{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T20:21:41Z","timestamp":1751660501219},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11607,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,6]]},"abstract":"<jats:p>The purpose of this paper is to study a formal system <jats:italic>PA(Q<\/jats:italic><jats:sup>2<\/jats:sup>) of <jats:italic>first order Peano arithmetic<\/jats:italic>, <jats:italic>PA<\/jats:italic>, augmented by a <jats:italic>Ramsey quantifier Q<\/jats:italic><jats:sup>2<\/jats:sup> which binds two free variables. The intended meaning of <jats:italic>Q<\/jats:italic><jats:sup>2<\/jats:sup><jats:italic>xx<\/jats:italic>\u2032\u03c6(<jats:italic>x, x<\/jats:italic>\u2032) is that there exists an infinite set <jats:italic>X<\/jats:italic> of natural numbers such that \u03c6(<jats:italic>a, a<\/jats:italic>\u2032) holds for all <jats:italic>a, a<\/jats:italic>\u2032 \u0404 <jats:italic>X<\/jats:italic> such that <jats:italic>a<\/jats:italic> \u2260 <jats:italic>a<\/jats:italic>\u2032. Such an <jats:italic>X<\/jats:italic> is called a <jats:italic>witness set<\/jats:italic> for <jats:italic>Q<\/jats:italic><jats:sup>2<\/jats:sup><jats:italic>xx<\/jats:italic>\u2032\u03c6(<jats:italic>x, x<\/jats:italic>\u2032). Our results would not be affected by the addition of further Ramsey quantifiers <jats:italic>Q<\/jats:italic><jats:sup>3<\/jats:sup>, <jats:italic>Q<\/jats:italic><jats:sub>4<\/jats:sub>, \u2026, Here of course the intended meaning of <jats:italic>Q<jats:sup>k<\/jats:sup>x<\/jats:italic><jats:sub>1<\/jats:sub> \u2026 <jats:italic>x<jats:sub>k<\/jats:sub><\/jats:italic>\u03c6(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026<jats:italic>x<jats:sub>k<\/jats:sub><\/jats:italic>) is that there exists an infinite set <jats:italic>X<\/jats:italic> such that \u03c6(<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>\u2026, <jats:italic>a<jats:sub>k<\/jats:sub><\/jats:italic>) holds for all <jats:italic>k<\/jats:italic>-element subsets {<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026 <jats:italic>a<jats:sub>k<\/jats:sub><\/jats:italic>} of <jats:italic>X<\/jats:italic>.<\/jats:p><jats:p>Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system <jats:italic>PA{Q<\/jats:italic><jats:sup>2<\/jats:sup>), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].<\/jats:p><jats:p>After some preliminaries in \u00a71, we begin in \u00a72 by giving a new completeness proof for <jats:italic>PA(Q<\/jats:italic><jats:sup>2<\/jats:sup>). A by-product of our proof is that for every regular uncountable cardinal <jats:italic>k<\/jats:italic>, every consistent extension of <jats:italic>PA(Q<\/jats:italic><jats:sup>2<\/jats:sup>) has a <jats:italic>k<\/jats:italic>-like model in which all classes are definable. 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