As far as I understand, experts already rate him better than Fischer. It seems one may still claim Kasparov is greater; Carlsen even says so himself. It will be very interesting to observe the performance the next 4 years!
I think he's very clearly "better" in that he plays objectively stronger chess moves and would beat 1972 Fischer if we had a time machine.
"Greater" is a different question, more one for sports fans than sports scientists. What do you need to be great at a sporting vocation? Pure playing strength? Longevity? If so, in years, or in number of championship matches won? Dominance over your peers? Cultural impact on the world outside of your sport? Some kind of adjusted playing strength where we hypothesize how strong players would be given all the same opportunities and training material? If so, should it be modern grandmaster games and engines, or that from an earlier era?
"All of the above" is a good answer, and depending how you weight them you could incontrovertibly claim any of Carlsen, or Kasparov, or Fischer is the "greatest". You could also try to squeeze out a case for Morphy, or Lasker, or Capablanca.
You can't compare the best in a given era because the players now learnt from the players in that era i.e. Carlsen grew up studying Kasparovs games so it's simply a flawed comparison.
Instead he suggests a metric of "how far ahead of the field" they where, So Morphy and Fischer do really well on that metric, Kasparov does as well, Lasker and most of the other greats.
Fischer was unique in so many ways but late 1980's Kasparov would have crushed him.
Carlsen may be the only player to have a reasonable claim to be his equal.
Kasparov's definition is pretty biased in his favour though. It's far easier to dominate through pure chess talent in an era before engine prep, because the skill difference expresses itself across many more moves than modern games.
Kasparov, to surprise of no one, comes up with a self-serving metric.
If anything it should be opposite. If you're so far ahead it usually means your opposition is quite weak. Being at the top for 10 years today is way more difficult than it was in Kasparov 's time for example and that was way more difficult than in Lasker's time. Especially taking into consideration that he enjoyed an enormous preparation advantage thanks to his team while today any idiot with a high end GPU or a CPU can see all the best opening moves in seconds.
There is still advantage in preparing the information in a way it's useful/easier to remember and that's what the teams of top GMs do but there are no purely chess secrets anymore. It's just much harder to be ahead of the field today.
"Greater" is a different question, more one for sports fans than sports scientists. What do you need to be great at a sporting vocation? Pure playing strength? Longevity? If so, in years, or in number of championship matches won? Dominance over your peers? Cultural impact on the world outside of your sport? Some kind of adjusted playing strength where we hypothesize how strong players would be given all the same opportunities and training material? If so, should it be modern grandmaster games and engines, or that from an earlier era?
"All of the above" is a good answer, and depending how you weight them you could incontrovertibly claim any of Carlsen, or Kasparov, or Fischer is the "greatest". You could also try to squeeze out a case for Morphy, or Lasker, or Capablanca.