Re: Dementia Redux
Jon, Edward, et al.,
There are many useful theorems that are
worth knowing for two reasons:
1. They provide useful criteria for dismissing
many kinds of bogus claims as pure BS.
2. They are important for self defense, when
somebody tries to use them to dismiss
what you're doing.
Jon's recent example is the theorem that all the points
on a two-dimensional surface can be put in a one-to-one
correlation with the points on a one-dimensional line.
Claim 2: Two versions of logic are equally expressive
iff there is a one-to-one correlation of statements
in one to statements in another.
Claim 3: Two computing systems are equivalent iff there
is a one-to-one correlation of the computable functions
in one to the computable functions of the other.
These statements are good to know, but it is also
important to know all the qualifications that are
not stated in those claims.
For example, every digital computer that has ever been
designed so far is equivalent, according to claim 3,
to a Turing machine. However, everybody who has ever
replaced a 5-year-old computer by a new one knows very
well that the new one can finish any computation in a
tiny fraction of the time taken by the old one.
Similarly, two different langauges that are equivalent
to first-order logic by claim 2 might be very different
in their ease of use, readability, etc.