Reddit reviews Models for Smooth Infinitesimal Analysis

thanks for the reply. i thought i might give some references here in case you or someone else is interested in smooth infinitesimal analysis (SIA) and synthetic differential geometry (SDG).

• an invitation to smooth infinitesimal analysis by john l. bell
  
  
• an introduction to smooth infinitesimal analysis by michael o'connor
  
  in the introduction by o'connor, he explains what i was getting at in my question (which was originally typed on mobiel without easy access to references). bell does the same in the below referenced primer. and that is, smooth infinitesimal analysis (and thus synthetic differential geometry) is a system which is a collection of some axioms, the adoption of intuitionistic logic, and then a model which says that the prior two things are consistent. with the logic and axioms, one can explore a lot of SIA and SDG without getting into the details of the model theory. but it's always been a curiosity of mine what that model theory is. i guess i could stop being lazy and learn it, but i'm almost not for sure i need to. i'm happy with someone saying that things are consistent and one get can on with doing SDG without having to worry about it. but there is some curiosity of what's actually going on, which i don't really understand.
  
  further references that aren't immediately available online:
  
  
• a primer of infinitesimal analysis by john l. bell
  
  
• toposes and local set theories: an introduction by john l. bell
  
  
• models for smooth infinitesimal analysis by moerdijk and reyes