What does incredulity mean?
1 : unwilling to admit or accept what is offered as true : not credulous : skeptical. 2 : expressing incredulity an incredulous stare.
Which pronunciation of either is correct?
Using English phonetics, [ EE-ther ] and [ AHY-ther ] are both correct. As a side note, giving “th” a phonetic Z (or S or F or D) sound is a common mispronunciation that can be corrected with practice.