integer sequence of intelligence

[AntonioLao]

Numerical sequences like 1, 2, 3, 4, 5, 6, 7, ...is an example of integer sequence where each term of the sequence is an integer. These can be classified into computable and definable sequences. A sequence is computable if there exists an algorithm that can predict any succeeding term of the sequence. On the other hand, a sequence is definable if there exists a true statement exclusively for one sequence but false for other sequences. Although both the set of computable sequences and the set of definable sequences are countable not all definable sequences are computable. For example, the sequence of perfect numbers. Nonetheless, the set of all integer sequence is uncountable which include sequences that cannot be defined. For example, random sequences whose terms are randomly distributed and not predictable. Random sequences can also be called irrational sequences although each term of any random sequence is an integer.

A multiple choice test, say of 5 choices for each of 10 questions would form a random irrational sequence of correct answers and one sequence of correct answers might look like: 2, 2, 3, 4, 5, 1, 3, 3, 1, 1. Another might look like this: 1, 1, 1, 1, 1, 2, 2, 3, 4, 5. For anyone taking this test it does not require any intelligence at all if the testee knew beforehand the random irrational sequence of correct answers in order to get a perfect score of 100%.
http://en.wikipedia.org/wiki/Integer_sequence

[SteveA]

Is there a proof that the set of all integer sequences is not in itself countable? I'm skeptical of that, though if there's some obvious paradox that can't be gotten around, it would be nice to know.

Notice that there is no random integer (which integer(s) is/are random? That would be a variable), though we could count through sets of elements in a random distribution and in that sense even count through arbitrary random distributions of integers as well.

The main issue appears to be whether or not there's a way to have a prespecified manner of describing any form of sequence - if we can list the class, there appears to always be a way to count through them, but counting through that ability to construct new classes in arbitrary manners appears something to have, at a minimum, no obvious generalization of and might be something that's impossible if it's something that could be cyclic or self redefining.

[AntonioLao]

Is there a proof that the set of all integer sequences is not in itself countable?

If there is then I am not aware of it. Countability, as I understood it, is only applicable to computable and definable integer sequences. The random sequence as I described it only means the random appearance of integer in a sequence, for example the correct answers to some multiple choice questions.

[SteveA]

Well here's a way that appears able to count through all finite sequences of rational numbers:

Write a number in binary format. Count the number of 1s between every 0 and represent this as a sequence of natural numbers.

From there interpret each of these natural numbers as a natural number using Cantor's method (we could use something like the Farey Series to remove reducible ratios):



So, for example, the sequence of numbers 2,1/3,1 could be rewritten as a count in Cantor's enumeration as the sequence of natural numbers 3,6,1 which could then be written as a binary number by using sequencial 1s as:

2,1/3,1->3,6,1->111011111101

There are ways we could even embed sets of numbers within this or includes pairs of numbers as upper and lower limits of random ranges etc.

Basically, my guess is that if there's a way to describe it as some deterministic structure, then there's a way to determine a system of counting that can reach it, but it seems difficult or maybe impossible to describe a single manner of counting that could derive all of these and any possible form or structure. Though if there was some finite/enumerable set of operations that could describe any possible structure, then there could be a way to construct a manner of counting through all of these ... all specific forms of "things" on a single number line? If not, then we need a second dimension and "free will" in the exploration process

Imagine if a probabilistic algorithm could potentially, in the limit, describe a broader range of forms with arbitrarily high probability than a single deterministic algorithm could ... just another "random" thought. There might be somethings that are qualitatively different in a randomized algorithm that could not be constructed by a deterministic one.

On another thread, the symmetry of left-hand and right-handed properties was pointed out and this appears to be a physical property that could be said to be entirely arbitrary in its selection. There might be no deterministic way to compute a specific selection of perfectly symmetric possibilities as there would be no asymmetric "handle" to grab hold of and compute a specific selection. A fundamental non-deterministic randomness appears possible.

[AntonioLao]

Using a fair die the probability of rolling a 1 is 1/6. The same probability for 2, 3, 4, 5, and 6. However, if there is such an infinite faces die each face marked by the positive integer starting with 1 then 2, 3, 4,5, and so on to infinity then the probability of rolling any integer is zero. On the other hand if there are countably many finite random integer sequences then the probability is 1/n where n the finite number of terms in the sequence but as n approaches infinity then the probability approaches zero.

[AntonioLao]

Given two magic triangles as the tessellations of spacetime charges, the first has sides add up to 9 while the second has sides add up to 12.