Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

tchow_at_lsa.umich.edu
Date: 01/27/05 

Date: 27 Jan 2005 15:02:12 GMT

In article <ct8928$ijm$1@ra.nrl.navy.mil>,
Ralph Hartley <hartley@aic.nrl.navy.mil> wrote:
>But earlier in this thread when I said:
>> Ralph Hartley <hart...@aic.nrl.navy.mil> wrote:
>>>when mathematicians make unqualified statements, with no other
>>>context, they usually *mean* "In ZFC".
>You said:
>> As a matter of sociological fact, this is definitely false.
>
>The only difference I can see between my interpretation and yours is that,
>you refer to a particular model V.

I feel that I didn't answer this adequately in my other article, so here's
some more comment.

The first point is that ZFC is a set of axioms whereas V is the class of all
sets, so they're fundamentally different kinds of entities, and so "in ZFC"
and "in V" are fundamentally different kinds of predicates. I can make a
claim of the form

    [1] "S" really means "S is true in V"

but to say

    [2] "S" really means "S is true in ZFC"

is a grammatical blunder. When you said "in ZFC" I think you probably meant
something like

    [3] "S" really means "S is provable in ZFC"

Now, [1] is somewhat controversial, but at least there's plenty of room for
debate on both sides. As I explained in my other article, I think [1] makes
sense for S = AC, but is less plausible for your average mathematical
statement that makes use of only very low levels of the cumulative
hierarchy.

On the other hand, [3] is much less tenable. For example, if [3] were true,
then you would expect that

    [4] S iff S is provable in ZFC

would be provable in a very weak theory, for any S. But this isn't the
case. For instance, if CH denotes the continuum hypothesis, then

    [5] CH -> CH is provable in ZFC

isn't provable even in ZFC (unless ZFC+Con(ZFC) is inconsistent), let alone
a weak theory. Conversely, if Con(ZFC) denotes "ZFC is consistent," then

    [6] Con(ZFC) is provable in ZFC -> Con(ZFC)

isn't provable in ZFC unless ZFC is inconsistent. (For Goedel's 2nd theorem
is provable in ZFC, so

    [7] Con(ZFC) is provable in ZFC -> ~ Con(ZFC)

is provable in ZFC. So if [6] were also provable in ZFC, then ZFC would
disprove "Con(ZFC) is provable in ZFC," i.e., ZFC would prove that something
is not provable in ZFC, so in particular ZFC would prove that ZFC is
consistent, which implies that ZFC is inconsistent by Goedel's 2nd theorem
again.) In fact, [6] is an instance of something called a "reflection
principle." It's very plausible informally, because if we "believe in"
ZFC then we believe that anything provable in ZFC is actually true.
However, one consequence of Goedel's results is that this plausible
principle is, logically speaking, an additional assumption over and above
the axioms of ZFC themselves.

Another way to remember that "S" and "S is provable in ZFC" are very
different statements is to think about what it takes to formalize each
of these statements. To formalize "S is provable in ZFC" we need to
formalize the concept of provability, which is a nontrivial exercise.
The provability predicate is a monstrously complicated thing if written
out in full gory detail. So "S is provable in ZFC" is necessarily a
monstrously long formal sentence, regardless of S. In contrast,
formalizing "S" itself could be easy if "S" is a simple statement.

What is probably generating confusion is the familiar sociological fact
that mathematicians do not confidently assert mathematical statements
unless they have a proof of that statement available. If someone tells
you confidently, "Every differentiable function is continuous," you can
rightly infer that (provided he is a trustworthy person) he knows a proof
(or at least knows *of* a proof) that every differentiable function is
continuous. But the fact that I won't assert S unless I know how to
justify S doesn't mean that "S" *means* "S is justifiable." This holds
not only in mathematics but in ordinary discourse. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

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