Apparently pointing out that things like the axiom of choice, the GCH, or other statements fall under Godel’s proof and are not self referential is not popular around here.
Let’s say for now that the proofs that AC and CH being undecidable in ZF—AC don’t have much to do with self-reference. (This seems sort of true to me but my understanding of the forcing technique is very poor.) How does this at all relate to the comments which got downvoted?
Those are statements that fall under what Godel proved, that is they are statements that are unprovable in ZF. So even though his statement doesn’t include self-reference it can still fall under Godel’s proof if his decoder is strong enough to determine what is an integer and what is not an integer. Self-referencing has nothing to do with it at all.
The best response that anyone has given to me about being wrong itself references back to the halting problem which is itself another formulation of Godel’s proof.
The correct response would be to point out that deciding if something is an integer can be accomplished with just addition and the example decoder proceeds in that manner rather than using any form of multiplication to determine what is an integer and what is not. Such a system is decidable but also infinite, so the string given is indeed decidable and given infinite time the decoder will halt (at infinity which isn’t part of the axiomization, which is where the problem lies). However, what throws me is “we can find such pathological inputs for any other encoding system,” which to me implies a stronger system is being thought of which would cause the system to hang for some inputs as it would fall under Godel’s proof.
It is very funny to me that my most downvoted comment isn’t about religion but is about Godel’s proof and no one gave a decent refutation of what I said.
Those are statements that fall under what Godel proved, that is they are statements that are unprovable in ZF. So even though his statement doesn’t include self-reference it can still fall under Godel’s proof if his decoder is strong enough to determine what is an integer and what is not an integer. Self-referencing has nothing to do with it at all.
The existence of specific undecidable statements in ZF or ZF—AC is a different sort of result than what Godel showed. That for example the continuum hypothesis is undecidable in ZFC is interesting because the continuum hypothesis is interesting. However, Godel’s theorems show that any consistent, axiomatizable systemn with all valid proofs recursively enumerable, that is strong enough to model a large chunk of N, must be incomplete. Exhibiting results like Cohen’s results about choice and the continuum hypothesis don’t give you the full result, they just show specific things about the system ZF. Indeed, if one didn’t know Godel’s theorems, and just knew about Cohen’s forcing results, you might be tempted to just throw in AC and GH as additional axioms if you didn’t have them already and then wonder if that system is complete.
It is very funny to me that my most downvoted comment isn’t about religion but is about Godel’s proof and no one gave a decent refutation of what I said.
People here have a very complicated set of attitudes. Even as many don’t want to bother talking about irrational aspects of religion, or engaging in religious individuals who are convinced that their religious view is somehow different from all the other religious views, people who are religious are that way due to a variety of very strong cognitive biases. So there’s some sympathy there. Getting math wrong and then insisting one is correct is just simple arrogance or at least, only Dunning-Kruger. There’s much less sympathy there. And yes, people have tried to explain why you were wrong albeit fairly succinctly.
However, what throws me is “we can find such pathological inputs for any other encoding system,” which to me implies a stronger system is being thought of which would cause the system to hang for some inputs as it would fall under Godel’s proof.
Let’s say for now that the proofs that AC and CH being undecidable in ZF—AC don’t have much to do with self-reference. (This seems sort of true to me but my understanding of the forcing technique is very poor.) How does this at all relate to the comments which got downvoted?
Those are statements that fall under what Godel proved, that is they are statements that are unprovable in ZF. So even though his statement doesn’t include self-reference it can still fall under Godel’s proof if his decoder is strong enough to determine what is an integer and what is not an integer. Self-referencing has nothing to do with it at all.
The best response that anyone has given to me about being wrong itself references back to the halting problem which is itself another formulation of Godel’s proof.
The correct response would be to point out that deciding if something is an integer can be accomplished with just addition and the example decoder proceeds in that manner rather than using any form of multiplication to determine what is an integer and what is not. Such a system is decidable but also infinite, so the string given is indeed decidable and given infinite time the decoder will halt (at infinity which isn’t part of the axiomization, which is where the problem lies). However, what throws me is “we can find such pathological inputs for any other encoding system,” which to me implies a stronger system is being thought of which would cause the system to hang for some inputs as it would fall under Godel’s proof.
It is very funny to me that my most downvoted comment isn’t about religion but is about Godel’s proof and no one gave a decent refutation of what I said.
The existence of specific undecidable statements in ZF or ZF—AC is a different sort of result than what Godel showed. That for example the continuum hypothesis is undecidable in ZFC is interesting because the continuum hypothesis is interesting. However, Godel’s theorems show that any consistent, axiomatizable systemn with all valid proofs recursively enumerable, that is strong enough to model a large chunk of N, must be incomplete. Exhibiting results like Cohen’s results about choice and the continuum hypothesis don’t give you the full result, they just show specific things about the system ZF. Indeed, if one didn’t know Godel’s theorems, and just knew about Cohen’s forcing results, you might be tempted to just throw in AC and GH as additional axioms if you didn’t have them already and then wonder if that system is complete.
People here have a very complicated set of attitudes. Even as many don’t want to bother talking about irrational aspects of religion, or engaging in religious individuals who are convinced that their religious view is somehow different from all the other religious views, people who are religious are that way due to a variety of very strong cognitive biases. So there’s some sympathy there. Getting math wrong and then insisting one is correct is just simple arrogance or at least, only Dunning-Kruger. There’s much less sympathy there. And yes, people have tried to explain why you were wrong albeit fairly succinctly.
Possibly.