Ultrafinitism is the antithesis of googology; it is the mathematical belief that the set of natural numbers is finite, and at some point numbers become large enough that they simply cease to exist. The philosophy is an extreme form of finitism, which states that infinite objects (such as the transfinite ordinals) are nonexistent. Ultrafinitists state that the natural numbers have an ending, because they have no physical realization and/or cannot be computed.
The primary argument for ultrafinitism is that large natural numbers are circularly defined. A natural number is defined by applying the successor function to zero a certain number of times. And here we run into a bootstrapping issue: a googolplex is
S...S0 where the number of copies of
S is googolplex.
- A googologist asks an ultrafinitist, "Do you believe in 1?"
- "Yes," the ultrafinitist replies.
- "Okay, so do you believe in 2?"
- The ultrafinitist pauses one second, and says, "Yes."
- "And do you believe in 3?"
- He thinks about it for two seconds and says, "Yes."
- "And do you believe in 4?"
- He pauses for four seconds and says, "Yes."