User blog comment:Boboris02/Ordinal Analysis of Theories/@comment-11227630-20180213035846/@comment-30754445-20180214110634

Boboris, you're forgetting something:

Taranovsky has provided an algorithm for generating the fundamental sequences in his notation.

So proving the well-foundedness of TON would be synonymous to proving that the function f LIMIT OF TON (n) (with these fundamental sequences) is a total function, and this cannot be done in any theory whose PTO is greater than TON.

This is exactly what a PTO means. It is a measure the recursion complexity that can be achieved from within the theory.

And remember: Even Peano arithmetic can talk indirectly about well-orders (and therefore - about ordinals). We can encode any finite sequence into the prime factors of a single integer, and then define certain well-orders as a family of such integers.

For example, we can prove in PA that the length of any sequence which obey the following rules:

(1) The number immediately after an odd number, is a smaller odd number

(2) The number immediately after an even number, is either a smaller even number or any odd number.

Is always less than N+M, where N is the first number in the sequence and M is the first odd number in the sequence (which we are guranteed to reach after less than N steps).

This is a theorem in PA, and it is equivalent to proving that the ordering 1,3,5,7,...2,4,6,8,... is a well-order. This of-course corresponds to the ordinal ωx2.

Similarly, PA can prove that more complicated well-orderings are - indeed - well orders. But it cannot do so for any well-order bigger than PTO(PA) = ε₀.