A few hours later, I find I remain physically pained when attempting to come to terms with the notion that a person could express the belief that only genetically modified tomatoes have genes. The very ability to formulate the statement requires knowledge that contradicts the belief.

I do not find this belief paradoxical. Folk science GMO in one sentence goes like this: you grab a normal tomato, and you add some genes to it, that make this tomato bigger, or less prone to mold, or something. One does not need to comprehend high-school genetics to get this stub of an idea.