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Jer
2015-08-24 09:43:11
An variation of a well know paradox.

I woke one morning with this idea bouncing around in my head. Despite my efforts I concede it doesn't lend itself to being an actual problem (it is essentially unsolvable) but I feel it is interesting enough to share. The following is as I submitted (and eventually withdrew it) from consideration as a problem.

In case you didn't know, The On-Line Encyclopedia of Integer Sequences or OEIS is a valuable source for mathematicians and recreational math buffs alike. Users can look up sequences for information about them and how they relate to other sequences. Users can even submit new sequences for inclusion. This data base currently has over 261,000 sequences.

When a new sequence is accepted it gets a reference tag consisting of a capital A followed by 6 digits. These digits could be treated as an integer, often with leading zeros. For example the Fibonacci sequence is A000045. We can say this sequence has reference number 45.

Notice that the Fibonacci sequence, though numbered 45, does not contain the number 45. There are many sequences whose number is not in their sequence. The first few are A000004, A000007, and A000009.

Question 1. Supposes a user was able to submit the sequence of all sequences whose reference number is not in the sequence (The sequence would begin 4, 7, 9, ...). Would this sequence contain its own number?
I woke up one morning with these thoughts in my head.


Some sequences in the Encyclopedia have a finite number of terms. For example A005188 was precisely 88 terms.

Question 2: Suppose the Encyclopedia currently has, say, 12 finite sequences with 12 terms. Suppose a user was able to submit the sequence of reference numbers of all sequences with 12 terms. Would this sequence contain its own number?

Incidentally, no such sequences would be allowed as the reference numbers are meaningless and are only to be used to identify sequences.

Jer
2015-08-24 09:48:39
Re: An variation of a well know paradox.

Ady Tzidon had some great thoughts on this while it was still in the queue. I cut and pasted to here so as not to lose them: (I also corrected some typos)

(sets that contain themselves as elements, like S, are definitely ruled out. )
(The above statement denies dealing with self referencing paradoxes (a.k.a. antimonies)
(since there is a long list of similar examples like Bhartharta paradox, Grelling-Nelson's paradox Russels paradox.)
(You are actually asking questios isomorphis to all those I have mentioned, for which Mathematics has no coherent answer other than amendment to the basic definition of classes,)
(which is equivalent to saying "do not touch it".)
(Antinomy (Greek ἀντί, antí, "against, in opposition to," and νόμος, nómos, "law") literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology, particularly etc)

Ady TZIDON
2015-08-25 14:52:10
Re: An variation of a well know paradox.

2 minor remarks:
a. You were right to withdraw the problem, which is not a puzzle in search of an answer. It is also ok to publish it here as another, no-doubt interesting, example of Russel's basic paradox.
b. Three misspelled words were still left in the above text: "..questios isomorphis..." s.b. "..questions isomorphic..." s.b. and "antimonies" s.b antinomies.
Sorry.

Ady TZIDON
2015-08-25 14:56:09
Re: An variation of a well know paradox.

CORRECTED:
2 minor remarks:
a. You were right to withdraw the problem, which is not a puzzle in search of an answer. It is also ok to publish it here as another, no-doubt interesting, example of Russel's basic paradox.
b. Three misspelled words were still left in the above text: "..questios isomorphis..." s.b. "..questions isomorphic..." and "antimonies" s.b antinomies.
Sorry.




Add to the thread:

Ady TZIDON
2015-08-26 15:00:56
Re: A variation of a well know paradox.

WHY: An variation of a well know paradox.

It s.b. : A variation of a well know paradox.

Jer
2015-08-26 22:36:27
Re: An variation of a well know paradox.

You aren't the only one prone to typos. I think I had put the word 'interesting' there, but removed it and missed changing 'An' to 'A'.

I figured readers could decide for themselves whether it was 'interesting'.

Charlie
2015-08-27 09:52:01
Re: An variation of a well know paradox.

I thought that n was just misplaced from the end of well-known.

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