The linked video is largely a critique of Dedekind cuts, arguing that they don't in general let us recognize, distinguish, or perform arithmetic on most real numbers. (Almost all of the informational input to a Dedekind cut for a randomly chosen real couldn't be written, remembered, or specified in any way by a human being.)

I think the presenter in the video is trying to justify a kind of finitist attitude based on the inaccessibility and unspecifiability of reals-in-general to us. This could also be advocating a position something like

https://en.wikipedia.org/wiki/Computable_number#Can_computab...

Edit: or perhaps https://en.wikipedia.org/wiki/Constructive_analysis (I didn't watch enough to understand exactly what alternative he proposes)