In this course the notions of Non-Standard Analysis are introduced starting from the notion of number and enumeration systems. This approach is certainly not the most direct, but it provides a reflection on the notion of number and more generally on the foundations of mathematics.

Lesson 1:

• Enumeration systems
• Real numbers, cardinals, ordinals and numerosities
• Construction of a theory of numerosity

Lesson 2:

• Numerosity and ordinal numbers
• The non-standard universe
• Numerosity and hypernatural numbers

Lesson 3:

• The continuum problem
• The empirical line, the rational line, the real line and the Euclidean line
• Hyperreal fields

Lesson 4:

• First applications
• The generalization of the concept of function
• Concluding considerations