The Mouse Set Theorem Just Past Projective

We identify a particular mouse, $M^{\text{ld}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{\sharp}$ for all $n$, and we show that $\mathbb{R}\cap M^{\text{ld}} = Q_{\omega+1}$, the set of reals that are $\Delta^1_{\omega+1}$ in a countable ordinal. Thus $Q_{\omega+1}$ is a mouse set. This is analogous to the fact that $\mathbb{R}\cap M^{\sharp}_1 = Q_3$ where $M^{\sharp}_1$ is the the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are $\Delta^1_3$ in a countable ordinal. More generally $\mathbb{R}\cap M^{\sharp}_{2n+1} = Q_{2n+3}$. The mouse $M^{\text{ld}}$ and the set $Q_{\omega+1}$ compose the next natural pair to consider in this series of results. Thus we are proving the mouse set theorem just past projective. Some of this is not new. $\mathbb{R}\cap M^{\text{ld}} \subseteq Q_{\omega+1}$ was known in the 1990's. But $Q_{\omega+1} \subseteq M^{\text{ld}}$ was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin's proof.Comment: 30 page

THE AREA AND CIRCUMFERENCE OF A

In grade school we were all given the formulas for the area and circumference of a circle: A = πr 2 and C =2πr where π ≈ 3.14159. Most likely these formulas were given with no justification, or even an intuitive explanation as to why they are true. In this article I will discuss the two formulas above, I will give elementary proofs of them, and I will discuss their history. To begin with, how should one interpret the formulas A = πr 2 and C =2πr? It is tempting to take the point of view that first there was this number π “out there, ” and subsequently it was discovered that π is useful for calculating the area and circumference of a circle. This of course would be misleading. The formulas A = πr 2 and C =2πr should be interpreted as both a theorem in geometry, and the definition of π. The theorem says that there is some constant k, such that for all circles, the area and circumference of the circle are given by: A = kr 2 and C =2kr. The definition says, let us agree to use the symbol π to refer to this constant. To be more precise, as I see it, there are at least seven important ideas associated with the area and circumference formulas given above. I describe these seven ideas below. (In order to understand what follows, it would be helpful to pretend for a moment that you have never heard of the number π. Then I will define π for you below.) Idea 1. The circumference of a circle is directly proportional to the radius of the circle. That is, there is some constant k such that for all circles, C = kr. This implies for instance that if you double the radius of a circle, then you double its circumference. Idea 2. The area of a circle is directly proportional to the square of the radius of the circle. That is, there is some constant h such that for all circles, A = hr 2. This implies for instance that if you double the radius of a circle, then you quadruple its area. Idea 3. The two constants of proportionality mentioned above are related to each other by the equation k =2h. Notation. Given Idea 3 above, let us agree to use the symbol π to refer to the constant h. Then Idea 1 says that C =2πr and Idea 2 says that A = πr 2. Idea 4. The number π is approximately equal to 3. To be more precise, 3