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Topic: Godel Incompleteness Theorem
Replies: 172   Last Post: May 2, 2013 2:49 AM

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Jose Carlos Santos

Posts: 4,872
Registered: 12/4/04
Re: Godel Incompleteness Theorem
Posted: Aug 21, 2008 4:56 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 21 Ago, 09:06, aatu.koskensi...@xortec.fi wrote:

> > I don't understand this statement. Are you saying that Gerhard
> > Gentzen' proof of the consistency of Arithmetic is trivial? Or do you
> > have something else in mind?

>
> Gentzen's proof is not at all trivial. What is trivial, by usual
> mathematical standards, is the consistency of Peano arithmetic,
> established by the following simple argument:
>
>  The axioms of Peano arithmetic are true, no contradiction is true
>  and the rules of inference of first-order logic preserve truth;
>  hence no contradiction is provable in Peano arithmetic.


Thanks.

> Here we use the notion of truth, a mathematical property of sentences
> in the language of arithmetic defined inductively. This notion, the
> definition of which has the form "A is true iff for all sets X of
> naturals ...", and the above argument, are not finitistically
> meaningful. Gentzen's proof, on the other hand, is finitistically
> meaningful, even though it contains an invocation of a principle,
> "quantifier-free transfinite induction up to epsilon-0", that is not
> finitistically justified.


Can you tell where can I find more details about this? I do not know
enough Logic to understand why is it that the definition of truth that
you mentioned is not finitistically meaningful.

>From such a proof we learn much more than
> just that Peano arithmetic is consistent -- about which there never
> were any real doubts -- e.g. a characterisation of the provably
> recursive functions of Peano arithmetic.


Again, thanks.

Best regards,

Jose Carlos Santos


Date Subject Author
7/21/08
Read Godel Incompleteness Theorem
thirdmerlin@hotmail.com
7/21/08
Read Re: Godel Incompleteness Theorem
Jesse F. Hughes
7/21/08
Read Re: Godel Incompleteness Theorem
G.E. Ivey
7/21/08
Read Re: Godel Incompleteness Theorem
Gerry Myerson
7/21/08
Read Re: Godel Incompleteness Theorem
Jesse F. Hughes
7/22/08
Read Re: Godel Incompleteness Theorem
Gc
7/22/08
Read Re: Godel Incompleteness Theorem
Paul Epstein
7/29/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
7/29/08
Read Re: Godel Incompleteness Theorem
Gerry Myerson
8/4/08
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
7/29/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/2/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/2/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/3/08
Read Re: Godel Incompleteness Theorem
Gerry Myerson
8/8/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/8/08
Read Re: Godel Incompleteness Theorem
Joshua Cranmer
8/9/08
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
8/9/08
Read Re: Godel Incompleteness Theorem
Joshua Cranmer
8/10/08
Read Re: Godel Incompleteness Theorem
Gerry Myerson
8/4/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/8/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/8/08
Read Re: Godel Incompleteness Theorem
Joshua Cranmer
8/9/08
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
8/12/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/12/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/18/08
Read Re: Godel Incompleteness Theorem
David R Tribble
8/19/08
Read Re: Godel Incompleteness Theorem
Jose Carlos Santos
8/19/08
Read Re: Godel Incompleteness Theorem
Gc
8/19/08
Read Re: Godel Incompleteness Theorem
Jose Carlos Santos
8/21/08
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
8/21/08
Read Re: Godel Incompleteness Theorem
Jose Carlos Santos
8/21/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/21/08
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
8/21/08
Read Re: Godel Incompleteness Theorem
Jack Markan
7/29/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
7/29/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
7/29/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/2/08
Read Re: Godel Incompleteness Theorem
Gc
8/3/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/3/08
Read Re: Godel Incompleteness Theorem
David Formosa (aka ? the Platypus)
8/12/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/3/08
Read Re: Godel Incompleteness Theorem
Gc
8/4/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/8/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/8/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/15/08
Read Re: Godel Incompleteness Theorem
thirdmerlin@hotmail.com
8/15/08
Read Re: Godel Incompleteness Theorem
Joshua Cranmer
8/15/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/18/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/18/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/18/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/19/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/20/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
5/2/13
Read Re: Godel Incompleteness Theorem
b9704084@hotmail.com
8/20/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/20/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/20/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/20/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/21/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/21/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/21/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/20/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/20/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/21/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/21/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/21/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/20/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/20/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/21/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/21/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/19/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
Jesse F. Hughes
8/20/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/20/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/20/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/20/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/21/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/21/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/22/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/22/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/28/08
Read Re: Godel Incompleteness Theorem
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8/29/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/18/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/18/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/19/08
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Herman Jurjus
8/19/08
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8/19/08
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8/18/08
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8/18/08
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8/18/08
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8/19/08
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LauLuna
8/19/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/19/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/19/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/19/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/20/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/20/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/20/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
8/20/08
Read Re: Godel Incompleteness Theorem
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8/21/08
Read Re: Godel Incompleteness Theorem
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8/21/08
Read Re: Godel Incompleteness Theorem
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8/21/08
Read Re: Godel Incompleteness Theorem
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8/21/08
Read Re: Godel Incompleteness Theorem
Jack Markan
8/19/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
8/19/08
Read Re: Godel Incompleteness Theorem
Mike Kelly
8/19/08
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8/19/08
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8/19/08
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LauLuna
8/19/08
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tchow@lsa.umich.edu
8/19/08
Read Re: Godel Incompleteness Theorem
Gc
8/19/08
Read Re: Godel Incompleteness Theorem
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8/28/08
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8/29/08
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8/29/08
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Zim Olson
8/30/08
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8/30/08
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9/2/08
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tchow@lsa.umich.edu
9/5/08
Read Re: Godel Incompleteness Theorem
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Read Re: Godel Incompleteness Theorem
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9/5/08
Read Re: Godel Incompleteness Theorem
tchow@lsa.umich.edu
9/7/08
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tchow@lsa.umich.edu
8/30/08
Read Re: Godel Incompleteness Theorem
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9/4/08
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Jack Markan
9/5/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
9/5/08
Read Re: Godel Incompleteness Theorem
Jack Markan
9/7/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
9/7/08
Read Re: Godel Incompleteness Theorem
contact080501@jamesrmeyer.com
9/8/08
Read Re: Godel Incompleteness Theorem
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9/8/08
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2/27/09
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contact080501@jamesrmeyer.com
2/27/09
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3/17/09
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3/17/09
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3/17/09
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Jesse F. Hughes
3/23/09
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3/23/09
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
3/24/09
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3/24/09
Read Re: Godel Incompleteness Theorem
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3/24/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/25/09
Read Re: Godel Incompleteness Theorem
Aatu Koskensilta
3/25/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/25/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/25/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/25/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/25/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/25/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/26/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/26/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/26/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/27/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/27/09
Read Re: Godel Incompleteness Theorem
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3/27/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/27/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/28/09
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barry barrett
3/30/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/30/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/31/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/31/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/31/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/31/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/31/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/31/09
Read Re: Godel Incompleteness Theorem
Jack Markan
3/31/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/31/09
Read Re: Godel Incompleteness Theorem
barry barrett
3/31/09
Read Re: Godel Incompleteness Theorem
Jack Markan
4/1/09
Read Re: Godel Incompleteness Theorem
barry barrett
4/1/09
Read Re: Godel Incompleteness Theorem
barry barrett

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