on the appointment setter widgets is there code to prevent two appointments being set at the same identical time and date?

http://www.jotform.com/form-templates/category/appointment-form

I think you can use Appointment Slots widget,  for showing your form users what times are available for an appointment. You can limit how many people choose an available slot because the widget reduces the number of selected slots each time the form is submitted. Please check the Appointment slots widget to the below URL:

https://widgets.jotform.com/widget/appointment_slots

You may also try "Gift Registry" widget. Simply add each slot as a separate entry and give them all (1) quantity.

http://widgets.jotform.com/widget/gift_registry

 Here is a screenshot:

If you need further assistance, feel free to revert back.