Section 3.3: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Show that the set of sequence numbers is representable (catalog item 10).

The set of sequence numbers is $\mathcal{S}=\{b|\forall x(x\le b\rightarrow x\in\mathcal{P}\rightarrow<x,b>\in\mathcal{D}$ $\rightarrow\forall y(y<x\rightarrow y\in\mathcal{P}\rightarrow<y,b>\in\mathcal{D})\}$ where $\mathcal{P}$ is the set of prime numbers, and $\mathcal{D}$ is the divisibility relation. In other words, if some prime number $x$ divides $b$ , then do all prime numbers $y$ less than $x$ . This, in particular, is vacuously true for $b=1=\langle\rangle$ and $b=2=\langle0\rangle$ . By catalog items 0, 3, 4 and 5, we conclude that $\mathcal{S}$ is representable.

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Section 3.3: Problem 5 Solution

Section 3.3: Problem 5 Solution