What Is the Order Type of Countable Ordinals?

Thread starter ForMyThunder
Start date Feb 3, 2011

AI Thread Summary

The order type of the set of all ordinals, denoted as Ω, cannot be defined since no set contains all ordinals. However, the notation Ω is frequently used to refer specifically to the set of all countable ordinals. This clarification is essential for understanding the context of the discussion. The distinction between all ordinals and countable ordinals is crucial in set theory. Ultimately, the conversation emphasizes the importance of precise terminology in mathematical discussions.

Feb 3, 2011

ForMyThunder

If \Omega is the set of all ordinals, what is the order type of \Omega?

Physics news on Phys.org

Feb 3, 2011

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

Any answer would suffice, since your question is vacuous -- no set contains all ordinals.

Feb 3, 2011

micromass

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Hurkyl is right, no set contains all the ordinals. But the notation \Omega is often used to denote the set of all countable ordinals. That's probably what you mean...

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

View full post »