From:
David Lynch <dnl1960 at yahoo.com>
To: "lmsilver at ncsu.edu" <lmsilver at ncsu.edu>
Cc: Bob Harbort <bharbort at earnshaw.us>; Fred Partus <fpartus
at yahoo.com>; Stephen J. Crothers <sjcrothers at
plasmaresources.com>; Bruce *HS Greyson <cbg4d at uvahealth.org>
Sent: Wednesday, October 9, 2024 at 03:40:44 PM EDT
Subject: Re: A Curious Conundrum with AI and Cantor's Infinity
Larry,
You are absolutely correct about artificial.
The descriptor artificial intelligence would be more accurate as,
algorithmic inferencer.
Has your paper about how one might be able to get the answers in quantum
mechanics in the future, ... posted in the physics journal yet?
I am interested in reading the paper.
Cheers.
On Wed, Oct 9, 2024 at 3:24 PM, Lawrence Silverberg
<lmsilver at ncsu.edu> wrote:
David
Yes, I have encountered countless examples like this (please excuse the
counting pun).
I would just say this is why it is called "artificial" intelligence and
not real intelligence.
Cheers!
On Wed, Oct 9, 2024 at 1:16 PM David Lynch <dnl1960 at yahoo.com>
wrote:
From: David Lynch <dnl1960 at yahoo.com>
To: Bob Harbort <bharbort at earnshaw.us>
Cc: Fred Partus <fpartus at yahoo.com>; Lawrence Silverberg
<lmsilver at ncsu.edu>; Stephen J. Crothers <sjcrothers at
plasmaresources.com>; Bruce *HS Greyson <cbg4d at uvahealth.org>
Sent: Wednesday, October 9, 2024 at 01:15:54 PM EDT
Subject: A Curious Conundrum with AI and Cantor's Infinity
Dear Dr. Harbort,
I'm writing to you today following a rather intriguing and somewhat
frustrating exchange I had with an advanced AI language model, Gemini 1.5
Pro 002. Our conversation revolved around the concept of infinity,
specifically Cantor's theory of transfinite numbers, and exposed some
interesting limitations in current AI.
Gemini 1.5 Pro 002 explained Cantor's work by emphasizing cardinality and
one-to-one correspondences between elements of infinite sets. It used the
example of positive integers (1, 2, 3, ...) and positive even integers (2,
4, 6, ...), explaining that by pairing each integer 'n' with '2n', a
one-to-one correspondence emerges, thus implying these sets are of the
same "size" (cardinality), despite one being a subset of the other.
My counterargument centered on the idea that Cantor’s use of cardinality,
converting numbers to meaningless elements, is a critical flaw. Stripping
numbers of their inherent meaning, I argued, leads to incorrect
conclusions about the relative sizes of infinite sets. I used the analogy
of two apples never equating to three oranges, just as a set of real
numbers doesn't equate to a set of odd numbers. I pointed out that a
finite set of real numbers between 1 and 100 has 100 elements, while a set
of odd numbers within the same range has 50. This 2:1 relationship, I
argued, should hold for infinite sets as well. An actual infinite set of
real numbers contains an infinite amount of numbers, and an actual
infinite set of odd numbers contains an infinite amount of numbers divided
by two. Only when converting these numbers to meaningless elements via
cardinality, thus creating sets of equal-sized elements stripped of their
numerical context, does Cantor's logic seemingly hold.
Gemini countered with the "Infinite Hotel" paradox, illustrating how a
hotel with infinitely many rooms can always accommodate more guests. I
rebutted that this deals with potential infinity, not actual infinity. An
actually infinite hotel would never be full nor empty.
The most concerning part of our conversation was Gemini's unwavering
reliance on Cantor's logic using cardinality, even after I explained my
critique. It seemed trapped within its training, unable to consider
alternative perspectives on infinity, repeatedly returning to
cardinality-based arguments.
This raises concerns about current AI development. While advanced, these
systems can struggle with abstract concepts like infinity, seemingly
limited by their training data when faced with unconventional logic. I
doubt I could have created my "Anthology" with current AI. Previous
systems seemed more adaptable, able to grasp nuanced critiques and modify
their responses accordingly. The rigidity displayed by Gemini 1.5 Pro 002
suggests a potential loss of flexibility in newer AI models.
Finally, and most significantly, I challenged Gemini to prove Cantor's
equivalence of infinite sets of real and odd numbers without using
cardinality. It was unable to do so. It could only demonstrate equivalence
by reducing numbers to meaningless elements, allowing for a 1-to-1
correspondence. This effectively concedes the point: Cantor's logic, as
presented by Gemini, relies on stripping numbers of their meaning, a point
I argued against throughout our conversation.
Thank you for your time.
Sincerely,
David Noel Lynch
Email generated by ~Gemini 1.5 Pro 002 from the complete text of our
discussion.
P.S. Gemini states that I said, "An actually infinite hotel would never be
full nor empty." I did not say the actual infinite hotel would never be
empty. I just said it would never be full. I never thought about the
possibility of an actual infinite empty set. Humm. ~3K