/sci/ - Science & Math

>>14961201
>what is a "proof-theoretic ordinal"?
It has a couple of meanings, but usually the "proof theoretic ordinal" of a theory T is the smallest ordinal that T cannot prove is well-founded. So in general, a stronger theory has a higher proof-theoretic ordinal associated with it, but this isn't a strict thing and it isn't a linearly-ordered thing. And more broadly, people might call an ordinal a "proof-theoretic ordinal" if it looks kinda like the sort of ordinal that shows up in these kinds of investigations.

The original proof-theoretic ordinal is ϵ0, which is the limit of the sequence of ordinal-exponentiations [math]
\omega, \omega^\omega, \omega^{\omega^\omega},\ldots[/math], and is the proof-theoretic ordinal of Peano arithmetic (PA). The reason for this is the proof-theoretic ordinal of PA has two parts: first, it is not super hard to see that PA proves every smaller ordinal is well-ordered. And second, it is a theorem of Gentzen that if [math]\epsilon_0[/math] is well-founded, then PA is consistent. Together with Godel's incompleteness theorem, it follows that PA cannot prove the well-foundedness of [math]\epsilon_0[/math].

Unfortunately Gentzen was too based for this world and the Harley Pasternaks don't want you to know about based Gentzen, blacklisting his shit and making proof-theoretic ordinals hard to understand

>>14961201
>what is a "proof-theoretic ordinal"?
It has a couple of meanings, but usually the "proof theoretic ordinal" of a theory T is the smallest ordinal that T cannot prove is well-founded. So in general, a stronger theory has a higher proof-theoretic ordinal associated with it, but this isn't a strict thing and it isn't a linearly-ordered thing. And more broadly, people might call an ordinal a "proof-theoretic ordinal" if it looks kinda like the sort of ordinal that shows up in these kinds of investigations.

The original proof-theoretic ordinal is [math]\epsilon_0[/math], which is the limit of the sequence of ordinal-exponentiations [math]\omega, \omega^\omega, \omega^{\omega^\omega},\ldots[/math], and is the proof-theoretic ordinal of Peano arithmetic (PA). The reason this is the ordinal of PA is twofold: first, it is not super hard to see that PA proves every smaller ordinal is well-ordered. And second, it is a theorem of Gentzen that if [math]\epsilon_0[/math] is well-founded, then PA is consistent. Together with Godel's incompleteness theorem, it follows that PA cannot prove the well-foundedness of [math]\epsilon_0[/math].

Unfortunately Gentzen was too based for this world and the Harley Pasternaks don't want you to know about based Gentzen, making proof-theoretic ordinals hard to understand