Then I would agree that intuitions don't provide evidence for such statements, although it is a little bit confusing to say that there is no correlation between intuitions and truth for this class of statements since people don't have any intuitions pertaining to members of this class of statements.
To further nitpick, one googol is small as compared to what exactly?
(I'd find it way more natural for "randomly constructed statement" to be selected from statements people actually construct and have intuitive opinions about, given the context of the discussion.)
Maybe I can get my point across using this analogy:
"You would expect a human to be able to remember a random natural number."
"No - because only an infinitessimal fraction of random numbers fall into the range of having a number of digits that fit into a typical human's memory."
"Ah, I thought we are only talking about typical natural numbers in human use, from which we randomly choose one."
The discussion we're leading has some similarities with talking about a "random mind" - discussing our little corner of the mindsp...
Closely related to some of Luke's recent discussions about philosophy, philosopher Paul Thagard has recently called for changes to the way we do philosophy:
In the same article, Thagard also lists eleven areas where modern philosophy goes awry. For example:
Source: Philosopher, Paul Thagard