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.