random (fwd)

[David C. Ullrich]

On Sat, 2 Jun 2001 21:07:57 +0200, "Alex Martelli" <aleaxit at yahoo.com>
wrote:
[...]
>
>
>> |I _think_ -- but am not a philosopher or mathematician
>> |so cannot feel sure -- that part of what Goedel shows is
>> |that arithmetic is powerful enough to model any other
>> |formal system.
>>
>> Well...  I *am* a philosopher, and have occassionally pretended to be a
>> mathematician (not the most convincing pretense though).  But I don't
>> think one needs to be to see what's wrong with the above.
>>
>> Assuming "Goedel" means his incompleteness theorem in the above, he
>> really doesn't show anything like the above.
>
>Sorry, I wasn't thinking specifically of his theorem but rather
>of his way of operating towards it.  Given any (finite) axiomatic
>system, even 'richer' than arithmetic, can't we MODEL that
>richer system with arithmetic, numbering well-formed
>formulas &c, by just the same process as Goedel uses to
>model arithmetic in itself? 

So my guess what you meant by that was correct. So was
my reply, which is still missing: No, but yes under
some weak technical conditions (like the language
needs to be countable. Uncountable languages actually
come up in logic.)

> The fact that we added a few
>primitive symbols and axioms to the set we started with
>for arithmetic (be that Peano's or whatever) doesn't appear
>to me to break down the process. 

Precisely. The words "a few" are important - if we
added really really a _lot_ of primitives we run out
of Godel numbers.


David C. Ullrich
*********************
"Sometimes you can have access violations all the 
time and the program still works." (Michael Caracena, 
comp.lang.pascal.delphi.misc 5/1/01)