Griceian Infinity and the Empty Set -- ø --

Speranza

Just for the record then I am pasting R. B. Jones's two commentaries under different posts, as they deal with the connection of the idea of 'infinity' with that of 'empty set'.

The first commentary by R. B. Jones concerns this:

The axiom of infinity re

(AI)

"There is a set I ( which is postulated to be infinite), such that
-- the empty set is in I
 & such that
-- whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I.

Jones comments:

"Note that this formulation ... presumes that we have a constant whose name is the usual symbol for the empty set"

Viz.

 ø

"It might appear that this enables the existence of the empty set to be proven, but this is an illusion."

"Why?"

"Well, first of all, if our language L contains a constant C [and not just specific "ø"]  then we can prove "there exists x such that x = C" in first order logic, without benefit of any set-theoretic axioms at all."

I guess this relates to Grice's Vacuous Names, although his example there is "Marmaduke Bloggs", rather than the empty set -- although I think I have discussed with R. B. Jones the few references to the empty set by Grice in his "Reply to Richards" (which I should recheck) -- in Grice's criticism to extensionalism.

Jones goes on:

"However, we can't prove anything about C without some axioms, so we can't prove that C has no members. Mentioning C in the pivotal role in [AI] above makes no difference. You still can't prove   ø has no members from [AI]. In fact, [AI] works perfectly well whatever set plays that pivotal role."

"[AI] can be simplified so that it only states the existence of a NON-empty set closed under succession: the function from x to x u {x}."

"[AI], as stated, does not prove the existence of an empty set in a context in which that is not already provable. Well, not in a NON-CONTRIVED context. It would, of course, in a context in which there was an axiom (or theorem) asserting that the existence of I (the infinite set) entailed the existence of the empty set."

which then connects with the other post I had submitted. This referred to there existing an inductive set, whose members are

(i) the empty set;

(ii) for every member y of x, is also a member of x.

Infinity can be formulated so as to imply the existence of the empty set."

On that remark, Jones had commented:

"I find this observation rather puzzling ... since the existence of the empty set is usually taken to be an axiom ... even though it can be derived from other axioms. ...  The existence of

 ø

follows from the Separation Axiom Schema, AND from the Replacement Scheme (from which Separation can be obtained)."

"To understand what is intended here ... one really needs to know in WHAT context infinity [the set I] SUFFICES to derive the existence of [ø], since in the context of ZFC or NBG - {empty set, infinity} the existence of the empty set is already provable."

Which are excellent points and which I should be able to connect with Grice's 'infinitely many stars' -- or not! while I try to retrieve what Grice said on the infamous (is it?) null set.