No, you cannot get Alzheimer's Disease from cooking with cookware that contains aluminum. The Food and Drug Administration has approved aluminum as a material for cooking, so rest assured that your aluminum cookware will not cause Alzheimer's Disease.
According to the nonprofit Alzheimer's Association, "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. Almost all scientists today focus on other areas of research, and most experts believe aluminum does not pose any threat."
Alzheimer's Disease & Aluminum
Can you get Alzheimer's Disease from cooking with aluminum product?