(Attemp to post) transfinite induction question.

[KakarotGoku]

So im still new at this but im pretty curious to how strong transfinite induction can get by default?

 

[Cat275]

So im still new at this but im pretty curious to how strong transfinite induction can get by default?

Since it can make a sequence it should be able to reach 1a to maybe 1a+ at best.

 

[KakarotGoku]

Since it can make a sequence it should be able to reach 1a to maybe 1a+ at best.

I see, may i know why not H1-A with a infinite sequence of this?

 

[Cat275]

I see, may i know why not H1-A with a infinite sequence of this?

H1-A requires the need to have a reference point of inaccessible ie be a inaccessible cardinal while in transfinite induction the sequence would be a set of hereditary sets(since it's a sequence) smaller than a large cardinal like inaccessible, another reason is that limitary ordinals like epsilon exist which is already the limit or the final product of a transfinite induction or recursion sequence but still smaller than inaccessible since it is still not a large cardinal or so.

(Also inaccessible itself is a limit cardinal bigger than the default ordinals transfinite induction could get.)

Though you could probably reach H1-A to 0 in Transfinite induction or recursion if you can use higher math somehow but by default it can only reach 1-A+ at best.

 

[KakarotGoku]

H1-A requires the need to have a reference point of inaccessible ie be a inaccessible cardinal while in transfinite induction the sequence would be a set of hereditary sets(since it's a sequence) smaller than a large cardinal like inaccessible, another reason is that limitary ordinals like epsilon exist which is already the limit or the final product of a transfinite induction or recursion sequence but still smaller than inaccessible since it is still not a large cardinal or so.

(Also inaccessible itself is a limit cardinal bigger than the default ordinals transfinite induction could get.)

Though you could probably reach H1-A to 0 in Transfinite induction or recursion if you can use higher math somehow but by default it can only reach 1-A+ at best.

I see, thanks for the info!