If we begin with a notion of number N that we denote F(N) as a function of time, can a decidable procedure exist on definability of the growth of numbers? Inspired by Tipler's Omega point and Thomson's lamp, what would be the bound when definability cease to have meaning?
Prologue: It all started after reading The Unimaginable Mathematics of Borges' Library of Babel and the review of it here. The problem arises when one sets to catalog the books as the number of the different books become approximately 10^10^6 (yet smaller than googoolplex), justifying the term "unimaginable". Susan Stepney points out in the review that when one wants to catalogue the number of books in Library:
[...] the problem of finding a "short" description of the book to put in the catalogue: there are not enough short descriptions. For the Vast majority of the books in the Library, the shortest description (that distinguishes it from other books) is the book itself. Most books cannot be "compressed" to a short description.
And then comes the punchline:
Or,as Bloch puts it, the Library is its own catalogue.
This brings to my thought experiment:
Thought Experiment: Suppose I type a single digit '1' and then I die with my thumb 'forever' locked on '0'. Is it possible that when the number
keeps increasing some interesting change happen to our understanding and philosophy of number system?
Recent edit: More specifically two points: 1) just like the Library becomes it's "own catalogue", if a number becomes inaccessibly large can there be self-reference leading to paradoxes? 2) what are then the implications of Poincare recurrence theorem? [latter being clarified already by Robert Munafo about non-literal meaning]
Background: This is related to my earlier question on Kunnen inconsistency in Math.SE. However, I am still having trouble grasping the behavior of large numbers even consulting a definitive website here on large numbers
I was reading a paper by Douglas Hofstadter on large numbers On Number Numbness, but again the argument veered towards philosophical interpretation.
Question: How to understand the behavior of large numbers? My motivation is from perspective of Poincare recurrence theorem a la Don Page's alternate universe count, or Skewes' number. Does logic as we know it 'break down'?
Here is the relevant portion from On Number Numbness that I had in mind while formulating OP:
If, perchance, you were to start dealing with numbers having millions or billions of digits, the numerals themselves (the colossal strings of digits) would cease to be visualizable, and your perceptual reality would be forced to take another leap upward in abstraction-to the number that counts the digits in the number that counts the digits in the number that counts the objects concerned. Needless to say, such third-order perceptual reality is highly abstract. Moreover, it occurs very seldom, even in mathematics. Still, you can imagine going far beyond it. Fourth- and fifth-order perceptual realities would quickly yield, in our purely abstraCt imagination, to tenth-, hundredth-, and millionth-order perceptual realities. By this time, of course, we would have lost track of the exact number of levels we had shifted, and we would be content with a mere estimate of that number (accurate to within ten percent, of course). "Oh, I'd say about two million levels of perceptual shift were involved here, give or take a couple of hundred thousand" would be a typical comment for someone dealing with such unimaginably unimaginable quantities. You can see where this is leading: to multiple levels of abstraction in talking about multiple levels of abstraction. If we were to continue our discussion just one zillisecond longer, we would find ourselves smack-dab in the middle of the theory of recursive functions and algorithmic complexity, and that would be too abstract. So let's drop the topic right here.
Relevant portion highlighted.