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.