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I thought that algorithm satisfies the finiteness property.Means for every input it will stop after sometime. So I did P1 = P2.I am confused now.

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$P_1$ is the set of all Problems such that every problem has an algorithm and nothing is defined about the algorithm.

So we don’t know anything about the algorithm. i.e. $P_1$ could be thought as set of all algorithms.An algo may or may not halt.


$P_2$ is the set of all problems that have an algo that halts. i.e. $P_2$ could be thought as set of all algorithms that halts.

So obviously $P_2 \subset P_1$

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