Vector spaces may have irrational dimensions. — Lionino

It turns to an uncountable dimension. Right? Or am I lost in something?

Now, I can't see the next step in your rule.

Interesting question. Usually, the dimension of a vector space is countable, but it can be finite or infinite.

A vector space of any given dimension, for example, with dimension π has a countable dimension of... exactly π. But under my definition we will have uncountably many subspaces of R2 that are not subspaces of R1, for example.

But under my definition we will have uncountably many subspaces of R2 that are not subspaces of R1, for example. — Lionino

I think I can see it. By uncountable in your definition, it means that there are infinite subspaces or dimensions. Right? It is not about to be countable but if the vector space has a dimension.

Uncountably many subspaces. There are uncountably many dimensions that a vector space may have, yes.