FOM: On "grasping" Con(ZFC)

Roger Bishop Jones rbjones at
Sat Aug 29 01:37:19 EDT 1998

I'm puzzled by the suggestion that Con(ZFC) is difficult to grasp,
but since I don't really have much idea what "grasp" is supposed to mean
in this context I thought a concrete example of how people might comprehend
it might be of interest.

Con(ZFC) asserts that it is impossible to derive a contradiction in a
particular first order theory.

Actually, quite a simple one, a handful of axioms, which are not at all
hard to memorise.

This can easily be described to an average programmer as the claim that
a program written to search for a derivation of a contradiction will not
His grasp of the conjecture will then depend on how hard he finds it to 
understand what this program is searching for.

It seems to me that this really is a trifling program, by comparison with
most the programs which a professional programmer works on every day of
his working life, and is expected to code up without too many errors and
eventually get to work correctly.  This is true even in the case
that this programmer is working by himself on programs simple enough for one
man to write.  Very often they are working with large numbers of other
programmers on programs which really are too complex for any individual to

He probably would scratch his head a bit if you started explaining this
in terms of Goedelisation, since computers use much more straightforward
methods for encoding text as numbers, and expect their compilers to sort
out that kind of thing.

For my part, I often gasp at the complexity (often quite unnecessary) which
programmers are happy to grapple with.
Con(ZFC) however, is nothing.

Roger Jones
email: rbjones at   www:

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