07/07/2020, 05:00 PM
When Hilbert (1926, but in a footnote, it says “Vortrag, gehalten am 4. Juni 1925”) reported about Ackermann’s result, he used a slightly different function from the one that Ackermann used in his 1928 paper. Hilbert’s \( \phi_n(a,b) \) is actually the same as \( \mathrm{hyper}n(a,b) \). (He started at \( n=1 \).)
Hermann Schubert (1899) already considered the general concept of hyperoperations (implied by “and so on”) and addition as a direct operation of first order (“Stufe”), anticipating Hilbert. Here is what he wrote (translated from German):
Hermann Schubert (1899) already considered the general concept of hyperoperations (implied by “and so on”) and addition as a direct operation of first order (“Stufe”), anticipating Hilbert. Here is what he wrote (translated from German):
Quote:Addition and subtraction are called basic arithmetic operations of first order, multiplication and division of second order. Furthermore, addition and multiplication are called direct, subtraction and division indirect basic arithmetic operations. […] In the same way as multiplication emerges from addition, exponentiation from multiplication, one could derive from exponentiation as the direct operation of third order a direct operation of fourth order, from it one of fifth order etc. But already the definition of a direct operation of fourth order is, although logically justified, unimportant to the progress of mathematics, because the commutative law already loses its validity at the third order.