Concrete examples of set theorists thinking independence proofs only determine provability rather than that a statement is neither true nor false?
I’m curious to know more about this quote from a paper by Joel David Hamkins.
The pervasive independence phenomenon in set theory is described on this view as a distraction, a side discussion about provability rather than truth — about the weakness of our theories in finding the truth, rather than about the truth itself — for the independence of a set-theoretic assertion from ZFC tells us little about whether it holds or not in the universe.
https://arxiv.org/abs/1108.4223
What is the “pervasive independence phenomenon”? Is he referring to the large number of statements independent of the axioms of ZFC, i.e. the continuum hypothesis, etc.?
In what way do the people he refers to think of independence phenomena as “a distraction”? I always thought mathematicians and set theorists thought of independence proofs as a huge deal with immense repercussions, such as Cohen’s proof regarding the continuum hypothesis. What are some examples of set theorists holding the “universe view” thinking independence proofs do not say anything deep about “the actual truth” of those independent statements?
…the independence of a set-theoretic assertion from ZFC tells us little about whether it holds or not in the universe.
How can this be? If it is independent, then it cannot be proved from the axioms. Thus, one has the freedom to assume it or assume the negation, as an axiom. Why would someone expect to know “whether it holds or not in the universe”, if it has been proven independent? Wouldn’t that answer the question?
1 answer
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| Julius H. |
Thread: Works for me He clarified the problem for me |
Feb 14, 2024 at 03:04 |
…the independence of a set-theoretic assertion from ZFC tells us little about whether it holds or not in the universe.
How can this be? If it is independent, then it cannot be proved from the axioms. Thus, one has the freedom to assume it or assume the negation, as an axiom. Why would someone expect to know “whether it holds or not in the universe”, if it has been proven independent? Wouldn’t that answer the question?
Hamkins' point is that it answers the question if you take a multiverse philosophical approach to set theory. However, if you believe that there is One True Set Theory it changes the question from "Does ZFC prove CH or ¬CH" to "Is ZFC+CH or ZFC+¬CH the correct choice of axioms?"
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