From wikipedia

Aspartame is rapidly hydrolyzed in the small intestine by digestive enzymes which break aspartame down into methanol, phenylalanine, aspartic acid

Phenylalanine and aspartice acid are amino acids and methanol is produced in smaller amounts than naturally present in the body / are produced by metabolism of common foods such as fruits. I'm absurdly confident that aspartame isn't harmful.

Nevertheless, this was retweeted by twitter-jack, which claims

Scientists, on the basis of early testing, were saying aspartame was a carcinogen in the late 1970s and early 1980s, but Donald Rumsfeld, who was CEO of the company that owned Aspartame, became part of the Reagan transition team and got to appoint the new head of the FDA, who then went on to cast the deciding vote in favour of allowing Aspartame into the American food supply

I am not including potential 'taste-based diet change' effects here - if the sweet taste that doesn't correspond to sugar intake causes some homeostasis on sugar intake to be knocked slightly off or something, that doesn't count. That is, imo, a marginal reason to not consume aspartame if you aren't already overweight!

In case anyone tries to rules lawyer, I mean 'significantly out of distribution in health harms of normal healthy foods' - after all 'All things are poison, and nothing is without poison; the dosage alone makes it so a thing is not a poison.'

edit: resolves PROB to confidence

MORE CLARIFICATIONS:

Aspartame is harmful to people with the inborn disorder of metabolism PKU. However those with PKU also have to avoid a wide variety of normal healthy foods, so I do not consider that to qualify.

-- offtopic --

The guy jack retweeted seems to be a bit racist.

The tweet also claims propionic acid is bad, which I also think is wrong, considering "Propionic acid is produced biologically as its coenzyme A ester, propionyl-CoA, from the metabolic breakdown of fatty acids containing odd numbers of carbon atoms, and also from the breakdown of some amino acids."