Absolute infinity

Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it with the Hebrew later for Tav and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for reflection principles, with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the Burali–Forti paradox), so large it almost "transcended" itself, and associated it metaphysically with God.

Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology.

As justification for reflection

Later authors have connected Cantor's remark that absolute infinity "can not be conceived" to reflection principles. For example, Maddy states:^p.503

Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \( V \) is already true of some [\( V_\alpha \)].