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.

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.