I can see two ways in which they "play the same role":

• \$$\text{false}\in\textbf{Bool}\$$ and \$$\infty\in\textbf{Cost}\$$ are both bottom elements with respect to the partial order.

ie \$$\text{false}\leq x\$$ for all \$$x\in\textbf{Bool}\$$ and \$$\infty\leq x\$$ for all \$$x\in\textbf{Cost}\$$

• they are both zero elements with respect to the monoidal product:

ie \$$\text{false}\wedge x = \text{false}\$$ for all \$$x\in\textbf{Bool}\$$ and \$$\infty + x = \infty\$$ for all \$$x\in\textbf{Cost}\$$