# Replaceable Axioms give more credence than irreplaceable axioms

*Epistemic status: hand waving but confidant.*

I was thinking about [reflective reasoning](https://www.lesswrong.com/tag/reflective-reasoning) and "[strange loops through the meta level](https://www.lesswrong.com/posts/C8nEXTcjZb9oauTCW/where-recursive-justification-hits-bottom)" and it lead me to this intuition that even if you have to make some assumptions to support your beliefs, if your beliefs can be justified by various different sets of assumptions, and not just one specific set of assumptions, then it gives those beliefs more credence.

What do I mean by justification by different sets of assumptions? Let's take peano arithmetic (PA) as an example (though any other formal system can work for this purpose as well).

1.  Is there an axiom of PA that can be replaced with a statement that is already true in PA such that PA would stay basically the same (the same statements would be true) and the old axiom would be provable as a regular statement in new-PA?
2.  If so, is it true for any PA axioms?
3.  If so, is it true for all PA axioms at once? Can you theoretically replace all PA axioms with statements that are already true in PA, prove the old axioms in the new system, and remain with a system where all the same statements are correct and no new ones are?

If (1) is correct, then it's not necessary to accept that axiom as long as you're willing to accept its replacement (and assuming that you're fine with PA but just don't like *assuming* stuff, you shouldn't have a specific objection with the replacement statement either).

If (2) is correct, then the same is true for *any* PA axiom.

if (3) is correct, then the same is true for any set of PA axioms. Which means if you think PA really does prove only correct things, you don't actually have to accept the axioms as mere assumptions, because you can prove them from their substitutes, which you already accept.

*(Question: Is this actually true about peano arithmetic?)*

So although at any moment you'd be using axioms to define what you're talking about and prove statements, none of them could be said to be *required and permanent* assumptions. There would be no assumptions you can be blamed for *always* assuming.

To bring it back to reflective reasoning, it would match the intuition that no belief, even the most fundamental ones like inductive and occemian priors, are beyond scrutiny under [the full power of our reasoning](https://www.lesswrong.com/posts/C8nEXTcjZb9oauTCW/where-recursive-justification-hits-bottom), and therefore can't be said to be merely assumed.

This also reminds me of [@So8res](https://www.lesswrong.com/users/so8res?mention=user)' [cross-entropy](https://www.lesswrong.com/posts/KcvJXhKqx4itFNWty/k-complexity-is-silly-use-cross-entropy-instead) idea - that instead of just prioritizing the hypothesis which has the shortest code, we should prioretize the hypothesis which has many different short codes.

In the same way, the more assumptions a belief requires, and the more complex these assumptions are, the more we discount that belief. But we should also look at how many different sets of assumptions can support that belief, and prioritize beliefs which can rely on more sets of assumptions rather than on ones that require very specific assumptions.

This also applies to beliefs supported by [circular justifications](https://www.lesswrong.com/posts/8yuQua2hgSiowv6qp/circular-reasoning). If a beliefs requires a very specific circle, it's less likely to be true than if it can fit in many different circles.

If there's no *particular* set of assumptions my worldview permanently depends on, if it can spring from many and various different sets of assumptions, then it's a stronger worldview. Even if it still requires assumptions or even circular reasoning.

I think this can be a step towards, or a part of, a solution to the regress problem or even [The Problem of the Criterion](https://www.lesswrong.com/tag/the-problem-of-the-criterion), because it alleviates the need to permanently rely on just one criterion.

This post only shows an intuition, so I would love to see someone formalise it.