4 ms·

There is a comparison table on the page, it says that in ZFC less than 50% of the statements are true. A quick google search gave me nothing, does anybody know

by gosub 15y ago

There is a comparison table on the page, it says that in ZFC less than 50% of the statements are true. A quick google search gave me nothing, does anybody know where I can learn more about this statement? Or is it a meta-joke?

omaranto 15y ago
It's a joke, but a slightly wrong one, I think: they should say "provable" instead of "true". For every true statement it's negation is false and viceversa, so exactly half of all possible statements are true. In a system like ZFC not all of the true statements are provable, so less than 50% are provable. (Also, less than 50% are refutable --this means provably false--, and the non-provable, non-refutable statements are called undecidable.)
- a3_nm 15y ago
  I don't understand your reasoning to justify that "half of all possible statements are true". It seems to me that of the undecidable statements, some (such as those of Gödel's construction) are true or false, but that some simply have no truth value. Did I get something wrong? [Indeed, this is all assuming that ZFC is consistent. Thanks mjw.]
  - p4bl0 15y ago
    That's also how I understand it. Anyway, you people should upvote a3_nm, he is the one who did the job (for Falso and the Security Password Checker), I only encourage him and host the pages :-).
  - omaranto 15y ago
    I wouldn't say undecidable statements have no truth value. In any given model of set theory each statement is either true or false. If something is provable it is true in every model, and if something is refutable it is false in every model. Undecidable statements can't be proved to be either true or false within a given proof system, but (1) in any given model are either true or false, as I said, and (2) can sometimes be shown to be true (or false) by arguments outside of the proof system considered.
- mjw 15y ago
  Assuming it's consistent, that is! Perhaps they shouldn't be so ready to belittle their competitors -- you never know, ZFC might yet turn out to share all the advantages of Falso :)
- chalst 15y ago
  It's not mathematically correct: both all statements of ZFC and the provable have the same cardinality: there's no sense to "half of statements" in ZFC.
  - omaranto 15y ago
    Again it's a joke, so let's not take it too seriously but I think the sense in which it is half is not cardinality (that would be silly, as you point out), but density. Look at the probability that a random statement of length at most n is true. Since for every statement it's negation has practically the same length and the opposite truth value, the probability should tend to 1/2 as n goes to infinity.
p4bl0 15y ago
Omaranto already answered, so I'll just add to links: https://secure.wikimedia.org/wikipedia/en/wiki/G%C3%B6del%27s_incompleteness_theorem https://secure.wikimedia.org/wikipedia/en/wiki/G%C3%B6del%27... https://secure.wikimedia.org/wikipedia/en/wiki/List_of_statements_undecidable_in_ZFC https://secure.wikimedia.org/wikipedia/en/wiki/List_of_state...

by gosub 15y ago

omaranto 15y ago

It's a joke, but a slightly wrong one, I think: they should say "provable" instead of "true". For every true statement it's negation is false and viceversa, so exactly half of all possible statements are true. In a system like ZFC not all of the true statements are provable, so less than 50% are provable. (Also, less than 50% are refutable --this means provably false--, and the non-provable, non-refutable statements are called undecidable.)

a3_nm 15y ago
I don't understand your reasoning to justify that "half of all possible statements are true". It seems to me that of the undecidable statements, some (such as those of Gödel's construction) are true or false, but that some simply have no truth value. Did I get something wrong? [Indeed, this is all assuming that ZFC is consistent. Thanks mjw.]
- p4bl0 15y ago
  That's also how I understand it. Anyway, you people should upvote a3_nm, he is the one who did the job (for Falso and the Security Password Checker), I only encourage him and host the pages :-).
- omaranto 15y ago
  I wouldn't say undecidable statements have no truth value. In any given model of set theory each statement is either true or false. If something is provable it is true in every model, and if something is refutable it is false in every model. Undecidable statements can't be proved to be either true or false within a given proof system, but (1) in any given model are either true or false, as I said, and (2) can sometimes be shown to be true (or false) by arguments outside of the proof system considered.
mjw 15y ago
Assuming it's consistent, that is! Perhaps they shouldn't be so ready to belittle their competitors -- you never know, ZFC might yet turn out to share all the advantages of Falso :)
chalst 15y ago
It's not mathematically correct: both all statements of ZFC and the provable have the same cardinality: there's no sense to "half of statements" in ZFC.
- omaranto 15y ago
  Again it's a joke, so let's not take it too seriously but I think the sense in which it is half is not cardinality (that would be silly, as you point out), but density. Look at the probability that a random statement of length at most n is true. Since for every statement it's negation has practically the same length and the opposite truth value, the probability should tend to 1/2 as n goes to infinity.

p4bl0 15y ago

Omaranto already answered, so I'll just add to links: https://secure.wikimedia.org/wikipedia/en/wiki/G%C3%B6del%27s_incompleteness_theorem https://secure.wikimedia.org/wikipedia/en/wiki/G%C3%B6del%27... https://secure.wikimedia.org/wikipedia/en/wiki/List_of_statements_undecidable_in_ZFC https://secure.wikimedia.org/wikipedia/en/wiki/List_of_state...