### Extending the time line by infinitesimals

- Concurrent event cascades:
- oversimplification
- real world systems
- → small delay times for message transfer or state
change
- → lots of additional (unknown) parameters

- modeling
- include delays (cures cascade problems)
- use abstract "infinitesimal" values

- Set of hyperreal numbers
^{*}ℝ:
- totally ordered field
- includes ℝ
- includes infinitesimals
- ε > 0, but smaller than any real positive
number
- large set (2ε, −ε, ε
^{2},..) around 0

- includes unlimited numbers
- ω := 1/ε
- large set (2ω, −ω, ω
^{2})

- can be explicitely constructed (as a set)
- uses advanced set theory and logic (axiom of
choice, ultrafilters)

- allows to simplify analysis (nonstandard
analysis)

- Hyperreals in the computer:
- only a finite subset (as for reals)
- sufficient for our purposes
- pair (t,r) of floating point numbers t, r
- represents t + rε