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Building on this question: Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem?

is the infinity of provable statements the same infinity of unprovable statements, or are there more unprovable statements?

I am wondering if there is anything we can say about the probability that a random true statement is provable....

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dashnick
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  • Each is $\aleph_0$ - there are only countably many statements, after all. – Noah Schweber Mar 31 '16 at 01:30
  • thanks! of course.. so I guess we don't have any clues to the "ratio" of provable to unprovable statements? – dashnick Mar 31 '16 at 01:31
  • There is not a very good way to define the ratio of the size of two sets with the same infinite cardinality. We can talk about the density of sets if there is a topology, but I don't know of any topology on the space of all statements. – Plutoro Mar 31 '16 at 01:33
  • I am naive in this area - so the argument that there are twice as many natural numbers as even numbers would be a topological argument? – dashnick Mar 31 '16 at 01:34
  • @dashnick There aren't twice as many natural numbers as there are even numbers. The evens are in bijection with the naturals, so their cardinality is the same – Exit path Mar 31 '16 at 01:37
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    Right, technically.. But in some sense of course there are twice as many - would that be a density argument? (In the sense that a random number has 2x the chance of being natural as being even) – dashnick Mar 31 '16 at 01:39

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