Countable Nonstandard Models and Cohesive Powers

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 [1] 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 [2] and used in [3] 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 [4]) and cohesive powers.  In particular, I will discuss the connection between the proof of Skolem's combinatorial lemma (Satz 1 in [4]) and proofs of the existence of cohesive sets.

[1] 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).

[2] 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).

[3] 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).

[4] 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).