>**Puzzle 186.** I said that \$$\infty\$$ plays the same role in \$$\textbf{Cost}\$$ that \$$\text{false}\$$ does in \$$\textbf{Bool}\$$. What exactly is this role?

Both play the role of showing that two objects are disconnected.

In the case of \$$\textbf{Bool}\$$, \$$\Phi(x,y) = \text{false}\$$ means \$$x \not\leq y\$$.

In the case of \$$\textbf{Cost}\$$, \$$\Phi(x,y) = \infty\$$ means \$$x \overset{\infty}\rightarrow y\$$ that is to say, that you'll never have enough of the required resource to get from \$$x\$$ to \$$y\$$, regardless of how must you try to acquire.

Both describe the situation, \$$x \not\rightarrow y\$$.