Sequences & Information Explained in 5 Minutes

Sequences & Information Explained in 5 Minutes

Suppose you have a set that contains just one object. The object can be anything, so let's use the number zero. Then our set is..

S={0}

Sets don't get much simpler than this! Now we ask a question, how many sequences can be made using the objects in S? A sequence in mathematics is simply an ordered list of items. So here's a few examples of sequences made from the object in S..

0
0,0,0,0
0,0
0,0,0,...

The first 3 examples are finite sequences, the last one is infinite. It's pretty clear that we can make a lot of different sequences, in fact we can make an infinite number. To be more precise, we can make a sequence for any positive integer n, we just list n zeros in the sequence. If you've read the post on Georg Cantor you'll realize that the number of sequences we can make is aleph0.

So, from a set containing just one object we've constructed an infinite number of different objects. Pretty neat. It's almost like getting something for nothing!

Can our systems of sequences be used to store information? They can in the sense that they can be used to label any group of objects, provided of course the number of objects is no bigger than aleph0.

This is good, but things get better. If S contains just 2 objects, S={0,1} we can build many more sequences. In fact the number of sequences is 2^aleph0=aleph1 and the "storage capacity" of our sequences increases dramatically. You may be familiar with the finite sequences built from S={0,1}, they're the binary representations used in computer science.

Content written and posted by Ken Abbott abbottsystems@gmail.com