Re: What is a proof, exactly?

From: Han de Bruijn (Han.deBruijn_at_DTO.TUDelft.NL)
Date: 11/25/04 

Date: Thu, 25 Nov 2004 13:27:26 +0100

J.E. wrote:

> [ ... ] As for your question of "how do you equal
> objects which are not sets?", the first thing I wonder is "what are
> you discussing that isn't a set?" A formula? Why not discuss the
> formalization of the formula into a set? A theory? Same thing with
> the formalization of the list of theorems. I really haven't come
> across a need for a wider equality in my work, so I'd be interested in
> your experience.

In conclusion, could I say that everything in (mainstream) mathematics
is a set? That only the equality of sets may be considered relevant?

OK. Back to equivalence relations. If I have an equivalence relation
which is related to a set, then that equivalence relation induces a
classification of that set. Right? Such a classification means that
the original set is partitioned into subsets where every subset is
non-empty, disjunct from any other subset in the partition and the
union of all these subsets is the original set. Am I still right?

You say that equality is always an equality between sets. Then any of
the members of a set must also be a set in itself. Now Cantor is saying
that a set is a collection of _distinct_ elements. Since elements are
just sets, this can be given a precise meaning: elements are distinct
iff their intersection is empty. All of the elements together make up
the whole set. And why not demand that the elements are "something",
which means not just empty.

The difference between a "true" equality and an equivalence relation
becomes quite subtle now. So subtle that I would even dare to ask if
these two concepts do not denote one and the same thing. Summarizing:

       Equality = Equivalence relation
       Member = Class in a partition

Terse mathematics, that would be a good thing ...

Han de Bruijn

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