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It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-zfc-ch https://mathoverflow.net/questions/273861/independence-over-... (Al

It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-zfc-ch https://mathoverflow.net/questions/273861/independence-over-... (Although its CH not the negation of it) what additional axioms be needed there?

JadeNB 4y ago
I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948 https://news.ycombinator.com/item?id=31423948).
- SemanticStrengh 4y ago
  I am making the distinction between natural statements and contrived statements. I claim godel only find contrived statements. However there exist non-contrived, natural statements that are unprovable in ZFC. My question is, what set of additional axioms are needed to have a maximal coverage/provability of those independent statements, while introducing the least possible axioms. And yes indeed ZFC + ¬CH seems like a good first step.
  - CaptainNegative 4y ago
    There are a large number of barriers to formalizing your question, such as (i) the definition of natural, (ii) the definition of "maximal" when comparing countably infinite sets, and (iii) the fact that asking about the provability of X in vanilla ZFC is still arguably interesting in the system ZFC+X (see: reverse mathematics), so it's not clear what adding it as an axiom does towards the "largeness" of the set of provable natural statements.
    - SemanticStrengh 4y ago
      About formalising (i) I guess it's about accepting everything that isn't a meta-token, I don't remember godel incompleteness theorems sufficently to be precise about it, anyway the useful list would be the wikipedia list of independent statements since most of them are not contrived (although very abstract). So yes my question can be specialized as to how to achieve maximum coverage of the wikipedia page. Otherwise, great comment and it's been a while since I hadn't heard the term reverse mathematics!

JadeNB 4y ago

I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948 https://news.ycombinator.com/item?id=31423948).

SemanticStrengh 4y ago
I am making the distinction between natural statements and contrived statements. I claim godel only find contrived statements. However there exist non-contrived, natural statements that are unprovable in ZFC. My question is, what set of additional axioms are needed to have a maximal coverage/provability of those independent statements, while introducing the least possible axioms. And yes indeed ZFC + ¬CH seems like a good first step.
- CaptainNegative 4y ago
  There are a large number of barriers to formalizing your question, such as (i) the definition of natural, (ii) the definition of "maximal" when comparing countably infinite sets, and (iii) the fact that asking about the provability of X in vanilla ZFC is still arguably interesting in the system ZFC+X (see: reverse mathematics), so it's not clear what adding it as an axiom does towards the "largeness" of the set of provable natural statements.
  - SemanticStrengh 4y ago
    About formalising (i) I guess it's about accepting everything that isn't a meta-token, I don't remember godel incompleteness theorems sufficently to be precise about it, anyway the useful list would be the wikipedia list of independent statements since most of them are not contrived (although very abstract). So yes my question can be specialized as to how to achieve maximum coverage of the wikipedia page. Otherwise, great comment and it's been a while since I hadn't heard the term reverse mathematics!