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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
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I assume we agree that the axiom says that there is at least one inductive set, i.e. a set X containing ∅ and with each set x the set x ∪ {x}. Well, I suppose Russell's reasons for adopting this axiom are those of the ordinary mathematician: It suits the purpose of set theoretically representing mathematical objects. With a little bit of Aussonderung the axiom gives you the smallest inductive set, which is a formidable stand-in for the set of natural numbers. Furthermore, the axiom is the basis for constructing the real numbers as Dedekind cuts. If these are Russell's reasons they seem sound to me.

Can the axiom be dispensed with? Well, if the question is whether there is a model of the set theory resulting from ZFC by replacing the axiom by its negation, then the answer is yes. This model is the set of the hereditarily finite sets, i.e. the sets Vi such that V0 is ∅ and Vn+1 is the power set of Vn.

But if the question is whether such a model has mathematically nice properties, then the answer is clearly no. Note that the above model is countable. But the real numbers, as Cantor has shown, are uncountable. So, there seems to be no way to represent the reals in such a universe set theoretically.

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    Principia Mathematica's theory is not set theory. PM's axiom of infinity asserts the existence of an infinite number of individuals. – Mauro ALLEGRANZA Apr 18 '14 at 6:31
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    I don't think we do agree on that. As @Mauro has pointed out, Russell didn't use the axiom is the context of the constructible hierarchy in the same way, for example, Godel did. Hence his quote - I interpret his claim more as a (general) metaphysical one... – Mathmo Apr 18 '14 at 10:35
  • Perhaps it has escaped your notice, but there exist countable models of the reals. The fact that the universe of hereditarily finite sets is countable does not disprove that it can represent the reals. Of course I do think it does not contain a standard model of the real numbers, but we must find another reason than its cardinality to prove this. – ziggurism Jan 18 '18 at 14:15

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