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Stephen MontgomerySmith1 science forum Guru
Joined: 01 May 2005
Posts: 487

Posted: Mon Jul 10, 2006 10:08 pm Post subject:
Re: Uncountable model of PA



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" Löwenheim
 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 

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Dave Seaman science forum Guru
Joined: 24 Mar 2005
Posts: 527

Posted: Mon Jul 10, 2006 6:37 pm Post subject:
Re: Uncountable model of PA



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" Löwenheim
 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, _Nonstandard Analysis_, Princeton University Press,
ISBN 0691044902.

Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia AbuJamal.
<http://www.mumia2000.org/> 

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ashu_1559@rediffmail.com1 science forum beginner
Joined: 10 Jul 2006
Posts: 3

Posted: Mon Jul 10, 2006 5:26 pm Post subject:
Re: Uncountable model of PA



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" Löwenheim
 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 AbuJamal.
http://www.mumia2000.org/ 


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Dave Seaman science forum Guru
Joined: 24 Mar 2005
Posts: 527

Posted: Mon Jul 10, 2006 5:22 pm Post subject:
Re: Uncountable model of PA



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" Löwenheim
 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 AbuJamal.
<http://www.mumia2000.org/> 

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ashu_1559@rediffmail.com1 science forum beginner
Joined: 10 Jul 2006
Posts: 3

Posted: Mon Jul 10, 2006 5:16 pm Post subject:
Uncountable model of PA



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" Löwenheim
 Skolem theorem. Thanks in advance. 

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