 A consequence of Gödel’s incompleteness theorems is that under the assumption that PA is consistent one can find a finitistic statement such that neither it nor its negation is provable from Peano Arithmetic. Gödel showed, through subtle coding of metamathematical notions in the language of arithmetic, how to express in the language of arithmetic a formula \(G\) that “says” of itself that it is not provable. One can also ascertain that the formula is true on the natural numbers. Since all the finitary reasoning is assumed to be included in PA, establishing the truth of the Gödel sentence and of the new incompleteness results requires appeal to some “infinitary” principles (when the truth of the Gödel sentence \(G\) is established through appeal to the statement expressing the consistency of PA, it is establishing the latter that requires some portion of infinitary reasoning, such as induction up to an infinite ordinal called \(\varepsilon_0)\).