I-notation (or |-notation) is a notation for large numbers by Denis Maksudov[1]. The creator stated that it reaches \(\psi_0(\Omega_{\omega})\) with respect to Buchholz's function in fast-growing hierarchy without a proof.[2]


The notation consists of a natural number, followed by the symbol "|" (vertical bar, possibly also writen as "I", a capital letter i), then a sequence of natural numbers and parentheses, which are in turn followed by subscripts consisting of natural numbers.

Let \(a\) be a natural number. Here are the rules for expansion:

  1. \(a|\ =\ a+1\)
  2. \(a|b()\ =\ a\underbrace{|b|b\cdots |b|b}_{a\textrm{ copies of }``|"}\), where \(b\) is the rest of the expression
  3. \(a|b(c())_de\ =\ a|b\underbrace{(c)_d(c)_d\cdots (c)_d (c)_d}_{a\textrm{ copies of }``(c)_d"}e\), where \(d\) is a natural number, \(b\) and \(c\) are parts of the expression that can include any parentheses with any subscript, and \(e\) is a part of the expression that includes only closing parentheses with any subscript.
  4. \(a\vert b(c()_{d+1}e)_fg=a\vert b(c(c(\cdots(c\underbrace{()_de)_d\cdots e)_de)_d}_{a\;d\textrm{'s}}e)_fg\), where \(a\), \(d\), \(f\) are natural numbers; \(b\), \(c\) can include left and right parentheses with any subscripts; \(g\) can include right parentheses with any subscripts; \(e\) can include only right parentheses with subscript \(d+1\) or more; \(d+1>f\)

Note: if some pair of brackets is not enclosed in any other then subscript of this pair should be zero, for example \((()_2()_2)_0(()_1()_0)_0()_0\). We use \((...)\) as an abbreviation for \((...)_0\).

See also

Online calculator for |-notation


  1. D. Maksudov, I-notation (accessed 2021-05-09)
  2. A difference page of the talk page of this article. (Note: The original source states that it reached TFBO, but it was based on confounding of the two ordinals according to the creator.)
Community content is available under CC-BY-SA unless otherwise noted.