I am sorry to appear rude, but this is a terrible muddle.

Let's see.....,.,We assume the existence of the natural numbers

, also known as the "counting numbers"

We assume 2 properties of this set

1. There exists a least element in

which by convention one calls as 1

2. We assume that every

has a successor element denoted by

Then, given an arbitrary set

, then the mapping

*defines* a sequence in

such that

is, by convention written as

for all

Thus the sequence

.

So given

, say, how hard is it to guess the next element in the sequence?

Your problem stems from the fact that you fail to distinguish between domains and codomains