The rational numbers do have the least upper bound property, and the real numbers are unnecessary.
The typical example of the rational number system being "flawed" is that there isn't a rational corresponding to sqrt(2) but that you can approach it from both sides. This argument is a fallacy, i think it might be circular but not quite. what is sqrt(2)? sqrt(2) is some made up concept to justify the creation of the rationals but in order to derive sqrt(2) you first need to construct the rationals. in other words, its a completely made up, imaginary concept, and the extension of the rationals, which is the only actually existing set of numbers, into the reals is unnecessary. I will make an analogy to illustrate my point. Modern academics decided that there were actually more than two genders. However, there is only XX and XY and gender corresponds to sex chromosomes. Therefore academics created the imaginary construct of a new kind of "gender" that is actually physically meaningless and is whatever someone wants it to be and which has no bearing on reality. The irrational numbers are the imaginary genders.
The proof of my original assertion is obvious once you realize that sqrt(2) doesn't exist. you can't define "the set of rationals < sqrt(2)" because sqrt(2) is meaningless.
The typical example of the rational number system being "flawed" is that there isn't a rational corresponding to sqrt(2) but that you can approach it from both sides. This argument is a fallacy, i think it might be circular but not quite. what is sqrt(2)? sqrt(2) is some made up concept to justify the creation of the rationals but in order to derive sqrt(2) you first need to construct the rationals. in other words, its a completely made up, imaginary concept, and the extension of the rationals, which is the only actually existing set of numbers, into the reals is unnecessary. I will make an analogy to illustrate my point. Modern academics decided that there were actually more than two genders. However, there is only XX and XY and gender corresponds to sex chromosomes. Therefore academics created the imaginary construct of a new kind of "gender" that is actually physically meaningless and is whatever someone wants it to be and which has no bearing on reality. The irrational numbers are the imaginary genders.
The proof of my original assertion is obvious once you realize that sqrt(2) doesn't exist. you can't define "the set of rationals < sqrt(2)" because sqrt(2) is meaningless.
