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what is the proof-theoretic strength (largest ordinal whose existence can be proved) for ZFC set theory?

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- Thread starter lolgarithms
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- #2

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So in fact you probably want

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So in fact you probably wantleast ordinal whose existence cannot be proved

oops, my bad. what is it?

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What is the least ordinal whose existence can't be proven in ZFC?

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what is the proof theoretic strenght of zfc? please help, i want to know!

plz, hurkyl, don't make me wait!!!

plz, hurkyl, don't make me wait!!!

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Why not research the question elsewhere on the Internet?

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Hurkyl

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What? All I can do is make the obvious guess, and I'm not even sure that's well-defined, let alone the answer you seek.plz, hurkyl, don't make me wait!!!

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What? All I can do is make the obvious guess, and I'm not even sure that's well-defined, let alone the answer you seek.

if the smallest ordinal that can't be proven is well-defined: what determines that? that a stronger set theory is not known?

so we just call the ordinal "the proof theoretic ordinal of zfc"? ok. might not be an oridnal with a name, like kripke-platek ordinal

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- #9

Hurkyl

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The obvious guess would (IMHO) be something like the supremum of all of the ordinals in the constructible hierarchy.if the smallest ordinal that can't be proven is well-defined: what is the guess?

However, I'm not sure the question is well-defined: why should "ZFC proves the existence of [itex]\beta[/itex]" and "[itex]\alpha < \beta[/itex]" should imply "ZFC proves the existence of [itex]\alpha[/itex]".

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CRGreathouse

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- #11

Hurkyl

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And that's where the extent of my knowledge ends. Although now that I think about it, the continuum hypotheses probably tells us some interesting information.

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