Post History
#3: Post edited
by
Julius H.
·
2024-02-10T16:16:33Z (9 months ago)
- 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 priced 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?
- 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?
#2: Post edited
by
Julius H.
·
2024-02-10T16:15:43Z (9 months ago)
- I’m curious to know more about this quote from a paper by Joel David Hamkins.
> The pervasive independence phe- nomenon 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 priced 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?
- 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 priced 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: Initial revision
by
Julius H.
·
2024-02-10T16:15:15Z (9 months ago)
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 phe- nomenon 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 priced 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?