[seqfan] Re: Two "dumb" sequences and a question

[Allan Wechsler]

I think the fast answer is "no". In response to Russell's paradox ("Let L
be the set of all sets that do not contain themselves. Does L contain L?")
the axioms for set theory were rigorized carefully by Ernst Zermelo (1908),
and Zermelo's system was critiqued and amended by Abraham Fraenkel and
Thoralf Skolem (c. 1922). The resulting system, the Zermelo-Fraenkel
axioms, explicitly forbid self-reference of this kind, largely as a
consequence of the brilliantly-lawyered "Axiom of Regularity". So in
standard set theory, no set can contain itself, and the issue does not
arise.

On Wed, Nov 30, 2022 at 5:50 PM Ali Sada <pemd70 at yahoo.com> wrote:

> Thank you very much Allan. And you are absolutely right. It was I who
> created an unnecessary paradox by ignoring some of the basic elements of
> the OEIS. Please let me explain.
>
> I lumped up two sequence in one. The correct sequences should have been:
>
> 1) Sequence AX contains all OEIS sequences where the A number is a term in
> the sequence itself;
> 2) Sequence AY contains all OEIS sequences where the A number is not a
> term in the sequence; and
> 3) Sequence AZ contains all OEIS sequences we don’t know if the A number
> is a term in the sequence or not.
>
> In my opinion, these sequences "existed" the minute Neil created the OEIS
> and I did not have the right to manipulate them.
>
> Is what I said above accepted within set theory? Can we create sets that
> defy the logic of already existed structures?
>
>
> Best,
>
> Ali
>
>
> On Wednesday, November 30, 2022 at 09:27:49 PM GMT+1, Allan Wechsler <
> acwacw at gmail.com> wrote:
>
>
> You don't have to go that far to create a sequence that is "paradoxical"
> in this sense. Suppose we try to add to OEIS a sequence -- say, A400000 --
> whose title is "List of nonnegative integers not included in A400000", or
> more simply, "Complement of A400000". Now we can't decide whether *any *integer
> belongs there. For example, should 17 go in? If the answer is yes, then it
> shouldn't, and if the answer is no, then it should.
>
> I think for the moment all we have to say is that there is a kind of
> noxious self-reference that makes a sequence unsuitable for inclusion in
> OEIS, and doing any kind of math with sequence numbers is a warning sign of
> this.
>
> Under that approach, the answer to Ali's actual question is, "This
> question won't arise, because that kind of sequence is not suitable for
> inclusion in OEIS."
>
> On Wed, Nov 30, 2022 at 8:32 AM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
> wrote:
>
> Hi everyone,
>
> Please consider the two sequences below:
>
> 1) Sequence AX contains all OEIS sequences where the A number is a term in
> the sequence itself. For example, A000027 since 27 is a positive integer.
>
> 2) Sequence AY contains all OEIS sequences where either:
> a) the A number is not a term in the sequence (e.g., A000040, since 40 is
> not a prime number),
> or
> b) we don’t know if the A number is a term in the sequence or not (e.g.,
> A329697).
>
> The question here is: Where should the number Y go? If we put it in
> sequence AY, then we know where it belongs and that contradicts the
> definition of AY.
> Also, it couldn’t be part of AX because Y is not a term of AY.
>
> I’m trying to have some basic understanding of set theory and I would
> really appreciate your feedback.
>
> Best,
>
> Ali
>
>
>
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