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A note on the Joint Embedding Property in Fragments of Arithmetic
Author(s) -
Otero Margarita
Publication year - 1992
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/24.5.417
Subject(s) - mathematics , peano axioms , property (philosophy) , extension (predicate logic) , embedding , arithmetic , fragment (logic) , block (permutation group theory) , bounded function , discrete mathematics , combinatorics , algorithm , computer science , mathematical analysis , philosophy , epistemology , artificial intelligence , programming language
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 theoryconsistent with PA and T ⊢ IE 1 −(bounded existential parameter‐free induction), then any two m dels 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 1 −fails to have JEP.