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.

New rules: A pot is stored on the shelf. To the pot there are other pots as well. All of them have the same size, weight, shape, and colour. Each pot contains the same proportion of whatever thing. They are located in the third step, so if one falls down, it gets broken. You are cooking (edit: dressing) Greek salad in your kitchen, but you notice that there isn't enough honey, so you ask your sister—who is a crackpot—to go to the pots and take some honey for the delicious salad that you are cooking up. When your sister enters the saloon, you hear a clatter sound, and your sister shouts, 'Ouch!'

How many pots got cracked considering there is already +1 crackpot and the rest of the pots are equivalent?

Please, elaborate.

Something closer to this:

pretty sure the cracked pots are an exponential function such that if you allow 3 or 4 it's containable, but 6 or 7 might make all the non-crackpots become pots that can be cracked.

F(x) = x^C where "C" is the cardinality of the set of "pots"

but 6 or 7 might make all the non-crackpots become pots that can be cracked. — Moliere

That is probably one of the best phrases I read here so far.

If only the sister wasn't a crackpot, the only pots susceptible to being cracked were the ones on the shelf. Everything here is very complex and tricky. Who is responsible for the cracking? The crackpot sister or the pots on the shelf? I am starting to think that they are opposite poles. Their orbital gravitating force only led them to the destruction. :sad:

You are cooking Greek salad... — javi2541997

Wait... cooking a Greek Salad...?

Wait... cooking a Greek Salad...? — Banno

OK... OK... sorry, my bad. You are dressing the Greek salad...

Ohh, thank god! The horror! I was having such visions...

Plot twist: I am actually the crackpot, not the imaginary sister mentioned in the "new rules post."

So, the counting of cracked pots starts at zero and not one.