Talk:Gentzen's consistency proof

message: The text used to start this article was cut from Gödel's incompleteness theorem and pasted here. --Trovatore 04:32, 15 February 2006 (UTC)
thread_title: Origin

message: Most other articles define ε0 as the least ordinal \alpha such that \omega^\alpha=\alpha, not such that \alpha^\omega = \alpha.  Are these two definitions equivalent? If we use the definition of ε0 at ordinal number, then isn't it true that: 
:{\varepsilon_0}^\omega=(\omega^{\varepsilon_0})^\omega=\omega^{\varepsilon_0\omega}=\omega^{(\omega^{\varepsilon_0})\omega}=\omega^{\omega^{\varepsilon_0+1}}>\omega^{\omega^{\varepsilon_0}}=\varepsilon_0
I don't really know anything about ordinals, I'm just using the rules at ordinal arithmetic, so I might have made a mistake. Can anyone verify this? BrentG 19:01, 17 August 2006 (UTC)
replies: text: Those expressions are not equivalent. \alpha^\omega > \alpha holds for all ordinals α except 0 and 1. However, I am not aware of anything to the contrary at ordinal number. Please name the section or subsection and paragraph number where you saw it. I do see "... the \gamma-th ordinal \alpha such that \omega^\alpha=\alpha is written \varepsilon_\gamma. These are called the 'epsilon numbers'." which is correct. JRSpriggs 03:36, 18 August 2006 (UTC)
replies: text: Nothing at ordinal number contradicts that. It's this article (Gentzen's consistency proof) that says "ε0 is the first ordinal \alpha, such that \alpha^\omega = \alpha".  I figured it was a mistake, but I didn't want to edit a technical detail on a subject I don't know that much about without getting someone else's opinion. I'll switch it to the correct definition. BrentG 06:15, 19 August 2006 (UTC)
replies: text: Sorry that I misunderstood you. I looked at this article, but I guess that I just saw what I expected to see. Especially since the line below it with the sequence of powers of omega looked OK. Thanks for fixing this article. JRSpriggs 08:09, 19 August 2006 (UTC)

text: The proof is not "very easy" but it is not carried out in set theory, it's carried out in a very weak fragment of second-order arithmetic. 66.127.55.192 (talk) 07:22, 10 February 2010 (UTC)
thread_title:  Definition of &epsilon;<sub>0</sub> 

message: In what sense is the proof based on Primitive Recursive Arithmetic? Gentzen's original papers prove the consistency of Peano Arithmetic (albeit using the axioms of PRA in the 1938 version). The proofs are completely unformalized and Gentzen does not say anything specific about formalization. Maybe it would be more appropriate to say that "Gentzen's proof can be carried out in" PRA+TI(epsilon0)?--141.70.13.27 (talk) 11:28, 6 January 2012 (UTC)
thread_title:  Primitive Recursive Arithmetic 

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