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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

missing the key road:

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

Date: Fri, 02 Jul 2004 19:15:29 +0200

Hi Hal,

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

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").

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

missing the key road:

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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