Polynomial rings and weak second-order logic. (English) Zbl 0592.03006

Bauval, Anne

J. Symb. Log. 50, 953-972 (1985).

Given a model M of a first order language L, extensions of M to four multisorted extensions of L are given; they deal with finite subsets, finite sequences and hereditarily finite sets on M, so the theory of each of these four extensions might be called ”the” weak second order theory of M. They are interpreted one in another as far as possible, and the weak second order theory of F is proved equivalent to the first order theory of the polynomial ring \(F[X_ i]_{i\in I}\), uniformly in any field F and any non-empty set I. Some results are also given about more general polynomial rings, especially F[G], where F is a field and G a commutative orderable monoid.

MSC:

03B15	Higher-order logic; type theory (MSC2010)
12L12	Model theory of fields
03C60	Model-theoretic algebra

Keywords:

interpretation; multisorted extensions

Citations:

Zbl 0563.12027

References:

[1]	DOI: 10.2307/2316784 · Zbl 0169.05403
[2]	Teilweise geordnete algebraische Strukturen (1966)
[3]	Algebra (1965) · Zbl 0193.34701
[4]	Model theory (1977)
[5]	DOI: 10.1007/BFb0090946
[6]	DOI: 10.1016/S0049-237X(08)71350-8
[7]	DOI: 10.1090/S0002-9947-1951-0041081-0
[8]	Le défi algébrique 2 (1976)
[9]	Introduction to mathematical logic (1968)
[10]	Notices of the American Mathematical Society 5 pp 673– (1958)