Re: How many different unit fractions are lessorequal than all unit fractions?

Moebius <invalid@example.invalid> wrote:

Am 05.10.2024 um 15:57 schrieb Alan Mackenzie:

I first came across the terms "potential infinity" and "actual infinity"
on this newsgroup, not in my degree course a few decades ago.  I'm not
convinced there is any mathematically valid distinction between them.

Actually, there is.

But in classical mathematics "infinity" means "actual infinity", and
"potential infinity" is of no significance here.

"Cantor's work was well received by some of the prominent
mathematicians of his day, such as Richard Dedekind. But his
willingness to regard infinite sets as objects to be treated in
much the same way as finite sets was bitterly attacked by others,
particularly Kronecker. There was no objection to a 'potential
infinity' in the form of an unending process, but an 'actual
infinity' in the form of a completed infinite set was harder to
accept."

(Herb Enderton, Elements of Set Theory)

There was no objection to a 'potential infinity' in the form of an
unending process, but an 'actual infinity' in the form of a completed
infinite set was harder to accept." [H.B. Enderton: "Elements of set
theory", Academic Press, New York (1977) p. 14f]

So the notion of "potential infinity" is a historical artifact due to
things being unclear at the time these ideas were being worked out.
Also there is no need for either of the prefixes "potential" or "actual"
to qualify "infinity" in modern mathematics.

Though, if I remember correctly, "infinity" was not much used in my
degree course, except in expressions such as "tends to infinity", but
"infinite" was used all the time.

If there were, I would have heard of it back then.

Right.

Does "actual infinity" create a logical system?

-> classical mathematics = ZFC (or something like that) + classical logic.

So why is Richard writing that "actual infinity" doesn't exist?

--
Alan Mackenzie (Nuremberg, Germany).