Crazy Sunday ideas…
Let’s consider the interval
[0, 1] of real numbers. I can pick a
and create for that number an (ordered) sequence:
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:
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:
|+|operation in which
P(a) |+| P(b) = P(c)where
a, b, cbelong 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.