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September 28th, 2018, 03:05 PM  #1 
Newbie Joined: Sep 2018 From: Poland Posts: 3 Thanks: 0  Do you have F. Waismann's: Intro. to Mathematical Thinking
Friedrich Waismann: Introduction to Mathematical Thinking. The Formation of Concepts in Modern Mathematics. (14. Ultrareal numbers) If you have it, would you be so kind as to quote what it states about ultrareal numbers? My email: htg@interia.pl 
September 28th, 2018, 08:22 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631 
Can't help with the book but found a reference on nLab. https://ncatlab.org/nlab/show/ultrareal+numbers where each operation is defined by iterating the previous one (the next operation in the sequence is pentation). The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of n, while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function. Addition (a + n) is the most basic operation, multiplication (an) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a, and exponentiation (an) can be thought of as a chained multiplication involving n numbers a. Analogously, tetration (na) can be thought of as a chained power involving n numbers a. The parameter a may be called the baseparameter in the following, while the parameter n in the following may be called the heightparameter (which is integral in the first approach but may be generalized to fractional, real and complex heights. So it's essentially about abstract tetration. nLab is a site devoted to ncategories so I'm pretty sure that it's tetration on steroids. OP is this definition of ultrareal numbers the same or different as the usage in the book? 
September 28th, 2018, 10:53 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2587 Math Focus: Mainly analysis and algebra  

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