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Using Formulas to Classify Interpretations (Optional)

16 March, 2015 - 11:47
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In the previous section, having a formula was rather useless until we had a particular interpretation for it. But we can view that same idea backwards: Given a formula, what are all the interpretations for which the formula is true?

For instance, consider a formula expressing that an array is sorted ascendingly: For all numbers i,j, (i<j) ⇒ (element (i) ≤ element (j)). But if we now broaden our mind about what relations/functions the symbols element, < , and ≤ represent and then wonder about the set of all structures/interpretations which make this formula true, we might find that our notion of sorting is broader than we first thought. Or equivalently, we might decide that the notion "ascending" applies to more structures than we first suspected.

Similarly, mathematicians create some formulas about functions being associative, having an identity element, and such, and then look at all structures which have those properties; this is how they define notions such as groups, rings, fields, and algebras.