Due Tuesday February 26 at the beginning of lecture.
116b- Homework 6
February 20, 2008
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116b: Computability and incompleteness | 
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A kind of library
Andrés E. Caicedo
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Mathoverflow- Answer by Andrés E. Caicedo for Riemann series theorem and uncountable number of sums which sum to every value November 1, 2019Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0Andrés E. Caicedo
- Answer by Andrés E. Caicedo for "Inner Regularity" of probability measure on totally ordered sets October 7, 2019Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Question about Jech's proof of V = L implies GCH June 3, 2019You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$ January 16, 2019Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact all initial ordinals, and make the contin […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Arguments against large cardinals October 29, 2010I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". Though some serious set theorists found it of possible interest immediately following it […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Riemann series theorem and uncountable number of sums which sum to every value November 1, 2019
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- Answer by Andrés E. Caicedo for How many integer solutions are there to $x+y+z=8$ February 12, 2020For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Cofinality of $2^{ February 10, 2020I assume you want to require that $\kappa$ is a limit cardinal; otherwise, if $\kappa=\rho^+$ then $2^{Andrés E. Caicedo
- Answer by Andrés E. Caicedo for What does it mean when we say that stationary sets are analogous to sets of non-zero measure? February 7, 2020Given a complete measure space $X$, consider the collection $F$ of full measure sets. These are the sets whose complement has measure 0. Note that if $Y\in F$ then any superset of $Y$ (that is, any $Z$ with $Y\subseteq Z\subseteq X$) is in $F$ as well. (This is why I require the measure to be complete, which means that any subset of a measure zero set is mea […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Help with proof of existence of $G_\delta$ sets $G$ such that both $G$ and its complement have uncountable intersection with any open interval January 30, 2020Although the $G_j$ are decreasing, their intersection is far from just being the rationals. As indicated in the original post, in fact $G$ meets each open interval in uncountably many points. A nowhere dense subset of $\mathbb R$ is a set $X\subseteq \mathbb R$ with the property that any nonempty open set $U\subseteq\mathbb R$ contains a nonempty open subset […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Countable and uncountable number December 12, 2019A set $G\subseteq \mathbb R$ is $G_\delta$ (that is, a countable intersection of open sets) if and only if there exists a function $f\!:\mathbb R\to\mathbb R$ that is continuous at each point of $G$ and discontinuous at each point of its complement. The result you are requesting follows from noticing that there are $G_\delta$ sets $G$ such that both $G$ and […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for How many integer solutions are there to $x+y+z=8$ February 12, 2020
