I don't know if this doubt is very silly or not but is really bothering me.
I went through this answer. Here, in the most upvoted answer, the logic is binary coding does not get rid of the nonsensical quantitative relationship. The answer further suggests that dummy variables do not suffer from the same problem.
For example, if we have an unordered categorical variable with 4 levels, we cannot use (0,0),(0,1),(1,0),(1,1) to show them as these are binary equivalent of 0,1,2,3 and numerical relationship is retained. But we can instead go with (0,0,0),(0,0,1),(0,1,0),(1,0,0).
My doubt is why can't we consider it as binary equivalent of 0,1,2,4 and hence consider it to have the similar sort of numerical relationship? How does this solve the problem which binary coding was not able to?