FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
Uncountable model of PA
Post new topic   Reply to topic Page 1 of 1 [5 Posts] View previous topic :: View next topic
Author Message
Stephen Montgomery-Smith1
science forum Guru


Joined: 01 May 2005
Posts: 487

PostPosted: Mon Jul 10, 2006 10:08 pm    Post subject: Re: Uncountable model of PA Reply with quote

ashu_1559@rediffmail.com wrote:
Quote:
Could someone define successor, addition and multiplication of the set
of real numbers in such a way so that it becomes a model for Peano's
Arithmetic? I hope this can be done in view of the "upward" L÷wenheim
- Skolem theorem. Thanks in advance.


One way would be to add uncountably many symbols to your first order
language, and then add axioms "s!=t" and "s!=n" for symbols s and t, and
natural numbers n. It is not hard to prove that Con(PA) implies
Con(this bigger system of axioms) (because any contradiction can only
involve finitely many of these axioms). The model of this bigger system
of axioms must also be uncountable.

Stephen
Back to top
Dave Seaman
science forum Guru


Joined: 24 Mar 2005
Posts: 527

PostPosted: Mon Jul 10, 2006 6:37 pm    Post subject: Re: Uncountable model of PA Reply with quote

On 10 Jul 2006 10:26:03 -0700, ashu_1559@rediffmail.com wrote:

Quote:
Dave Seaman wrote:

On 10 Jul 2006 10:16:57 -0700, ashu_1559@rediffmail.com wrote:
Could someone define successor, addition and multiplication of the set
of real numbers in such a way so that it becomes a model for Peano's
Arithmetic? I hope this can be done in view of the "upward" L÷wenheim
- Skolem theorem. Thanks in advance.

The Robinson construction of the hypernaturals using an ultraproduct
surely gives an uncountable model for PA. The cardinality is the same as
that of the standard reals.

Could you also suggest a reference, please?

Abraham Robinson, _Non-standard Analysis_, Princeton University Press,
ISBN 0-691-04490-2.


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Back to top
ashu_1559@rediffmail.com1
science forum beginner


Joined: 10 Jul 2006
Posts: 3

PostPosted: Mon Jul 10, 2006 5:26 pm    Post subject: Re: Uncountable model of PA Reply with quote

Dave Seaman wrote:

Quote:
On 10 Jul 2006 10:16:57 -0700, ashu_1559@rediffmail.com wrote:
Could someone define successor, addition and multiplication of the set
of real numbers in such a way so that it becomes a model for Peano's
Arithmetic? I hope this can be done in view of the "upward" L÷wenheim
- Skolem theorem. Thanks in advance.

The Robinson construction of the hypernaturals using an ultraproduct
surely gives an uncountable model for PA. The cardinality is the same as
that of the standard reals.

Could you also suggest a reference, please?


Quote:


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
http://www.mumia2000.org/
Back to top
Dave Seaman
science forum Guru


Joined: 24 Mar 2005
Posts: 527

PostPosted: Mon Jul 10, 2006 5:22 pm    Post subject: Re: Uncountable model of PA Reply with quote

On 10 Jul 2006 10:16:57 -0700, ashu_1559@rediffmail.com wrote:
Quote:
Could someone define successor, addition and multiplication of the set
of real numbers in such a way so that it becomes a model for Peano's
Arithmetic? I hope this can be done in view of the "upward" L÷wenheim
- Skolem theorem. Thanks in advance.

The Robinson construction of the hypernaturals using an ultraproduct
surely gives an uncountable model for PA. The cardinality is the same as
that of the standard reals.



--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Back to top
ashu_1559@rediffmail.com1
science forum beginner


Joined: 10 Jul 2006
Posts: 3

PostPosted: Mon Jul 10, 2006 5:16 pm    Post subject: Uncountable model of PA Reply with quote

Could someone define successor, addition and multiplication of the set
of real numbers in such a way so that it becomes a model for Peano's
Arithmetic? I hope this can be done in view of the "upward" L÷wenheim
- Skolem theorem. Thanks in advance.
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [5 Posts] View previous topic :: View next topic
The time now is Sun Nov 18, 2018 5:04 pm | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts Model of Reality and magnetism Fabrizio J Bonsignore Electromagnetics 5 Sun Jul 09, 2006 11:47 pm
No new posts new model of numbers dinar num-analysis 0 Sun Jul 02, 2006 8:23 am
No new posts What is this probability tree model called? news.cuhk.edu.hk Probability 0 Fri Jun 30, 2006 6:18 pm
No new posts What is a good model to analyze unbalanced hierarchical n... news.cuhk.edu.hk Probability 0 Thu Jun 29, 2006 4:48 pm
No new posts how to adjust the SAM in a CGE model wzt num-analysis 2 Thu Jun 29, 2006 3:04 pm

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0141s ][ Queries: 20 (0.0023s) ][ GZIP on - Debug on ]