This is imprecise. It is more useful to say that it happens because we literally have Z≠R.
Indeed, randomness and countability have little to do with this situation. Consider S=(R∖Q)∪B, where B is a random set of nonzero rational numbers such that P(q∈B)=12,∀q∈Q∗. Then S is a random set (i.e., a set-valued random variable) that is not uncountably smaller than R (the difference between R and S is included in the countable set Q), and yet we know for sure that not all real numbers are in S (because, for example, 0 cannot be an element of S).
Saying every finite combination of atoms exists an infinite number of times in an unbounded universe is more like saying every finite sequence of digits exists an infinite number of times in the digits of pi.
Note that the last property (aka, the idea that pi is normal) is not something that has been proven. So if you are trying to use the notion that it has to be true in an analogy with the idea that “every finite combination of atoms exists an infinite number of times in an unbounded universe” must also be true, this would not be a sound argument.
In light of this particular example, I also don’t really understand why you focused on randomness in your previous comment. After all, pi is not a “random” number under the most natural meaning of that term; it and its digits are fully deterministic.
Indeed, randomness and countability have little to do with this situation.
No, Z literally cannot contain R because R>Z. On the other had, we know that the universe is made up of atoms and that many finite arrangements of atoms can be found in the universe.
Note that the last property (aka, the idea that pi is normal) is not something that has been proven. So if you are trying to use the notion that it has to be true in an analogy with the idea that “every finite combination of atoms exists an infinite number of times in an unbounded universe” must also be true, this would not be a sound argument.
You maybe confusing “sound” with “proven”. Most mathematicians believe pi is normal, so assuming this is a perfectly reasonably assumption to make when reasoning even if the conclusions of that reasoning won’t have the rigor of mathematical proof.
I don’t see what this has do to with randomness or countability? You are the one who brought those two notions up, and that part my response only meant to deal with them.
You maybe confusing “sound” with “proven”.
No, I am using “sound” in the standard philosophical sense as meaning an argument that is both valid and has true premises, which we do not know holds here because we do not know that the premise is correct.
Right, but such an argument would not be “sound” from a theoretical logical perspective (according to the definition I mentioned in my previous comment), which is the only point I meant to get across earlier.
This is imprecise. It is more useful to say that it happens because we literally have Z≠R.
Indeed, randomness and countability have little to do with this situation. Consider S=(R∖Q)∪B, where B is a random set of nonzero rational numbers such that P(q∈B)=12,∀q∈Q∗. Then S is a random set (i.e., a set-valued random variable) that is not uncountably smaller than R (the difference between R and S is included in the countable set Q), and yet we know for sure that not all real numbers are in S (because, for example, 0 cannot be an element of S).
Note that the last property (aka, the idea that pi is normal) is not something that has been proven. So if you are trying to use the notion that it has to be true in an analogy with the idea that “every finite combination of atoms exists an infinite number of times in an unbounded universe” must also be true, this would not be a sound argument.
In light of this particular example, I also don’t really understand why you focused on randomness in your previous comment. After all, pi is not a “random” number under the most natural meaning of that term; it and its digits are fully deterministic.
No, Z literally cannot contain R because R>Z. On the other had, we know that the universe is made up of atoms and that many finite arrangements of atoms can be found in the universe.
You maybe confusing “sound” with “proven”. Most mathematicians believe pi is normal, so assuming this is a perfectly reasonably assumption to make when reasoning even if the conclusions of that reasoning won’t have the rigor of mathematical proof.
I don’t see what this has do to with randomness or countability? You are the one who brought those two notions up, and that part my response only meant to deal with them.
No, I am using “sound” in the standard philosophical sense as meaning an argument that is both valid and has true premises, which we do not know holds here because we do not know that the premise is correct.
In philosophy it is perfectly normal to use arguments that are merely “likely” to be true (as opposed to mathematically proven)
Right, but such an argument would not be “sound” from a theoretical logical perspective (according to the definition I mentioned in my previous comment), which is the only point I meant to get across earlier.