3 ms·

I don't know if it is quite what you are looking for, but with the normal mathematical axioms, it isn't possible to prove whether or not the there are any sets

by thayne 4y ago

I don't know if it is quite what you are looking for, but with the normal mathematical axioms, it isn't possible to prove whether or not the there are any sets with a cardinality between the cardinality of the natural numbers and the cardinality of the real numbers. But one of those two must be true, you just can't prove it. Of course you can add a new axiom that allows you to prove one or the other (or accept one of those statements as an axiom), but you will still have other statements that you can't prove.

DontchaKnowit 4y ago
This is super interesting. Was not aware of this. Is there a proof that this is not provable?
- thayne 4y ago
  Yes, Paul Cohen proved that the continuum hypothesis (that there is no set with cardinality between that of natural numbers and real numbers) was independent of the ZFC formulation of set theory (the most standard set of axioms used in today's set theory) in 1963.
  - schoen 4y ago
    More about this on Wikipedia: https://en.wikipedia.org/wiki/Continuum_hypothesis#Independence_from_ZFC https://en.wikipedia.org/wiki/Continuum_hypothesis#Independe... Also: https://en.wikipedia.org/wiki/Axiom_of_choice#Independence https://en.wikipedia.org/wiki/Axiom_of_choice#Independence And thanks to Wikipedia for this elaborate list: https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC https://en.wikipedia.org/wiki/List_of_statements_independent...
schoen 4y ago
> But one of those two must be true, you just can't prove it. Three alternative accounts: * Both the CH and ¬CH mathematical universes really exist, so we just have to choose which one we're more interested in at a given time. Like one might say there are the "reall numbers" and the "realle numbers", both valid and interesting constructions which humanity was just slow to recognize the distinctions between (because they were initially less relevant to our interests and our day-to-day lives). * We are actually ultimately thinking about one or the other of them, or are in some sense in one or the other mathematical universe, but we don't know enough about our intuition about the reals to be able to specify or explain which one. (Maybe we need other properties whose obviousness or relevance humanity is not smart enough to notice?) * Some finitist or ultrafinitist approach is actually right: the real numbers are a formalism that, while reasonably motivated by historical attempts to "complete" mathematics in various ways, doesn't correspond to anything Platonically real or to anything intellectually relevant to humanity. (In this account, there is potentially no answer to the question because the real numbers don't exist at all. Neither CH nor ¬CH refers to a mathematical reality, just to games about formalisms.)
- thayne 4y ago
  sure, but if you ignore any connection to the real world or platonic truth and are just talking about within the ZFC formalism, then `CH ∨ ¬CH` is a true statement, even though it is impossible to prove either `CH` or `¬CH`.
  - chriswarbo 4y ago
    > within the ZFC formalism, then `CH ∨ ¬CH` is a true statement It's also a provable statement (via excluded middle)

by thayne 4y ago

DontchaKnowit 4y ago

This is super interesting. Was not aware of this. Is there a proof that this is not provable?

thayne 4y ago
Yes, Paul Cohen proved that the continuum hypothesis (that there is no set with cardinality between that of natural numbers and real numbers) was independent of the ZFC formulation of set theory (the most standard set of axioms used in today's set theory) in 1963.
- schoen 4y ago
  More about this on Wikipedia: https://en.wikipedia.org/wiki/Continuum_hypothesis#Independence_from_ZFC https://en.wikipedia.org/wiki/Continuum_hypothesis#Independe... Also: https://en.wikipedia.org/wiki/Axiom_of_choice#Independence https://en.wikipedia.org/wiki/Axiom_of_choice#Independence And thanks to Wikipedia for this elaborate list: https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC https://en.wikipedia.org/wiki/List_of_statements_independent...

schoen 4y ago

> But one of those two must be true, you just can't prove it. Three alternative accounts: * Both the CH and ¬CH mathematical universes really exist, so we just have to choose which one we're more interested in at a given time. Like one might say there are the "reall numbers" and the "realle numbers", both valid and interesting constructions which humanity was just slow to recognize the distinctions between (because they were initially less relevant to our interests and our day-to-day lives). * We are actually ultimately thinking about one or the other of them, or are in some sense in one or the other mathematical universe, but we don't know enough about our intuition about the reals to be able to specify or explain which one. (Maybe we need other properties whose obviousness or relevance humanity is not smart enough to notice?) * Some finitist or ultrafinitist approach is actually right: the real numbers are a formalism that, while reasonably motivated by historical attempts to "complete" mathematics in various ways, doesn't correspond to anything Platonically real or to anything intellectually relevant to humanity. (In this account, there is potentially no answer to the question because the real numbers don't exist at all. Neither CH nor ¬CH refers to a mathematical reality, just to games about formalisms.)

thayne 4y ago
sure, but if you ignore any connection to the real world or platonic truth and are just talking about within the ZFC formalism, then `CH ∨ ¬CH` is a true statement, even though it is impossible to prove either `CH` or `¬CH`.
- chriswarbo 4y ago
  > within the ZFC formalism, then `CH ∨ ¬CH` is a true statement It's also a provable statement (via excluded middle)