On Thu, 28 Feb 2008, Pat Hayes wrote:
> At 3:55 PM -0500 2/28/08, John F. Sowa wrote:
>> ...
>> But if you look at what Goedel did, he took arithmetic as
>> the foundation and mapped logic to arithmetic because most
>> mathematicians at that time (and I believe even at present)
>> have more faith in arithmetic than they have in logic.
>
> I think you are just historically wrong about this. Goedel didn't map
> logic to arithmetic for foundational reasons: he did it to show that
> arithmetic couldn't be given a foundation, in a sense. The reason for
> choosing arithmetic was (1) it was rich enough to support the mappings
> he needed and (2) it was - and still is - regarded as the most
> 'obvious' and simple part of mathematics, so his result was all the
> more shocking. (01)
Yes, I'd have to agree that John's history is a little off the mark
here. To expand on Pat's point: Gödel's mapping of the proof theory of
predicate logic into arithmetic had nothing to do with foundational
issues. Rather, it was the mechanism that enabled him to prove the
essential incompleteness of number theory. Specifically, he (and, in a
somewhat better form a couple years later, Rosser) showed that, for any
proposed axiomatization of arithmetic, one can always construct a
sentence that, in virtue of the mapping in question, in a certain
precise sense "says of itself" that it is not provable. Assuming the
consistency of the axioms, it follows that the sentence is neither
provable nor refutable by those axioms. (02)
-chris (03)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (01)
|