🚀 go-pugleaf

In article <728@bsu-cs.UUCP> dhesi@bsu-cs.UUCP (Rahul Dhesi) writes:

> Doesn't the paradox exist because we force it to exist despite knowing
> better?

I did not see any paradox.  As I interpreted Descartes' argument, he was
saying: There is at least one thing we can know for certain, namely that it
is absurd to claim as certain knowledge that nothing can be known for
certain.

> Consider a set that contains itself, and contains every other set that
> does not contain itself.  Where's the paradox?

Here: Call that set S.  We have

    x in S <=> x = S or (x != S and x !in x).         (*)

Now let T consist of the elements of S, except S itself.  So

    x in T <=> x in S and x != S
           <=> (x = S or (x != S and x !in x)) and x != S
           <=> x != S and x !in x.

In particular (putting x:=T),

    T in T <=> T != S and T !in T.

By the construction of T, we know that T != S (since S in S
but S !in T), and so

    T in T <=> T !in T.

An axiom set that allows the construction of S but at the same time
prevents the construction of T would have to be pretty weird.

> Paradoxes arise when you choose an inconsistent set of axioms.  "Doctor,
> it hurts when I do that!"  "Don't do that!"

This misses the whole point that there is no agreed way to make sure that a
set of axioms is consistent.  Is ZFC consistent?  If you say "Yes, that has
been proved", then what about the reasoning used in consistency proofs of
ZFC?  How do you know with mathematical certainty that such methods cannot
"prove" inconsistent systems consistent?

Patient: Doctor, it hurts when I try to give consistent axiomatic
    foundations for mathematics!
Doctor Brouwer: Don't try to axiomatize mathematics!
Doctor Wiener: Stop worrying!  It is just a natural thing on the way to
    mathematical adolescence we have all had.  Just concentrate on the
    beauty of ZFC, my child, and soon you will outgrow these qualms.

-- 

Lambert Meertens, CWI, Amsterdam; lambert@cwi.nl