As I understood it radioactivity was the means of transference of the the genetic characteristics of the spider. Spiderman was quoted as originally quoted as having the proportional strength of a spider i.e. able to lift 30 40 times body weight (pete weighs 165 therefore 6600lbs or 3 metric tons)
spidermans power has sadly increased over the years like superman (although not to the same extent). For some unknown reason writers sometimes feel the need to push the envelope on the reality of their character. I think spider man would be just as cool if he could only lift 3 tons.