At 14:15 into the movie, Colossus displays its first message. The President turns to look at the small monitor, and the display reads "WARN HERE IS ANOTHER SYSTEM". On the big ceiling monitor however it reads "WARN THERE IS ANOTHER SYSTEM", the correct message.
When Colossus is learning math, Harrison is standing across from Forbin. He is asked to get a printout. When he comes back to Forbin he stands next to him, yet there is a quick shot of him standing across from Forbin and then in the next shot he is standing next to Forbin.
Just after the television transmission during the press conference announcing the activation of Colossus, Professor Forbin calls the CPO and Blake answers the "phone". When he goes to get the phone, some of the pool cues behind him are misplaced in the on-wall rack. A moment later, when everyone else comes over to say hello to Forbin, they are all properly stored.
At the start The President states Colossus has control of ALL Allied defenses yet later in the movie when it chooses new targets for missiles it targets London, Rome & Denmark which have all been NATO members since 1949.
In a conversation near the film's start, Dr. Cleo Markham states that a four sided, three dimensional triangle is a pyramid. A pyramid, having a square base, is five sided. An actual four sided, three dimensional triangle is a tetrahedron.
When the equations are scrolling past when Colossus and Guardian are "talking" to each other, you can see that they do not actually get ever more complex as the dialogue suggests. Instead, they repeat the same sequence of calculus and trig identities on a repeating loop.
In an effort to prevent Colossus from launching America's ICBMs at other countries, Air Force personnel secretly replace the activating modules in the missiles' nuclear warheads with non-operative dummy modules. Later, when Colossus discovers a plan to overload him and thereby render him powerless, he detonates the warheads in two of the missiles. But detonation would have been impossible with the dummy modules installed.
The fascinating and imaginative opening scene in which Dr. Forbin activates Colossus, all the overhead lights come on, punch tape is seen running, displays are on: all of which is for human interface and use. Yet, Dr. Forbin will be the last person to ever be inside Colossus. All of those things should have been turned off as Colossus would have no need for them.