Many of us are in the habit of saying that any system strong enough to
support arithmetic must be incomplete. But what about the theorem that any
consistent first-order theory has a consistent and complete extension?

Many of us are in the habit of saying that any system strong enough to
support arithmetic must be incomplete. But what about the theorem that any
consistent first-order theory has a consistent and complete extension?

This is proven on p. 159 of Angelo Margaris' _First Order Mathematical
Logic_.

Let Q_1, Q_2, ... be an enumeration of all the statements of T. Add Q_1 as
an axiom of T if T + Q_1 is consistent; otherwise, do not. Then do the same
with Q_2, Q_3, ... We easily arrive at a consistent and complete extension
of first-order arithmetic in which no propositions are left undecidable.


But its not r.e., because "T+Q_i consistent" is not semi decidable.

Godel was concerned to show that systems in nature, like the human mind,
can be expressed completely by an arithmetic syntax.

Godel was concerned to show that the reason why nature was not a
machine
was that the syntax of arithmetic (on which nature was grounded)
required more than a purely formal element. It required a subjective
component which gave meaning to terms. The subjective component could
not be presented by a formal presentation.

On Wed, 18 Jun 2008 14:45:55 +0100, John Jones <jonescardiff at (no spam) aol.com
wrote:

Scott H wrote:
Many of us are in the habit of saying that any system strong enough to
support arithmetic must be incomplete.
More precisely, what is meant by that is that any system, such as the
natural world, that can be expressed through an arithmetic syntax is
completely expressed by that syntax if we endorse Godellian syntax.

My god. It's no wonder you're confused about the consequences
of the incompleteness theorems if you think that this is what they
say.

David C. Ullrich