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

On 10/05/2024 09:05 AM, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

Am 05.10.2024 um 15:57 schrieb Alan Mackenzie:

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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.

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Actually, there is.

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

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"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."

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(Herb Enderton, Elements of Set Theory)

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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]

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

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Right.

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Does "actual infinity" create a logical system?

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

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So why is Richard writing that "actual infinity" doesn't exist?
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The "classical 'quasi-modal' logic", of today,
is _not_ classical logic, where here,
"modern classical logic", is fundamentally
"exactly and only De Morgan's rules".
I.e. "material implication" after "ex falso quodlibet",
the "quasi-modal" part with false antecedent / false consequent
being "materially implied", is neither material nor implication,
so "classical logic" is perhaps better renamed
"Compte's Boole's Russell's logical positivism's not-quite classical
quasi-modal logic".
So, if you have material implication in your logic,
I think that's wrong or that it's simple to break it,
and if you don't, that's good, and if you don't know
whether you have "material implication" in your logic,
I imagine you assume you don't, yet, referring to yon
latest textbook might surprise whether or not
it does _not_ include the quasi-modal as 'classical',
or rather that it _qualifies_ what it is instead
of the burdening a rather more clear definition
of 'classical logic' with it.
Like, reading Langer's logic, it's like, "there here
is material implication, it's wrong-ish yet for
induction it's quite simple, so, know what it is
and know what it isn't, largely the rest of this
text will assume it's immaterial".
Then, "infinity", gets involved, "infinitary reasoning".