# Proof theoretic ordinal of zfc, and other formal systems

1. Jun 27, 2009

### lolgarithms

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

Last edited: Jun 28, 2009
2. Jun 28, 2009

### g_edgar

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

3. Jun 28, 2009

### lolgarithms

oops, my bad. what is it?

4. Jun 29, 2009

### lolgarithms

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

5. Jun 29, 2009

### lolgarithms

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

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

Last edited: Jun 29, 2009
6. Jun 30, 2009

### g_edgar

Why not research the question elsewhere on the Internet?

7. Jun 30, 2009

### Hurkyl

Staff Emeritus
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.

8. Jun 30, 2009

### lolgarithms

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

Last edited: Jun 30, 2009
9. Jun 30, 2009

### Hurkyl

Staff Emeritus
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 $\alpha$".

10. Jun 30, 2009

### 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?

11. Jun 30, 2009

### Hurkyl

Staff Emeritus
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.