Section 2.2: Problem 21 Solution
Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
We could consider enriching the language by the addition of a new quantifier. The formula (read “there exists a unique such that ”) is to be satisfied in by iff there is one and only one such that . Assume that the language has the equality symbol and show that this apparent enrichment comes to naught, in the sense that we can find an ordinary formula logically equivalent to .
Let . Suppose that variable is not a free variable of the wff , and let be where we substituted for all free occurrences of (note, that is not a free variable of ). Consider . We show that and are logically equivalent. For a structure and , iff there is unique such that iff there is such that and for any other , iff there is such that and for every , or iff there is such that and for every , or iff there is such that and for every , or iff there is such that and for every , ( is not a free variable of ) or iff there is such that and for every , iff there is such that and iff there is such that iff iff iff .