14 research outputs found
Vaught's conjecture for theories of discretely ordered structures
Let be a countable complete first-order theory with a definable,
infinite, discrete linear order. We prove that has continuum-many countable
models. The proof is purely first-order, but raises the question of Borel
completeness of
Around Podewski's conjecture
A long-standing conjecture of Podewski states that every minimal field is
algebraically closed. It was proved by Wagner for fields of positive
characteristic, but it remains wide open in the zero-characteristic case.
We reduce Podewski's conjecture to the case of fields having a definable (in
the pure field structure), well partial order with an infinite chain, and we
conjecture that such fields do not exist. Then we support this conjecture by
showing that there is no minimal field interpreting a linear order in a
specific way; in our terminology, there is no almost linear, minimal field.
On the other hand, we give an example of an almost linear, minimal group
of exponent 2, and we show that each almost linear, minimal group
is elementary abelian of prime exponent. On the other hand, we give an example
of an almost linear, minimal group of exponent 2, and we show that
each almost linear, minimal group is torsion.Comment: 16 page