1.1 The axiomatical system of Hrbacek set theory 

    1.1a The universe of HST 
    1.1b Axioms for the external universe
    1.1c Axioms for standard and internal sets
    1.1d Well-founded sets 
    1.1e The structure of internal and well-founded sets  
    1.1f Axioms for sets of standard size  
    1.1g Putting it all together 
    1.1h Zermelo-Fraenkel theory ZFC 

1.2 Basic elements of the nonstandard universe 

    1.2a How to define fundamental set theoretic notions in HST 
    1.2b Closure properties and absoluteness 
    1.2c Ordinals and cardinals 
    1.2d Natural numbers, finite and *-finite sets 
    1.2e Hereditarily finite sets 
    
1.3 Sets of standard size 
    
    1.3a Cardinalities of sets of standard size 
    1.3b Saturation and the Hrbacek paradox
    1.3c The principle of Extension    
    
1.4 The class $\Delta^{ss}_2$

    1.4a Basic properties of $\Delta^{ss}_2$
    1.4b Cuts (initial segments) of *-ordinals 
    1.4c Monads and transversals 
    1.4d On non-well-founded cardinalities 
    1.4e Small and large sets 

1.5 Some finer points 

    1.5a Von Neumann hierarchy and Reflection in ZFC 
    1.5b Von Neumann hierarchy over internal sets in HST
    1.5c Classes and structures 
    1.5d Interpretations 
    1.5e Models 
    1.5f Simulation of models of ZFC 
    1.5g Asterisk is an elementary embedding 

 Historical and other notes to chapter 1