Planescape is Jeremy Crawford's favourite D&D setting. "It is D&D", he says, as he talks about how in the 2024 core rulebook updates Planescape will be more up front and center as "the setting of settings".
100%.No, “we” have very reliable, consistently verifiable descriptions of how these things work (which is what a “theory” is in a scientific context). And we have a very good understanding of infinity, which is a mathematical concept. It is absolutely possible for a set to be infinite and to still not include everything that exists or could conceivably exist.
I’m not 100% sure about this statement. I’m not a mathematician, but aren’t all even numbers integers by definition? All even integers would be “countably infinite,” as would all integers, so both sets would be equally large. Whereas all real numbers is not countable, making it a larger set than the set of integers (or the set of even integers). All three are infinite, but the real numbers are a higher “class” of infinity, so to speak, than the other two. “All even real numbers,” if I’m not mistaken, would be the same set as “all even integers,” because there are no even real numbers that aren’t also integers.The ‘set of all real numbers’ is larger than the ‘set of all even real numbers’. Both are equally infinite.
I'm going to explicitly preface this with the statement: "If we go much farther down this train of thought, I'm going to have to resort to digging out my copy of Halmos and doing some remedial reading", but: wouldn't sets of integers always be larger than sets of real numbers, as integers include negative numbers and sets of real numbers don't?I’m not 100% sure about this statement. I’m not a mathematician, but aren’t all even numbers integers by definition? All even integers would be “countably infinite,” as would all integers, so both sets would be equally large. Whereas all real numbers is not countable, making it a larger set than the set of integers (or the set of even integers). All three are infinite, but the real numbers are a higher “class” of infinity, so to speak, than the other two. “All even real numbers,” if I’m not mistaken, would be the same set as “all even integers,” because there are no even real numbers that aren’t also integers.
Integers are whole numbers - both positive and negative, but no fractions/decimals.I'm going to explicitly preface this with the statement: "If we go much farther down this train of thought, I'm going to have to resort to digging out my copy of Halmos and doing some remedial reading", but: wouldn't sets of integers always be larger than sets of real numbers, as integers include negative numbers and groups of real numbers don't?
Err… Again, I’m not a mathematician, but I’m pretty sure negative numbers are real numbers. My understanding is that all whole, rational, and irrational numbers are real numbers. Imaginary numbers are the only type of number not included in the set of real numbers.I'm going to explicitly preface this with the statement: "If we go much farther down this train of thought, I'm going to have to resort to digging out my copy of Halmos and doing some remedial reading", but: wouldn't sets of integers always be larger than sets of real numbers, as integers include negative numbers and sets of real numbers don't?
All of these statements are true, but the set of all rational numbers (the integers and fractions/decimals) is also countably infinite, so it’s the same size as the set of all integers (which is the same size as the set of all positive integers, etc).Integers are whole numbers - both positive and negative, but no fractions/decimals.
Real Numbers contain all Integers (positive and negative) and also include fractions/decimals.
So the set of Real Numbers is larger.
This is definitely “time to reread Halmos” territory for meThe reason the set of all real numbers is larger is because it includes irrational numbers (like Pi and friends), which makes it not countable.
If this were a scientific debate, I’d be interested in this line of argument. It is not, however.I’m sorry, but you are just wrong about this. This isn’t a hypothetical scenario, real physicists do really talk about multiple universes right now, and they don’t use the word the way you are trying to use it here. We can’t currently verify or falsify the existence of other universes, but there are valid models that describe them, and those models treat them as separate universes from the one we inhabit.