# Numbers with non-denumerable decimal expansion

Posted in: Math , Idea

Crazy Sunday ideas…

Let’s consider the interval [0, 1] of real numbers. I can pick a number, say:

    0.25


and create for that number an (ordered) sequence:

    <2, 5>


I can also create a Power Sequence (derived from the definition of the Power Set) that for the previous sequence would give me something like:

    P(<2, 5>) = <<2>, <5>, <2, 5>>


(note that there is no empty set in the new Power Sequence). We can finally flatten the sequence to obtain:

    <2, 5, 2, 5>


Now all real numbers have a denumerable decimal expansion (i.e. we can create a one-one correspondence between any sequence of numbers that describe the decimal representation of a number and the sequence of natural numbers). In our (previous) 0.25 case, we could make it easier by appending infinite zeros after the number, to obtain:

    <2, 5, 2, 5, 0, 0, ...>


and then build a one-one correspondence with <1, 2, ...>.

Ok, let’s now consider ∏/10:

    <3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, ...>


And now let’s take for that decimal expansion the flattened sequence of it’s Power Sequence P(s). We obtain a decimal expansion which is non-denumerable and which represents a number I’ll call:

c

All real numbers have a representation in this new set of non-denumerable decimal expansions, just like each integer, rational, etc have a representation in the Dedekind (cuts) definition of the real numbers.

It’s interesting to note however that this “new set” of “numbers” is a superset of the real numbers, since there is no one-one correspondence between denumerable and non-denumerable sets, then there is no one-one correspondence between denumerable sequences and non-denumerable ones.

I think it’s a crazy but interesting idea. Some questions and things to consider:

• Is this equivalent to a transfinite ordinal theory like the Cantor one?
• I’d like to define a closure with a |+| operation in which P(a) |+| P(b) = P(c) where a, b, c belong to R. I wonder how that would be.

I have plenty of other questions, but I’ll just daydream with this until I find it’s a stupid idea.