Can there be an NFA that decides on real numbers ?
A Decidability Question
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Nope.
A real number can have an infinite number of digits behind the decimal point. There may not be a system in those digits (i.e., they may be generated by a random process). In that case there cannot be a description of this sequence of digits that is significantly shorter than the sequence itself.
Now take such a real number r. Since any NFA has only a finite number of states and can be described finitely, it will be inadequate to accept only the real number r (for otherwise that would contradict the fact that there cannot be a finite description of r).