# On the Axiom of Infinity

Russell, in Principia Mathematica, says the following of his Axiom of Infinity:

"The axiom of infinity will be true in some possible worlds and false in others"

He is notoriously sheepish about its validity as an axiom and its use in his logical system has been largely rejected as an ad hoc manoeuvre in secondary literature (some Russell scholars do offer a defence of sorts).

My question is this:

What were Russell's reasons for adopting the axiom - were they sound, and can the axiom be dispensed of?

• In PM's system the axiom was of course necessary, in order to derive the basic law of numbers (natural and real), and this was the main goal of PM's enormous project; thus, in PM it cannot be dispensed of. Of course, the axiom was debatable (and debated) according to original W&R point of view regarding the possibility of deriving all of math starting only from logical principles (logicism). Axiom of infinity (as well as reducibility axiom) is hardly "logical". – Mauro ALLEGRANZA Apr 18 '14 at 15:26