Solve the following problems from Chapter 3 of the textbook: 2, 4, 5, 7, 9, 10, 26.

In addition, solve the following problem:

Let be a linear map between finite dimensional vector spaces over For we denote by the preimage of under This is the set of all vectors such that

Recall that a subset of a vector space over is convex iff whenever and then also Show that is convex iff is convex. Is it necessary that is linear for this to hold, or does it suffice that is continuous?

Assume that is a subspace of and show that is a subspace of

43.614000-116.202000

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When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.

I am not sure which statement you heard as the "Ultimate $L$ axiom," but I will assume it is the following version: There is a proper class of Woodin cardinals, and for all sentences $\varphi$ that hold in $V$, there is a universally Baire set $A\subseteq{\mathbb R}$ such that, letting $\theta=\Theta^{L(A,{\mathbb R})}$, we have that $HOD^{L(A,{\ma […]

A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathbb R)$, then $\Gamma$ is a Wadge initial se […]

Craig: For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the current records are. Perhaps the first pape […]

Yes. Consider, for instance, Conway's base 13 function $c$, or any function that is everywhere discontinuous and has range $\mathbb R$ in every interval. Pick continuous bijections $f_n:\mathbb R\to(-1/n,1/n)$ for $n\in\mathbb N^+$. Pick a strictly decreasing sequence $(x_n)_{n\ge1}$ converging to $0$. Define $f$ by setting $f(x)=0$ if $x=0$ or $\pm x_n […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

I feel this question may be a duplicate, because I am pretty certain I first saw the paper I mention below in an answer here. You may be interested in reading the following: MR2141502 (2006c:68092) Reviewed. Calude, Cristian S.(NZ-AUCK-C); Jürgensen, Helmut(3-WON-C). Is complexity a source of incompleteness? (English summary), Adv. in Appl. Math. 35 (2005), […]

The smallest such ordinal is $0$ because you defined your rank (height) inappropriately (only successor ordinals are possible). You want to define the rank of a node without successors as $0$, and of a node $a$ with successors as the supremum of the set $\{\alpha+1\mid\alpha$ is the rank of an immediate successor of $a\}$. With this modification, the smalles […]

The perfect reference for this is MR2562557 (2010j:03061) Reviewed. Steel, J. R.(1-CA). The derived model theorem. In Logic Colloquium 2006. Proceedings of Annual European Conference on Logic of the Association for Symbolic Logic held at the Radboud University, Nijmegen, July 27–August 2, 2006, S. B. Cooper, H. Geuvers, A. Pillay and J. Väänänen, eds., Lectu […]

Consider $A=\{(x,y)\in\mathbb R^2\mid x\notin L[y]\}$. Check that this set is $\Pi^1_2$ (this is similar to the proof that there is a $\Delta^1_2$ well-ordering in $L$). The point is that $A$ does not admit a projective uniformization. It does not really matter that the number of Cohen reals you added is $\aleph_2$; any uncountable number would work. The rea […]

When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.

Yes, I am thinking of as and I would like a proof or a counterexample in this context.