$\newcommand{\Con}{\operatorname{Con}}$The intention of Hilbert's program was to start with a simple logic and then justify more complex logics from there. So, we get a sequence of logics:
$$
L\_1 → L\_2 → L\_3 \to \cdots
$$

His initial idea was to justify a succession by $L\_n \vdash \Con(L\_{n+1})$. This idea was derailed by Gödel's second incompleteness theorem. You can't prove the consistency of a more complex system if you can't prove your own consistency.

However, we might think of other ways of succession. The next obvious one is to use relative consistency, $L\_n \vdash \Con(L\_n) → \Con(L\_{n+1})$. Unfortunately this also fails due to Gödel, because more complex logics can prove the consistency of the lesser complex logics. If $L\_{n+1} \vdash \Con(Ln)$ and $L\_n \vdash \Con(L\_n) → \Con(L\_{n+1})$ and $L\_{n+1}$ can prove everything of $L\_n$, then that would imply $L\_{n+1} \vdash \Con(L\_{n+1})$.

A number of times is suggest to have the succession as follows: $L\_{n+1} = L\_n + \Con(L\_n)$. The funny thing about this is that it defies Gödel, now we can make consistency proofs in the succession (although Gödel still applies for the whole system). Unfortunately the successor logic can not do much more except for the consistency proof.

A closer look learns that the strength of a system is often determined by the expressiveness allowed for the induction hypothesis. An example is $I\Sigma\_1$ where the induction scheme is limited to $\Sigma\_1$ sentences. $I\Sigma\_2$ can prove the consistency of $I\Sigma\_1$. Also the classifications in reverse mathematics are partly based on what is allowed for the induction hypothesis.

The means that if we have a logic and extend it with additional expressiveness and that expressiveness is also allowed for the induction hypothesis, then the system will get stronger.

*More expressiveness leads to more strength*

This devastating for Hilbert's program. A more complex logic is not just a handy way to deal with certain concepts, as it is with higher level computer languages compared to the Turing machine, but it also adds more truth.

However, we humans accept easily higher level constructs for induction hypothesis in a succession. When we create a succession:

1) We prove that the new logic is a conservative extension of the original one, not taking induction into account.

2) Well, for the induction we just “intuitively” accept it.

Let's call this the “universal induction” principle. This principles informally says we can just accept anything as induction hypothesis. As far I know we have never encountered a paradox in this area.

My question is, is there a way to formalize this universal induction principle? I don't think this question can directly be answered. A positive answer would be a major development in Hilbert's program. A negative answer is probably not possible, because the question is not a precise mathematical question.

Is there some literature that thinks the same way?

If not and if you think it is possible, what are the restrictions we must set and in which we must formulate this?

If you think it is not possible, what are the arguments against?

Although the question is not mathematical precise, I hope it is not voted down for this reason. I am thinking about this, made a little progression and like to hear the opinion of the community.