# Proof theoretic ordinal of zfc, and other formal systems

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

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If an ordinal $$\alpha$$ can be proved to exist, so can $$\alpha+1$$

So in fact you probably want least ordinal whose existence cannot be proved

If an ordinal $$\alpha$$ can be proved to exist, so can $$\alpha+1$$

So in fact you probably want least ordinal whose existence cannot be proved

oops, my bad. what is it?

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

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

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Why not research the question elsewhere on the Internet?

Hurkyl
Staff Emeritus
Gold Member
plz, hurkyl, don't make me wait!!!
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.

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

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

CRGreathouse
Homework Helper
Perhaps it isn't well defined. What is known, then, about the supremum of the set of definable ordinals in ZFC, and the infimum of undefinable ordinals in ZFC?

Hurkyl
Staff Emeritus