“Iterated ultrapowers for the masses”, part 2

Comparing Near-linearity Notions in Open Induction

There have been works in number theory on characterizing the class of Beatty sequences (integer parts of natural multiples of a fixed nonnegative real slope). The same is true for the inhomogeneous case when a fixed intercept is added before taking the integer part. We consider some notions of multiplicative or additive near-linearity and elaborate on the extent to which they charecterize various such sequences. We show some implications from standard number theory carry over to Open Induction and some do not. [In a second talk we could relate this to the weak fragment allowing the standard integers as a direct summand of a model. That second talk would include two more multiplicative vs. additive topics, details to follow.]

Some additive vs. multiplicative issues in subrecursivity, maximality, and near-linearity

We deal with three topics around addition without or with multiplication.

We first present algorithms to compute a certain real, generating its Beatty sequence or base 2 expansion. The former calculates in integers with addition, in conjunction with the counting operator. The latter calculates in integers with addition and multiplication. Motivation comes from subrecursive reals.

Next, let F be an ordered field, D a maximal discrete subring of F, and G a maximal discrete additive subgroup of F. We point out that although there are examples where F has elements of infinite distance to D, it can never realize any gaps of G. If F is countable, then G can be constructed Delta^0_2 relative to F.

Finally we finish and extend the talk of last week by considering some nonstandard models M of weak arithmetic which have the integers as an additive direct summand. We present functions f and g from M to M whose value at a sum minus sum of values is always 0 or 1 yet for some x,y,u,v ≥ 1in M, we have f(xy) ug(v) + u – 1.

Assessing the Reverse Mathematical Strength of Gratzer-Schmidt Theorem
Gratzer-Schmidt theorem in lattice theory states that all complete and compactly generated lattices are isomorphic to the congruence lattice of an algebra. There has been an effort to assess the strength of this theorem in the reverse mathematical setting. I will discuss my recent progress on this topic and its potential implications.