Hartog's theorem

1. Jan 6, 2010

mathshelp

Write out the proof of Hartog's Theorem again carefully highlighting how the Axiom of Replacement is used

How can you highlight the axiom of replacment?

Last edited: Jan 7, 2010
2. Jan 6, 2010

JSuarez

In the proof of Hartog's theorem, you reach a point where you just proved that the class of all well-orders of the set X (using the separation axiom schema). Now you have to prove that each the class of all order-types (isomorphic ordinals) of each of these well-orders is also a set, and it's here that you use replacement, by constructing a first-order formula that expresses the isomorphism of each (Y,<), where Y is a subset of X and "<" is a well-order in Y, to an ordinal.
This is the key step in Hartog's proof, the one that allows him to sidestep AC, so I think the "highlighting" is a careful, step-by-step explanation of how the replacement schema is used.

3. Jan 7, 2010

mathshelp

That makes sense, but how do you construct a first order formula?

4. Jan 7, 2010

JSuarez

I assume that, if you're studying formal Set Theory, that you are familiar with first-order logic (otherwise, it's impossible to understand anything of Set Theory).