I’ve been thinking about the set theory multiverse and its philosophical implications, particularly regarding mathematical truth. While I understand the pragmatic benefits of the multiverse view, I’m struggling with its philosophical implications.
The multiverse view suggests that statements like the Continuum Hypothesis aren’t absolutely true or false, but rather true in some set-theoretic universes and false in others. We have:
Gödel’s Constructible Universe (L) where CH is true
Forcing extensions where CH is false
Various universes with different large cardinal axioms
However, I find myself appealing to basic logical principles like the law of non-contradiction. Even if we can’t currently prove certain axioms, doesn’t this just reflect our epistemological limitations rather than implying all axioms are equally “true”?
To make an analogy: physical theories being underdetermined by evidence doesn’t mean reality itself is underdetermined. Similarly, our inability to prove CH doesn’t necessarily mean it lacks a definite truth value.
Questions I’m wrestling with:
What makes certain axioms “true” beyond mere consistency?
Is there a meaningful distinction between mathematical existence and consistency?
Can we maintain mathematical realism while acknowledging the practical utility of the multiverse approach?
How do we reconcile Platonism with independence results?
I’m leaning towards a view that maintains objective mathematical truth while explaining why we need to work with multiple models pragmatically. But I’m very interested in hearing other perspectives, especially from those who work in set theory or mathematical logic.
[Question] Set Theory Multiverse vs Mathematical Truth—Philosophical Discussion
I’ve been thinking about the set theory multiverse and its philosophical implications, particularly regarding mathematical truth. While I understand the pragmatic benefits of the multiverse view, I’m struggling with its philosophical implications.
The multiverse view suggests that statements like the Continuum Hypothesis aren’t absolutely true or false, but rather true in some set-theoretic universes and false in others. We have:
Gödel’s Constructible Universe (L) where CH is true
Forcing extensions where CH is false
Various universes with different large cardinal axioms
However, I find myself appealing to basic logical principles like the law of non-contradiction. Even if we can’t currently prove certain axioms, doesn’t this just reflect our epistemological limitations rather than implying all axioms are equally “true”?
To make an analogy: physical theories being underdetermined by evidence doesn’t mean reality itself is underdetermined. Similarly, our inability to prove CH doesn’t necessarily mean it lacks a definite truth value.
Questions I’m wrestling with:
What makes certain axioms “true” beyond mere consistency?
Is there a meaningful distinction between mathematical existence and consistency?
Can we maintain mathematical realism while acknowledging the practical utility of the multiverse approach?
How do we reconcile Platonism with independence results?
I’m leaning towards a view that maintains objective mathematical truth while explaining why we need to work with multiple models pragmatically. But I’m very interested in hearing other perspectives, especially from those who work in set theory or mathematical logic.