According to the Alzheimer's Association in the U.S.:

"Myth 4: Drinking out of aluminum cans or cooking in aluminum pots and pans can lead to Alzheimer’s disease.

Reality: During the 1960s and 1970s, aluminum emerged as a possible suspect in Alzheimer’s. This suspicion led to concern about exposure to aluminum through everyday sources such as pots and pans, beverage cans, antacids and antiperspirants. Since then, studies have failed to confirm any role for aluminum in causing Alzheimer’s. Experts today focus on other areas of research, and few believe that everyday sources of aluminum pose any threat."

From the Alzheimer's Society of Canada:

"Studies have not provided strong evidence of aluminum being a risk factor for the development of dementia. ... It would be difficult to significantly reduce exposure to aluminum simply by avoiding the use of aluminum cookware, foil, beverage cans and other products. Use of aluminum in pots and pans only contributes to a very small percentage of the average person's intake of aluminum."

From a 2013 Washington Post article:

"Naturally occurring aluminum is the third-most-abundant element on Earth, so it really is everywhere. Because it’s present in the soil, it can be found in certain foods, such as spinach and tea. It’s also used in a number of industrial processes that bring it into close contact with humans. Although many studies of aluminum exposure have focused on drinking water — utilities often use aluminum salts to clarify and purify their water — it is also found in cookware and food packaging, in antacids, antiperspirants and a handful of medications, and in some processed foods. That makes teasing out people’s exposures to aluminum over a lifetime, and the effect of those exposures, “very hard to do,” Perl says."

I won't be getting rid of any aluminum cookware any time soon.