[JohnKemeny]

It should simply say:

for any j>0 there exists an algorithm multiplying nxn matrices in time O(n^{2+j}).

[Lucasoato]

You are correct, I apologize for the confusion! :)

[hnfong]

Assuming your claim is not equivalent to "matrix multiplication is in O(n^2)", then it is false assuming conventional mathematics (i.e. uncountable sets exist), because j in R is uncountable, and algorithms are countable...

[ernesth]

You do not need a different algorithm for each real. Just take the algorithm that proves the statement for rational p. It proves the statement for all reals bigger than p. (hence having as many algorithms as there are rational (countably many) is enough)

[hnfong]

Then that’s basically saying there is a O(n^2) algorithm which I dealt with in my first sentence.

[akoboldfrying]

Are algorithms on a Real RAM [1] countable? That would mean that there are constants you can't express.

[1] https://en.wikipedia.org/wiki/Real_RAM