I saw him do that vs I remember him doing that

[Alexey86]

One thing is still not clear to me in terms of logic: the combination of the progressive and the infinitive verb forms (I've found many examples of this combination). Let's consider "I'm watching him cross the street." Both actions are contemporaneous. But how is it possible for somebody to be in an ongoing process of watching a complete action? As you mentioned earlier, "I see him arrive" is incorrect. If I understand correctly, the verb "see" doesn't have to end with -ing to mean an ongoing process.

[Charlie Bernstein]

One thing is still not clear to me in terms of logic: the combination of the progressive and the infinitive verb forms (I've found many examples of this combination). Let's consider "I'm watching him cross the street." Both actions are contemporaneous. But how is it possible for somebody to be in an ongoing process of watching a complete action? As you mentioned earlier, "I see him arrive" is incorrect. If I understand correctly, the verb "see" doesn't have to end with -ing to mean an ongoing process.

Crossing the street is a continuous action. You could just as well say, "I'm watching him crossing the street."

That's slightly ambiguous, because it could mean that you're watching him while you're crossing the street. So using cross instead of crossing is clearer. (But not much, since the context will probably be understood. Either word is really fine.)

[Alexey86]

So using cross instead of crossing is clearer.

Does it mean that the infinitive form can describe incomplete actions, and I can use cross even if in fact I didn't see somebody get to the other side?

[jutfrank]

One thing is still not clear to me in terms of logic: the combination of the progressive and the infinitive verb forms (I've found many examples of this combination). Let's consider "I'm watching him cross the street." Both actions are contemporaneous. But how is it possible for somebody to be in an ongoing process of watching a complete action?

Yes, good question. Like any other verb phrase, cross the street is necessarily coded in the mind as a complete action. That means that the essential meaning of it is that somebody goes all the way from one side to the other side. This sense of 'whole action' is the default meaning of all verb phrases in an infinitive form.

In this case, the speaker fully expects to see the action complete even if it isn't complete at the exact moment of utterance. If there is doubt in the speaker's mind that the person being watched will not make it to the other side, the -ing form may be used to suggest this.