Description: The existence of omega (the class of natural numbers). Axiom 7 of
TakeutiZaring p. 43. This theorem is proved assuming the Axiom of
Infinity and in fact is equivalent to it, as shown by the reverse
derivation inf0 .

A finitist (someone who doesn't believe in infinity) could, without
contradiction, replace the Axiom of Infinity by its denial
-.om e. V ; this would lead to _om = On by omon and
Fin = _V (the universe of all sets) by fineqv . The finitist
could still develop natural number, integer, and rational number
arithmetic but would be denied the real numbers (as well as much of the
rest of mathematics). In deference to the finitist, much of our
development is done, when possible, without invoking the Axiom of
Infinity; an example is Peano's axioms peano1 through peano5 (which
many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994)