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From: Charlie-Boo on 24 Jun 2010 07:52
Consider the Venn Diagram that consists of an overall universal set of
all sets of sentences, and two overlapping sets within it. One inner
set is the set of r.e. sets, and the other is the set of all co-r.e.
sets (i.e. its complement is r.e.) Now place these four sets within
the Venn Diagram: PR=provable sentences, DIS=refutable sentences,
TW=true sentences, TRUE=all sentences. That is, each of these 4 sets
will occur somewhere within the Venn Diagram.

Notice the pattern! Now, what can we prove from the fact that this
pattern occurs?

TONS!

C-B
From: George Greene on 24 Jun 2010 11:06
On Jun 24, 7:52 am, Charlie-Boo <shymath...(a)gmail.com> wrote:
> Consider the Venn Diagram that consists of an overall universal set of
> all sets of sentences, and two overlapping sets within it.  One inner
> set is the set of r.e. sets, and the other is the set of all co-r.e.
> sets (i.e. its complement is r.e.)  Now place these four sets within
> the Venn Diagram:

OK.

> PR=provable sentences,
These are r.e. but not co-r.e. (if the theory is rich enough for G1
to apply).

> DIS=refutable sentences,
This is exactly the same as PR, up to the toggling of "~" at the
front.
These two sets are not only in the same region, they might as well
be the SAME DOT (same set) in that region.

> TW=true sentences,

This is outside of both sets (neither r.e. nor co-r.e.) if the theory
is as rich
as, e.g, true arithmetic (or should that be "as poor" -- it is
indviidual MODELS
of the theory that get to have truths;in this case, however, ALL the
models
will have non-r.e. truth-sets).

> TRUE=all sentences.
This set,being trivial, is both r.e. and co-r.e.

> That is, each of these 4 sets will occur somewhere within the Venn Diagram.
>
> Notice the pattern!

I honestly doubt that you noticed this pattern.

> Now, what can we prove from the fact that this pattern occurs?

Obviously it was more important to prove THAT the pattern occurs,
if a pattern is what you want to call it. I don't.

> TONS!

Well, you could've proved the SAME tons from whatever machinery you
used to prove the original 4 results above.
From: Charlie-Boo on 24 Jun 2010 12:00
On Jun 24, 7:52 am, Charlie-Boo <shymath...(a)gmail.com> wrote:
> Consider the Venn Diagram that consists of an overall universal set of
> all sets of sentences, and two overlapping sets within it.  One inner
> set is the set of r.e. sets, and the other is the set of all co-r.e.
> sets (i.e. its complement is r.e.)  Now place these four sets within
> the Venn Diagram: PR=provable sentences,

"DIS=refutable sentences,"

Oops, make that ~DIS (where DIS=refutable sentences)

> TW=true sentences, TRUE=all sentences.  That is, each of these 4 sets
> will occur somewhere within the Venn Diagram.
>
> Notice the pattern!  Now, what can we prove from the fact that this
> pattern occurs?
>
> TONS!
>
> C-B

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