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Defining Certain Subsets of the Natural Numbers

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Let L be the language of addition (with equality) in first-order logic. That is, let L be the first-order language that allows for use of the equality symbol ("=") and whose only non-logical symbol is a binary function symbol "+". (That is, L has no constant symbols and no predicate (relation) symbols.)

Now consider the L-structure that has as universe the set N of all the natural numbers (0, 1, 2, ...) and where the function symbol "+" is interpreted as the usual addition on the natural numbers.

(a) Use an L-formula of first-order logic to define the set E_1 of all the even natural numbers.

(b) Use an L-formula of first-order logic to define the set E_2 of all the even natural numbers that are not divisible by 4.

(c) Use an L-formula of first-order logic to define the set S={1} (i.e., S is the one-element subset of the natural numbers whose only element is the number 1).

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Solution Summary

A certain language of first-order logic is given, and formulas in that language are then used to define certain subsets of the set of all the natural numbers. In addition to provision of the actual formulas, a detailed explanation of the choice of those formulas is provided.

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