# Is every formula of LΩ equivalent to a formula of L1 modulo T1?

Q: Is every formula of $L_\Omega$ equivalent to a formula of $L_1$ modulo $T_1$?

The notation comes from the following question: Is the following theory countably axiomatizable?

Edit: I mean $T_\Omega$.

• Okay, it looks like $T_0 \not= T_1$ and $T_1 = T_2 = T_\Omega$. Dec 14, 2016 at 2:44
• In what sense is $T_0\not=T_1$, if $T_1=T_2$? (Note that in my original answer where I said that the hierarchy collapses immediately in a sense, my indexing was wrong: $T_0$ is already "essentially the same as" $T_1$, in a precise sense; and I've edited that.) Dec 14, 2016 at 4:48
• $\exists v\,\forall x\, x\in v$ is not true in all models of $T_0$ but it is in all models of $T_1$. Dec 14, 2016 at 15:45

Noah is correct that over the theory $T_1$, we don't know about the new constants. But meanwhile, over the final theory $T_\Omega$, or what I would call $T_{\omega_1}$, every formula in the extended language is equivalent to a formula in the language of (class) set theory itself. You have added constants $c_\phi$ and axioms $\forall x\ (x\in c_\phi\iff \phi(x))$, where $\phi$ uses only earlier-mentioned constants. So by induction, any instance of $x\in c_\phi$ that appears in a formula can be systematically eliminated by this means, and so in fact we don't need any of these new constants.
• +1. Note for the OP that we don't have to wait until $T_\Omega$ to see this for a given sentence - each sentence at level $\alpha$ is immediately seen to be "redundant" by $T_\alpha$. Dec 14, 2016 at 2:02
• Indeed. I just wanted to point out to the OP that nothing "special" happens at $T_\Omega$. (By the way, in my original answer I meant to say that, but I got my indexing messed up - I fixed it now.) Dec 14, 2016 at 2:09
No, they are not. There are formulas in $L_2$, for example, involving constant symbols in $L_2$ but not in $L_1$. $T_1$ has nothing to say about those constant symbols, so no formula involving such a constant symbol (in a nontrivial manner) can be $T_1$-equivalent to an $L_1$ formula.
The sense in which $T_2$ is "no more than" $T_1$ is interpretability (and conservativity) - any model of $T_1$ interprets an expansion of itself which is a model of $T_2$.