My Number is the Bigger One!

First you can read what happened?

Recalling what Ali and Zoe said - 

Ali: "I don't know whose number is bigger." 

Zoe: "I don't know whose number is bigger either." 

Ali: "I still don't know whose number is bigger." 

Zoe: "Now I know that my number is bigger!" 

First statement of Ali indicates that she doesn't have either 1 or 9. If she had 1 (or 9) then she would have an idea that Zoe must have bigger (or smaller) number.

Now Zoe is smart enough to know that Ali doesn't have 1 or 9 which is clear from Ali's first statement. Zoe's first statement indicates that she doesn't have 2 or 8 (& neither 1 or 9). If she had 1/2 (or 8/9) then she could have concluded that Ali has bigger (or smaller) number.

Till now Ali has an idea that the Zoe doesn't have 1,2,8,9. So still Ali can't have 2/8 as in that case too she could have made a different statement. Further if Ali had 3 or 7 (and knowing the fact that Zoe doesn't have 1,2,8,9); Ali could have an idea whose number is bigger as 3 is smallest while 7 is biggest among remaining numbers. That means she doesn't have 3 or 7 ( and 1,2,8,9).

From all the statement Zoe can conclude that Ali doesn't have 1,2,3,7,8,9. In short, Ali must have either 4,5,6. 

Now when Zoe says she has bigger number then it must be either 6 or 7 and Ali having 4 or 5. It can't be 5 as in that case Zoe wouldn't be confident as Ali could have 6.

So the Zoe's smallest possible number is 6.