Notation: "infinity" superscript on universal quantifier

Hey guys,

I've come across a some notation that I have not seen before and since it's all symbols, I couldn't really find anything on Google about it. I don't think it's necessary/relevant to explain the entire context here, but the formula read like this:

$\displaystyle P(e) \rightarrow (\forall^{\infty}y)(f(e,y) \in COINF)$

where COINF if the index set of all coinfinite recursively enumerable sets.

It's probably not all that complicated, but I've never seen the $\displaystyle \forall^{\infty}$ quantifier notation before. What does it mean? The variable $\displaystyle y$ is supposed to range over the natural numbers.

Thank you!

Selinde

Re: Notation: "infinity" superscript on universal quantifier

I know what recursively enumerable sets are, but I don't think $\displaystyle \forall^\infty$ is a universally accepted notation. I believe it should have been defined earlier in your text. If it were $\displaystyle \exists^\infty$, it could mean that there exist infinitely many objects. Maybe $\displaystyle \forall^\infty y$ means "for all y's that are indices of infinite recursively enumerable sets"? If your text is available online, I could look into it.

Re: Notation: "infinity" superscript on universal quantifier

The text is available here: http://arxiv.org/pdf/1302.7069v1.pdf -- the notation first occurs on the bottom of page 8 and is not defined earlier. That's why I assumed I was just missing some knowledge (as is usually the case when I'm reading these things ;) ) The meaning that you suggested is a possibility, but I do not yet understand the proof enough to check this.

Edit: I think it might mean "for all but finitely many", could this be correct?