On 10 June 2010, at 12.30, in room 20 of the building of Collegium Civitas of Opole University there was held an interdisciplinary lecture delivered by Professor Jerzy Pogonowski of Adam Mickiewicz University of Poznań. The lecture, under the title Inexpressible longing for the intended model, was organized by the Group of Logic, Language and Information.
Professor Urszula Wybraniec-Skardowska welcomed all the guests, especially the lecturer of the day – Prof. Jerzy Pogonowski. Then, Professor Janusz Czelakowski presented the silhouette of the lecturer, underlining the fact that the research profile of the latter is very broad. It includes both problems pertaining to linguistics and the foundations of metalogic, set theory, theory of recursion, logical pragmatics, and the history of logic.
Professor Jerzy Pogonowski gave his lecture relating to an intended model of the theory as the one for the sake of which the theory is built. Limitations which logic of the first order carries with itself, indicate that a number of mathematical theories do not characterize – in an explicit way – their own models and that these theories possess non-standard models. The question of characterization of intended models is still being discussed in an intense manner, as – for instance – on the level of metalanguage in the context of studies on the categorical nature of theories. Another formulation is connected with acceptance of the so-called extreme axioms, e.g. Fraenkel restriction axiom, or Suszko canonical axiom within set theory. The notion of the intended model can not be characterized for certain important mathematical theories with the aid of either syntactic means (expressed in the language of the theory) or semantic means; in order to do so one needs certain decisions that make references to metalanguage – certain pragmatic decisions. The latter were presented using examples of both mathematical theories and empirical ones. The key question is that of delineating the subject reference of the theory.
The lecture was followed by a discussion which was continued in the second part of the meeting, during the seminar held by the Group. That was planned to be held in the same room. Professor J. Pogonowski introduced the subject matter being “The ontology of set theory”. During the seminar a number of interesting themes were raised, which were connected with the existence of many well-known and mutually contradicting extensions of Zermelo-Fraenkel theory, or with the status of the theory of category as the basis of mathematics. Another aspect of the discussion related to the development, shaping and passing on intuitions concerning set-theoretic notions in the secondary and higher school system.