Incompleteness thru Venn Diagrams
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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: 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 |