{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T11:20:31Z","timestamp":1772450431020,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":20830,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1957,3]]},"abstract":"<jats:p>This paper contains examples T<jats:sub>1<\/jats:sub> and T<jats:sub>2<\/jats:sub> of theories which answer the following questions:<\/jats:p><jats:p>(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?<\/jats:p><jats:p>(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)<\/jats:p><jats:p>The theory T<jats:sub>1<\/jats:sub> represents a modification of a theory described by Myhill [7]. The common feature of theories T<jats:sub>1<\/jats:sub> and T<jats:sub>2<\/jats:sub> is that in both of them pleonasms are essential in the construction of the axioms.<\/jats:p><jats:p>Let T<jats:sub>1<\/jats:sub> be a theory with identity = which contains one binary predicate <jats:italic>R<\/jats:italic>(<jats:italic>x, y<\/jats:italic>) and is based on the axioms A<jats:sub>1<\/jats:sub>, A<jats:sub>2<\/jats:sub>, A<jats:sub>3<\/jats:sub>, B<jats:sub>1<\/jats:sub>, B<jats:sub>2<\/jats:sub>, B<jats:sub>3<\/jats:sub>, B<jats:sub>4<\/jats:sub>, C<jats:sub><jats:italic>nm<\/jats:italic><\/jats:sub> which follow.<\/jats:p><jats:p>A<jats:sub>1<\/jats:sub>: <jats:italic>x = x<\/jats:italic>. A<jats:sub>2<\/jats:sub>: <jats:italic>x = y<\/jats:italic> \u2283 <jats:italic>y = x<\/jats:italic>. A<jats:sub>3<\/jats:sub>: <jats:italic>x<\/jats:italic> = <jats:italic>y<\/jats:italic> \u2227 <jats:italic>y = z<\/jats:italic> \u2283 <jats:italic>x = z<\/jats:italic>.<\/jats:p><jats:p>(Axioms of identity.)<\/jats:p><jats:p>B<jats:sub>1<\/jats:sub>: <jats:italic>R<\/jats:italic>(<jats:italic>x, x<\/jats:italic>). B<jats:sub>2<\/jats:sub>: <jats:italic>R<\/jats:italic>(<jats:italic>x, y<\/jats:italic>) \u2283 <jats:italic>R<\/jats:italic>(<jats:italic>y, x<\/jats:italic>). B<jats:sub>3<\/jats:sub>: <jats:italic>R<\/jats:italic> (<jats:italic>x, y<\/jats:italic>) \u2227 <jats:italic>R<\/jats:italic>(<jats:italic>y, z<\/jats:italic>) \u2283 <jats:italic>R<\/jats:italic>(<jats:italic>x, z<\/jats:italic>).<\/jats:p><jats:p>(Axioms of equivalence.)<\/jats:p><jats:p>B<jats:sub>4<\/jats:sub>: <jats:italic>x = y<\/jats:italic> \u2283 [<jats:italic>R<\/jats:italic>(<jats:italic>z, x<\/jats:italic>) \u2261 <jats:italic>R<\/jats:italic>(<jats:italic>z,y<\/jats:italic>)].<\/jats:p><jats:p>Let <jats:italic>\u03c6<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> be the formula<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200070638_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>which express that there is an abstraction class of the relation <jats:italic>R<\/jats:italic> which has exactly <jats:italic>n<\/jats:italic> elements.<\/jats:p><jats:p>Let <jats:italic>f<\/jats:italic>(<jats:italic>n<\/jats:italic>) and <jats:italic>g<\/jats:italic>(<jats:italic>n<\/jats:italic>) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets <jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>X<\/jats:italic><jats:sub>2<\/jats:sub>.<\/jats:p><jats:p>We now specify the axioms <jats:italic>C<jats:sub>mm<\/jats:sub><\/jats:italic>.<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200070638_eqnU2\"\/><\/jats:disp-formula><\/jats:p><jats:p>It is obvious that the set composed of the formulas A<jats:sub>1<\/jats:sub>\u2212A<jats:sub>3<\/jats:sub>, B<jats:sub>1<\/jats:sub>\u2212B<jats:sub>4<\/jats:sub>, C<jats:sub><jats:italic>nm<\/jats:italic><\/jats:sub> (<jats:italic>n,m<\/jats:italic> = 1,2, \u2026) is recursive.<\/jats:p><jats:p>The theory T<jats:sub>1<\/jats:sub> is essentially undecidable; for if there were a complete and decidable extension T\u2032<jats:sub>1<\/jats:sub> (of it, then the recursive sets <jats:italic>Z<\/jats:italic> = {<jats:italic>n<\/jats:italic>: <jats:italic>\u03c6<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is provable in T\u2032<jats:sub>1<\/jats:sub>} and <jats:italic>Z<\/jats:italic>\u2032 = {<jats:italic>n<\/jats:italic>: \u223c<jats:italic>\u03c6<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> is provable in T\u2032<jats:sub>1<\/jats:sub>} would separate the sets <jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>X<\/jats:italic><jats:sub>2<\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2964056","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:47:42Z","timestamp":1146944862000},"page":"36-38","source":"Crossref","is-referenced-by-count":5,"title":["Two theories with axioms built by means of pleonasms"],"prefix":"10.1017","volume":"22","author":[{"given":"Andrzej","family":"Ehrenfeucht","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200070638_ref002","first-page":"92","volume":"20","author":"Henkin","year":"1955","journal-title":"On a theorem of Vaught"},{"key":"S0022481200070638_ref006","first-page":"384","volume":"15","author":"Kreisel","year":"1954","journal-title":"Mathematical reviews"},{"key":"S0022481200070638_ref005","first-page":"244","article-title":"A symmetric form of G\u00f6del's theorem","volume":"12","author":"Kleene","year":"1950","journal-title":"Indagationes mathematicae"},{"key":"S0022481200070638_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(55)50046-1"},{"key":"S0022481200070638_ref004","doi-asserted-by":"crossref","first-page":"131","DOI":"10.4064\/fm-40-1-131-139","article-title":"Undecidability of some simple formalized theories","volume":"40","author":"Janiczak","year":"1953","journal-title":"Fundatnenta mathematicae"},{"key":"S0022481200070638_ref008","first-page":"98","volume-title":"Undecidable theories","author":"Tarski","year":"1953"},{"key":"S0022481200070638_ref007","first-page":"49","volume":"21","author":"Myhill","year":"1956","journal-title":"Solution of a problem of Tarski"},{"key":"S0022481200070638_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457985"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200070638","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T04:51:39Z","timestamp":1559883099000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200070638\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1957,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1957,3]]}},"alternative-id":["S0022481200070638"],"URL":"https:\/\/doi.org\/10.2307\/2964056","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1957,3]]}}}