What makes something exist?

According to the result in this module, there's an infinity bigger than the universe, and we can know that using our minds. But it doesn't show up anywhere in the real world, only in our minds.

And there are infinities bigger than that, and bigger and bigger and bigger.

Did these infinities exist before Cantor proved they did? What makes something real?

Everyone who understands the construction and proof knows these infinities; we bring them into existence every time we contemplate them. They exist in our collective consciousness, as a part of our intellectual heritage, our culture. These are mathematical infinities, which may or may not correlate with the infinities of our lived experience.

On the one hand, the construction of these infinite sets is so explicit and concrete and objective that there's no ambiguity or subjectivity about them; they seem to be really real.

On the other hand, these infinite sets are pure creative constructs like justice or love. Can there be love without minds and hearts to think and feel it?