The set of integers is a countable infinity. This is pretty obvious, since you can easily count from one member to the next.
The set of irrational numbers is an uncountable infinity. This is because if I give you one member, you can’t give me an objectively “next” one. There’s infinitely many choices.
Example: I say what’s the next member of the set of irrational numbers after 1.05? Well, there’s 1.050001, 1.056, etc.
Can a member of an infinite set be mapped to a corresponding member of another infinite set? And if so, how?
In this situation, the sets are both countable. QA can open bug #1, bug #2, etc. It’s also - for now - at least a surjective mapping of Starfield bugs -> Skyrim bugs. Because they’re both countable, for each bug in Starfield you can find at least one bug in Skyrim (because it’s a known bigger set at the moment).
This is pretty obvious, since you can easily count from one member to the next.
I’d just like to chip in that it isn’t necessary for a countably infinite set to have an obvious method of counting. Listing all of the rationals in numerical order isn’t possible (what’s the smallest fraction above 0?) but it is nevertheless possible to create a bijection with the naturals.
I’m not great at high-level maths like that, but can infinities be sized different in a way that makes a comparison of quantity valid?
Yes! There’s actually two facets to consider:
Infinities can be countable or uncountable:
The set of integers is a countable infinity. This is pretty obvious, since you can easily count from one member to the next.
The set of irrational numbers is an uncountable infinity. This is because if I give you one member, you can’t give me an objectively “next” one. There’s infinitely many choices.
Example: I say what’s the next member of the set of irrational numbers after 1.05? Well, there’s 1.050001, 1.056, etc.
Can a member of an infinite set be mapped to a corresponding member of another infinite set? And if so, how?
Spoiler, there are three different ways: surjective, injective, and bijective.
In this situation, the sets are both countable. QA can open bug #1, bug #2, etc. It’s also - for now - at least a surjective mapping of Starfield bugs -> Skyrim bugs. Because they’re both countable, for each bug in Starfield you can find at least one bug in Skyrim (because it’s a known bigger set at the moment).
But we don’t know more than that right now.
I love that this comment represents more work into the issue of bugs than Bethesda bothers with.
I’d just like to chip in that it isn’t necessary for a countably infinite set to have an obvious method of counting. Listing all of the rationals in numerical order isn’t possible (what’s the smallest fraction above 0?) but it is nevertheless possible to create a bijection with the naturals.
Yeah, it’s quite the rabbit hole.
https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
There is the same number of even numbers as there are even and odd numbers.
I made a detailed reply here about how the infinite set theory to the OOP. I don’t know how to tag you, so here’s a link