The Formalization of Arithmetic and its Limitations
We now construct an axiom system A for arithmetic or, rather, for that part of elementary arithmetic which is concerned with the addition, multiplication, and exponentiation of natural numbers 0, 1, 2, 3, … . Although the scope of this axiomatization is rather narrow, we shall find that in a sense it encompasses a larger part of arithmetic than could have been anticipated. In addition, we shall find that there is a more or less uniform method for extending A whenever the need for a larger scope should arise. [In this respect the character of the system A is related to that of purely implicational logic.]
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