No, you cannot get Alzheimer's Disease from cooking with aluminum product. In fact, aluminum is present in a wide variety of everyday items: toothpaste, soda cans, baking soda, antacids and aspirin all contain aluminum. Even the air we breathe, the food we eat and the soil we grow our foods contain aluminum. Product made of aluminum has been used for over a hundred years with no proven connection to Alzheimer's disease.
Additionally, nonstick coated cooking surfaces prevent food from coming in contact with the aluminum surface. As to hard-anodized aluminum cookware, the process of hard-anodization changes the molecular structure of the pan's surface and prevents food reactions that can occur with plain aluminum cooking. For example, plain aluminum may affect the taste of acidic foods like tomatoes.
The Food and Drug Administration has approved aluminum as a material for cooking, so rest assured that your aluminum cookware is safe.