Einer

I don't think I have to be involved in this argument of yours. Could you please continue it elsewhere as to avoid notifying me about every added comment?

@Su20200 While that's true, it's in everyone's interest for you to ask good questions. I think you could do better on that front.

@Su20200 I've changed my previous comment to a (hopefully) more helpful one.

@Su20200 Well, there are such properties but it's impossible to tell which one might help you with the given information. If you want to post the problem, I'd suggest posting it as a separate question.

@Null What do you mean when you say that $\mathbb{N}$ is defined by peano? Do you mean by Peano Arithmetics? Because that's definately not the case - there are many non-standard models of PA

I mean... It's not uncommon that reading a given paper takes months, if you have to catch up on all the background first

If you hate it and have enough time at your hands, read everything you need to know to thoroughly understand it.

@aerdna91 ccc forcings don't preserve $\omega$ sequences. Adding a Cohen real is the generic counterexample of that. (A real is just a sequence of natural numbers.)