Infinite tetration and superroot of infinitesimal

[Ivars]

Before proceeding further, I would like to be precise about what an "infinitesimal" number is.

• There exists no z such that \( 0 < z < a \) for all real \( 0 < a \)
  
• There exists dz such that \( 0 < dz < a \) for all real \( 0 < a \)
  

An infinitesimal is a number that satisfies the second property, is this correct?
How does this not form a contradiction with the first property?

Andrew Robbins

Infinitesimal is NOT a Real number at the scale we are looking at things. At the scale where Reals are Reals, Infinitesimals are not Real. Their Real part is 0, but there is more in them. Infinitesimal is definitely closer to zero, as well as closer (surrounding) to any Real than any other Real. It may be that application of relations < ; > is misleading here.

Considering Infinitesimal to be somehow Real just smaller than any Real , or have any Real part ( if it is multidimensional) in the scale we are in leads to contradiction You pointed out.