corresponding constant. If A is a submodel of B and (B, a)a∈A Diagn(A), we say that A is an nelementary submodel of B (i.e., that B is an nelementary extension of A), in notation A ≺n B. In general, A is nembedded in B if for some embedding f of A into B, f(A) is an nelementary submodel of B. A Σn+1chain of models is a chain of models A0 < A1 < · · · < Aα < · · · , α < γ, where for each