[FOM] Is Godel's Theorem surprising?
Rob Arthan
rda at lemma-one.com
Mon Dec 11 18:12:08 EST 2006
On 10 Dec 2006, at 07:19, joeshipman at aol.com wrote:
> -----Original Message-----
> From: rda at lemma-one.com
>>> ...
>>> I'm also wondering, though this is a separate point, whether today
>>> the theorem is not only not surprising, but perhaps even intuitively
>>> obvious.
>> But then there must be some intuitively obvious difference between a
> complete
>> system like the first-order theory of the reals and an incomplete
> system like
>> PA or ZF. What would that intuition be?
>
>
>
> The intuition is simply that the first-order theory of the reals is
> "only about the reals', while it is easy to frame any question of
> mathematical interest in ZF, and easy to frame any "finitary' question
> using PA, so these theories are about mathematics in general.
>
My point (which I'm sorry to say I didn't make very clear) was that,
whatever your intuitions about PA or ZF might be, complete theories are
not, in general, easy to recognise. Just saying "the first-order theory
of the reals is only about the reals" doesn't really help you design a
quantifier-elimination procedure for it!
Regards,
Rob.
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