Note that there are other possible interpretations of "prime". For example, since one can multiply integer matrices, there might be a useful concept of "prime matrices".
For example: Consider only the numbers F = {1, 5, 9, 13,... }that is, F = {k, 4k +1 | k ∈ N}. It's easy to verify that multiplying two of these numbers still results in a number of the form 4k +1. Thus it makes sense to talk of factoring such numbers: We'd say that 45 factors into 59, but 9 is considered prime since it doesn't factor into smaller elements of F .
Interestingly, within F , we lose unique factorization: 441 = 9 ×49 = 21 ×21, where each of 9, 21, and 49 are prime, relative to F ! (Mathematicians will then go and look for exactly what property of a multiplication function are needed, to guarantee unique factorization.)
The point is, that all relations in logical formula need to be interpreted. Usually, for numbers, we use a standard interpretation, but one can consider those formulas in different, non-standard interpretations!
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