Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,…,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if $y<n$, then $x*(y+1)=(x*y)*(x*1)$. The algebra $(S_{n},*)$ satisfies the self-distributivity law $x*(y*z)=(x*y)*(x*z)$ if and only if $n$ is a power of $2$, and if $n$ is a power of $2$, … Read more
Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 6 years ago. Improve this question In the thread “Is platonism regarding arithmetic consistent with the multiverse view in set theory?“, Prof. Hamkins writes: The view you are suggesting is … Read more
I am curious what braid groups (strings in R3) are NIP. Consider the following: Let BN be braid group with “braids” indexed by the natural numbers (alternatively, the direct limit of braid groups on finitely many braids with inclusion maps between them). This structure (in the language of groups) is not NIP. Proof: There is … Read more
Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M’\subsetneq M$ such that: $M’\models PA^-$ (or $Q$) $a\not \in |M’|$ there exists $b\in M’$ such that $M\models a<b$ $M’\models T$ Q1. For which $M\models PA$, $a\in … Read more
Background: In “Stability under integration of sums of products of real globally subanalytic functions and their logarithms”, by R. Cluckers and D.J. Miller, it is shown that the integral of a semialgebraic function f:Rm×Rn→R over Rn may be represented as a log-analytic function over Rm. (A log-analytic function is a function belonging to the algebra … Read more
One natural way to show that a forcing adds no new reals is to show that it is countable closed (EDIT: this is somewhat misleading, see Joel’s comment below). However, it turns out that this is overkill: there are forcings not adding any reals which are not countably closed, in particular, which do add a … Read more
Is there any model of NF (New Foundations) on HoTT (homotopy type theory)? Because there is a model of ZF(C) on HoTT (The HoTT book, Section 10.5) and NF on ZFC (by this Wikipedia articile), I think anyway we have a model of NF in that way. I am looking for easier ways to define … Read more
My question is about two types of consequence of large cardinals, considered over ZFC on their own. First, we have statements of the form, “The club filter on ω1 is an ultrafilter when restricted to ‘reasonably nice’ sets of ordinals.” For Γ a class of formulas, let “CD(Γ)” denote the statement Every Γ-definable set of countable … Read more
There is a notion of “proof theoretic ordinal” for a formal theory https://en.wikipedia.org/wiki/Ordinal_analysis Can we go backwards? That is, we are given some recursive ordinal notation (we don’t know if it’s indeed ordinal notation). Is there some canonical way to construct formal theory corresponding to this ordinal? I realize that probably not every ordinal is … Read more
In our background universe V – satisfying ZFC – we say that an ordinal δ is a Woodin cardinal iff it satisfies one of the following equivalent properties: For all A⊆Vδ there is a cardinal κ<δ such that for all ν<δ there is a definable elementary embedding j:V≺M, M transitive, such that crit(j)=κ, j(κ)>ν and … Read more
