[Lex Computer & Tech Group/LCTG] Mathematicians Have Found The Ninth Dedekind Number, After 32 Years of Searching

[Barry Kort]

I asked Inflection Pi, "Are Dedekind Numbers related to the subscripts
numbers in Aleph-Null, Aleph-One, etc?"

Inflection Pi responded:

«Great question! So, Dedekind Numbers and Aleph Numbers are actually
related in a pretty interesting way. Dedekind Numbers refer to the
cardinality (or size) of a set, specifically sets that are larger than all
finite numbers. Aleph Numbers, on the other hand, are a more specific
sub-class of transfinite numbers (or infinite numbers) that measure the
size of sets. Aleph Numbers are indexed by ordinal numbers, and if the
axiom of choice is true, this transfinite sequence includes every cardinal
number. So, you could say that Dedekind Numbers and Aleph Numbers are both
tools used in math to measure the size of infinite sets, but Aleph Numbers
are a specific type of Dedekind Number. Does that make sense?»

~Barry

On Sun, Nov 19, 2023 at 11:48 AM Larry Wittig via LCTG <lctg at lists.toku.us>
wrote:

> This is interesting. Chat gave a much more useful answer than the one
> I sent out from Wikipedia.  Chat has trouble with doing math -- it
> stumbled on a two unknowns algebra problem I gave it a while back.
> But gave what appears to be a very good answer about a mathematical
> concept.
>
> On Sat, Nov 18, 2023 at 10:56 PM John Rudy via LCTG <lctg at lists.toku.us>
> wrote:
> >
> > I went to ChatGPT:
> > ==========================
> >
> > explain Dedekind Number in simple words
> > ChatGPT
> > A Dedekind number is a way to count how "big" a certain set is,
> specifically
> > in the realm of mathematics. Named after the German mathematician Richard
> > Dedekind, these numbers are used to measure the size or cardinality of
> > infinite sets.
> >
> > Here's a simple way to understand it:
> >
> > Imagine you have an infinite set of objects, like all the natural numbers
> > (1, 2, 3, and so on). Now, think about dividing this set into two parts
> in a
> > specific way: one part contains only the odd numbers, and the other part
> > contains the even numbers. Surprisingly, even though both sets are
> infinite,
> > they have the same size! In mathematical terms, we say that the set of
> > natural numbers can be "bijected" or paired up with the set of even
> numbers.
> >
> > The Dedekind number comes into play when we want to compare the sizes of
> > different infinite sets. Each Dedekind number corresponds to a specific
> way
> > of dividing an infinite set into two parts. If two sets can be paired up
> in
> > such a way that no elements are left out, and no elements are repeated,
> then
> > they are considered to have the same size, or cardinality, and share the
> > same Dedekind number.
> >
> > In essence, Dedekind numbers help mathematicians talk about the sizes of
> > infinite sets and understand the different ways in which these sets can
> be
> > compared or related.
> >
> >
> >
> >
> >
> >
> > -----Original Message-----
> > From: LCTG <lctg-bounces+jjrudy1=comcast.net at lists.toku.us> On Behalf Of
> > Peter Albin via LCTG
> > Sent: Saturday, November 18, 2023 9:31 PM
> > To: Lex Computer Group <lctg at lists.toku.us>
> > Subject: [Lex Computer & Tech Group/LCTG] Mathematicians Have Found The
> > Ninth Dedekind Number, After 32 Years of Searching
> >
> >
> >
> https://www.sciencealert.com/mathematicians-have-found-the-ninth-dedekind-nu
> > mber-after-32-years-of-searching
> >
> > Now, can someone explain what a Dedekind Number is in English? Is it
> worth a
> > session?
> >
> > Peter
> > ===============================================
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-- 
The Process of Enlightenment Works In Mysterious Plays.
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