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116b- Homework 9

March 11, 2008

Due Tuesday March 18 at 1pm.

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116b- Homework 8

March 5, 2008

Due Tuesday March 11 at the beginning of lecture.

Important update: In problem 4.(a), recall that $U^1$ is a universal $\Sigma_1$ predicate for unary formulas, so if $x$ is the Gödel number of a $\Sigma_1$ formula $\psi(v)$ in one free variable $v$ , then $U^1_x(y)$ holds iff $\psi(y)$ holds. Hence, asking that $U^1_x$ is finite is the same as asking that $\{n:{\mathbb N}\models \psi(n)\}$ is finite. Actually, this is a serious typo:

$\{x:U^1_x\ \mbox{is finite}\}$ is $\Sigma_{\bf 2}$ -complete. The set $\{x\colon U^1_x\ \mbox{is cofinite}\}$ is $\Sigma_3$ -complete.

Sorry about this. Either ignore 4.(a), or try to show (for extra credit) that the set is $\Sigma_2$ -complete, or (for a much more challenging problem) that the corresponding set with “cofinite” is $\Sigma_3$ -complete.

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116b- Homework 7

February 25, 2008

Due Tuesday March 4 at the beginning of lecture.

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116b- Homework 6

February 20, 2008

Due Tuesday February 26 at the beginning of lecture.

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116b- Homework 5

February 13, 2008

Due Tuesday February 19 at the beginning of lecture.

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116b- Homework 4

February 4, 2008

Due Tuesday February 12 at the beginning of lecture.

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116b- Homework 3

January 29, 2008

Due Tuesday February 5 at the beginning of lecture.

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116b- Homework 2

January 22, 2008

Due Tuesday January 29 at the beginning of lecture.

Update: There is a typo in problem 2, it must be $z<y$ .

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116b- Homework 1

January 15, 2008

Due Tuesday January 22 at the beginning of lecture.

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117b – Homework 8

March 1, 2007

Due Tuesday, March 13 at noon.

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