Calendar

Sep
10
Tue
Section 1.5
Sep 10 @ 9:00 am – 10:00 am
Section 6.7
Sep 10 @ 10:30 am – 11:30 am
Sep
11
Wed
Logic seminar: David Webb
Sep 11 @ 2:30 pm – 3:30 pm
Sep
12
Thu
Section 1.6
Sep 12 @ 9:00 am – 10:00 am
Section 6.8
Sep 12 @ 10:30 am – 11:30 am
Sep
17
Tue
Section 2.1
Sep 17 @ 9:00 am – 10:00 am
Section 7.1
Sep 17 @ 10:30 am – 11:30 am
Sep
18
Wed
Talk to Karen Yamamoto
Sep 18 @ 8:30 am – 9:30 am
Logic seminar: David Webb
Sep 18 @ 2:30 pm – 3:30 pm

“Iterated ultrapowers for the masses”, part 2

Sep
19
Thu
Section 2.2
Sep 19 @ 9:00 am – 10:00 am
Section 7.2
Sep 19 @ 10:30 am – 11:30 am
Section 7.3
Sep 19 @ 10:30 am – 11:30 am
Sep
24
Tue
Section 2.3
Sep 24 @ 9:00 am – 10:00 am
Section 7.4
Sep 24 @ 10:30 am – 11:30 am
Sep
25
Wed
Logic seminar: Mojtaba Moniri
Sep 25 @ 2:30 pm – 3:30 pm

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.]

Sep
26
Thu
Section 2.4
Sep 26 @ 9:00 am – 10:00 am
Section 7.5
Sep 26 @ 10:30 am – 11:30 am
Oct
1
Tue
Midterm 1 MATH 307
Oct 1 @ 9:00 am – 10:00 am
Midterm 1 MATH 252A
Oct 1 @ 10:30 am – 11:30 am
Oct
2
Wed
Logic seminar: Mojtaba Moniri
Oct 2 @ 2:30 pm – 3:30 pm

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.

Oct
3
Thu
Section 2.5
Oct 3 @ 9:00 am – 10:00 am
Section 7.7
Oct 3 @ 10:30 am – 11:30 am
Oct
8
Tue
Section 5.1
Oct 8 @ 9:00 am – 10:00 am
Section 7.8
Oct 8 @ 10:30 am – 11:30 am
Section 8.5
Oct 8 @ 10:30 am – 11:30 am
Oct
9
Wed
Logic seminar
Oct 9 @ 2:30 pm – 3:30 pm
Oct
10
Thu
Section 5.1/5.2
Oct 10 @ 9:00 am – 10:00 am
Section 9.1
Oct 10 @ 10:30 am – 11:30 am
Oct
15
Tue
Section 5.2
Oct 15 @ 9:00 am – 10:00 am
Section 9.2
Oct 15 @ 10:30 am – 11:30 am