Question

The “Possible Missions” domain Let the constant m mean “the current mission”. Let the predicate M(x) mean “x is a mission”, and the predicate P(x) mean “x is possible”. We will call this the “Possible Missions” domain.

1a) Consider the set of predicate logic sentences we can write about the "Possible Missions" domain. Will this set be finite; infinite but countable; or not countable? Explain your answer.

1b) Let S be the set of predicate logic sentences from Q1a. Suppose we take the union of S and the real numbers, R. Is S ∪ R countable or uncountable? Explain your answer.

Answer 1

a) This set is finite and so countable. This is because a predicate statement is made by some combination of symbols (there exists, for all, negation) and in a certain order. The number of permutations of such symbols is finite so this is finite

b) R is uncountable so that S U R is a union of a countable and an uncountably infinite set. So this set is uncountably infinite

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