Countable Nonstandard Models and Cohesive Powers
- Date: Wednesday 21 November 2018, 16:15 – 17:15
- Location: Mathematics Level 8, MALL 1, School of Mathematics
- Type: Logic, Seminars, Pure Mathematics
- Cost: Free
Rumen Dimitrov, Western Illinois University. Part of the Logic Seminar Series.
In the study of the structure of the lattice L^*(V_infty) of computably enumerable subspaces of V_infty modulo finite dimension, we encountered interesting nonstandard fields. These fields were used in  to characterize the principal filters, in L^*(V_infty), of the equivalence classes of quasimaximal spaces with extendable bases. Cohesive powers of computable structures were formally introduced in  and used in  to classify the orbits of such quasimaximal spaces in L^*(V_infty).
In this talk I will prove various model-theoretic properties of cohesive powers. I will also discuss the connection between Skolem's countable nonstandard model of PA (see ) and cohesive powers. In particular, I will discuss the connection between the proof of Skolem's combinatorial lemma (Satz 1 in ) and proofs of the existence of cohesive sets.
 Dimitrov, R.D. A class of Sigma^0_3 modular lattices embeddable as principal filters in L^*(V_infty). Arch. Math. Logic 47, pp. 111-132 (2008).
 Dimitrov, R.D. Cohesive powers of computable structures. Annuare De L'Universite De Sofia "St. Kliment Ohridski", Fac. Math. and Inf., tome 99, pp. 193-201 (2009).
 Dimitrov, R.D. and Harizanov, V. Orbits of maximal vector spaces. Algebra and Logic 54, pp. 680-732 (2015) (Russian); pp. 440-477 (2016) (English translation).
 Skolem, T. Über die nicht-charackterisierbarkheit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler aussagen mit ausschliesslich Zahlenvariablen. Fund. Math. 23, pp. 150-161 (1934).