[FOM] As to strict definitions of potential and actual infinities.

[Dean Buckner]

Alexander Zenkin's posting only confirms my view that expressions like
"actual infinity" "potential infinity" "completed infinity" and the like are
vague, imprecise, incoherent &c.

The problem is the dependence on temporal notions or physical processes
(Aristotle's discussions focus either on cutting things up into parts, or
for waiting for things to be completed).  Thus non-logical elements obtrude
into what should be a perfectly logical definition.

Is it possible for Alexander to restate his definition without using
expressions like "arrives at" "continue [for]ever" and such like?

And there is nothing _obviously_ wrong with the non-naive definition of
actual infinity, which (as I understand it) simply postulates the existence
of a set of which every object in Zenkin's series is a member.  There is no
contradiction in every member n of the series being such that there exists
another member n+1, and that every such object belongs to the set, which is
not itself a member of the series.  And according to set theory such a set
is itself a number (but not a finite one).  There is no appeal to time or
process or becoming in this formulation.  On the face of it, there is no
contradiction

As John Mayberry has said, if there is something wrong with set theory,
which has been the subject of intense research over the last 100 years, then
it is nothing gross or obvious.

More subtle is Wittgenstein's point that "series" is ambiguous between an
extension, and a law (an ambiguity that lurks in Zenkin's definition, when
he talks of "the common *series* of the common finite natural numbers" -
what, all of them?).

W. imagines a machine that makes coiled springs, in which a wire is pushed
through a helically shaped tube to make as many coiled springs as needed.
What is called an infinite helix need not be anything like a finite piece of
wire: it is the law of the helix.  Hence the expressions "infinite helix"
and "infinite series" are misleading for the same reason.

Nowadays, the analogue for the helical tube that is the matrix for an
infinite long spring, is the bit of source code that loops to produce an
"unending" sequence of integers.

Wittgenstein's objections (in the final sections of Philosophical Grammar)
are neither gross nor obvious, more of which later.

Dean



Dean Buckner
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