Half the people who go into math instead of physics do so in order to avoid worrying about units.

If we want to be persnickety about this, I actually think Michael's original formula was correct given my statement of the problem. After all, I said \\(\Psi\\) was a feasibility relation from \\(\lbrace{0,1,2\rbrace}\\) to \\(\mathbb{N}\\). So, for \\(\Psi(x,y)\\) to make sense we need \\(x \in \lbrace0,1,2\rbrace \\). So it's possible to have \\(x = 1\\), but not possible to have \\(x = 1 \text{ bread}\\).

However, you could argue that in my problem I should have used the set \\(\lbrace 0 \text{ bread}, 1 \text{ bread}, 2 \text{ bread}\rbrace \\), not the set \\(\lbrace{0,1,2\rbrace}\\).

That's fine; I wouldn't argue with that. In applications, sets of numbers are 'typed'. That is, the set \\(\lbrace 0 \text{ bread}, 1 \text{ bread}, 2 \text{ bread}\rbrace \\) is not equal to \\(\lbrace 0 \text{ dollar}, 1 \text{ dollar}, 2 \text{ dollar} \rbrace \\), just isomorphic. I'm just too lazy to include units in all my sets of numbers. This amounts to treating various isomorphic sets as equal.