proof theoretic ordinal of zfc, and other formal systems

**lolgarithms** · Jun27-09, 11:52 PM

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

**g_edgar** · Jun28-09, 08:05 AM

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

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

**lolgarithms** · Jun28-09, 03:43 PM

Originally Posted by g_edgar

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

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

oops, my bad. what is it?

**lolgarithms** · Jun29-09, 03:41 PM

could someone answer this question? please?
What is the least ordinal whose existence can't be proven in ZFC?

**lolgarithms** · Jun29-09, 07:50 PM

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

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

**g_edgar** · Jun30-09, 09:27 AM

Why not research the question elsewhere on the Internet?

**Hurkyl** · Jun30-09, 09:46 AM

Originally Posted by lolgarithms

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.

**lolgarithms** · Jun30-09, 06:25 PM

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

**Hurkyl** · Jun30-09, 06:33 PM

Originally Posted by lolgarithms

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 $LaTeX Code: \\beta$ " and " $LaTeX Code: \\alpha < \\beta$ " should imply "ZFC proves the existence of $LaTeX Code: \\alpha$ ".

**CRGreathouse** · Jun30-09, 06:41 PM

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** · Jun30-09, 07:00 PM

Originally Posted by CRGreathouse

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?

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.