# Re: Mathematical Logic, Podnieks'page ...

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Fri, 02 Jul 2004 19:15:29 +0200

Hi Hal,

At 12:44 02/07/04 -0400, Hal Ruhl wrote:

>By the way if some systems are complete and inconsistent will arithmetic
>be one of them?
>
>As I understand it there are no perfect fundamental theories. So if
>arithmetic ever becomes complete
>then it will be inconsistent.

Yes, if by "arithmetic" you mean an axiomatic system, or a formal theory,
or a machine.
No if by arithmetic you mean a set so big that you cannot define it in any
formal theory, like
the set of all true arithmetical sentences. That set cannot be defined in
Peano arithmetic for exemple. Some logician use the word "theory" in that
generalized sense, but it is misleading.
Now the set of true sentence of arithmetic is that large sense is obviously
consistent gieven that it contains only the true proposition! (but you
cannot defined it "mechanically").

>In the foundation system which I believe contains mathematics from the
>beginning arithmetic is complete so its inconsistent.

No, because if it is complete, it will not be a mechanical or formal
system. Only a theory will
be inconsistent if both complete and enough rich. Not a model.

To borrow Boolos title, I would like to say I get the feeling this list is
Logic, logic and logic ....
BTW an excellent introduction to elementary logic is the penguin book by
Wilfried Hodges :
http://www.amazon.co.uk/exec/obidos/ASIN/0141003146/qid=1088787942/sr=1-2/ref=sr_1_26_2/026-1716457-4246007

Only the first sentence of the book is false. (will say more on that book
later ...)

Bruno

http://iridia.ulb.ac.be/~marchal/
Received on Fri Jul 02 2004 - 13:13:08 PDT

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