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Jan 27, 2016 at 16:11 comment added John Gowers I talked about the difference between internal and external quantification in my question. But ZFC has $\omega_1$ as an internal ordinal, and FC can prove that $\omega_1$ is larger than any countable ordinal, so internal vs external quantification doesn't tell the whole story.
Jan 27, 2016 at 12:15 history answered Peter LeFanu Lumsdaine CC BY-SA 3.0