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On 8/9/2011 12:38 PM, Patrick Juola wrote:
<blockquote
cite="mid:CA+YKVPOup7-r1P4teAMoXHXv_Q7ixL1B8n15-HoKiOSE03SpGw@mail.gmail.com"
type="cite"><br>
<br>
<div class="gmail_quote">On Tue, Aug 9, 2011 at 12:31 PM, Angus
Grieve-Smith <span dir="ltr"><<a moz-do-not-send="true"
href="mailto:grvsmth@panix.com">grvsmth@panix.com</a>></span>
wrote:
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div class="im">
<br>
<blockquote class="gmail_quote" style="margin:0 0 0
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Godel showed that a formal system capable of doing
arithmetic perfectly cannot be both complete and
consistent.<br>
</blockquote>
</div>
Gödel showed that a formal system cannot be both complete
and consistent. His results were not specific to arithmetic.</blockquote>
<div><br>
</div>
<div>He did not. Aristotelian logic is a formal system that is
both complete and consistent; anything true and expressible in
the system is also provable in the system -- anything
(expressible) and provable in the system is also true. </div>
<div><br>
</div>
<div>It just happens to be a very weak system, since you can't
express basic arithmetic in it.</div>
<div><br>
</div>
Kleene's description of Godel's results (quoted in Wikipedia) is
relevant here: "Any effectively generated theory <i>capable of
expressing elementary arithmetic</i> cannot be
both consistent and complete. In particular, for any consistent,
effectively generated formal theory that proves certain basic
arithmetic truths, there is an arithmetical statement that is
true but not provable in the theory." (Italics mine.) That's
one of the key things that a lot of people don't understand
about Godel.</div>
<br>
</blockquote>
<br>
I didn't mention systems that were too weak to express basic
arithmetic truths, because I thought they were irrelevant to the
discussion, but yes, Gödel did use arithmetic to weed out complete,
consistent formal systems that were not powerful enough, and to
encode the syntax of the formal systems. That doesn't mean that his
theorem was all about arithmetic. The theorem makes a general point
about formal systems, and his genius was using arithmetic to make
that point.<br>
<br>
Saying this is about arithmetic is like saying that because he
wrote in German, his work only applies to formal systems when
they're being discussed in German.<br>
<br>
<pre class="moz-signature" cols="72">--
-Angus B. Grieve-Smith
<a class="moz-txt-link-abbreviated" href="mailto:grvsmth@panix.com">grvsmth@panix.com</a>
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