 Some mathematicians and philosophers have adopted finitism not merely as a methodological viewpoint, but also as a metaphysical one. Finite objects, like numbers and sentences, exist (in whatever sense mathematical objects exist), but infinite objects (like the completed set of all the natural numbers, or even arbitrary irrational numbers represented by Dedekind cuts) don’t. Versions of this view are often attributed to the 19th century number theorist and algebraist, Leopold Kronecker, who is quoted as saying “The dear God created the whole numbers; everything else is the work of man.” Kronecker criticized Cantor’s work as theology rather than mathematics. Hilbert started his program with the intent of defending Cantor while working within a framework dialectically acceptable to Kronecker’s allies. But when Kurt Gödel proved that no finitary theory for arithmetic and syntax can even prove its own consistency, let alone prove the consistency of a stronger theory for talking about completed infinities, Hilbert’s program was taken to have failed at disabusing the metaphysical finitists. Gödel’s incompleteness theorems apply most notably to Peano Arithmetic. The language of Peano Arithmetic is given by \(\{0,\,',\, +,\,\times\}\), where \('\) is the successor function (it adds 1 to each number). Within it one can express ordinary arithmetical claims such as the commutativity of addition and the infinitude of prime numbers. The axioms of Peano Arithmetic tell us that the function \('\) is one-one; that 0 is not the successor of any number; that \(+\) and \(\times\) satisfy the usual recursive definitions; and finally we have a schema of induction for every formula \(A(x)\) expressible in the language, i.e. if \(A(0)\) and for all \(x, A(x) \rightarrow A(x')\), then for all \(x, A(x)\).