If it helps, don't think about operations on infinite sets as "things that are true (or false)". Rather, think about it as "an abstract model with certain rules, that can produce interesting results if applied in careful ways."
Like √-1. Does that exist? Well, not really. But if we make up a pretend number i that we say is equal to √-1, we can perform interesting mathematical operations on it that can still produce useful results in the real world. But only if we follow a given set of rules carefully and consistently.
"All models are wrong, but some models are useful."
Infinity isn't real - at least, not in our universe. It certainly isn't a number. But it's a concept we can use to produce useful results sometimes, if we handle it carefully. But it's also a concept that can produce some weird results, even if we follow all the rules.
Like √-1. Does that exist? Well, not really. But if we make up a pretend number i that we say is equal to √-1, we can perform interesting mathematical operations on it that can still produce useful results in the real world. But only if we follow a given set of rules carefully and consistently.
"All models are wrong, but some models are useful."
Infinity isn't real - at least, not in our universe. It certainly isn't a number. But it's a concept we can use to produce useful results sometimes, if we handle it carefully. But it's also a concept that can produce some weird results, even if we follow all the rules.