FOM: surreal numbers; Conway's foundational ideas

Thomas Forster T.Forster at
Thu May 27 03:10:38 EDT 1999

I think we take them for granted too much.  A common and
important example is the Von Neumann ordinals.  All the
textbooks write as if ordinals were being *defined* when
they are only being implemented.  I often find myself
having to point out to my students that the proofs they
offer me of facts in ordinal arithmetic (in the exercises
they are set) are not proofs of those facts, but are 
merely proofs of implementations of those facts in Von
Neumann ordinals.     Of course, people will say, we
know that *really* but it doesn't matter beco's etc etc.
Well,  it does matter, and unfortunately students often
don'y have this dinned into them sufficiently for them
to justly say that they know that *really* these proofs
aren't proofs at all but it's OK beco's we can prove that
Von Neumann ordinals are faithful in the appropriate sense.

   There is a chap called `Visser' in the Netherlands somewhere
who writes about translations between formal languages.
I suspect that we need to pay more attention to this sort
of thing than we do.  And, i suppose, when i say `we',
i mean `they'!


More information about the FOM mailing list