> The hyper-reals of non-standard analysis are, in fact, a strict superset of the reals.
And .9999==1 in the hyper-reals
>>If it were, then any operation involving only real numbers
>>would behave identically to the real number system.
>Why is that a problem? Seems to me that that's desirable.
Consider the expression ".999... - 1" in the real number system. As has been established earlier, this simplifies to 0.
Consider the same expression ".999... - 1" in the hyper-real number system. As all the numbers are real, this expression simplifies just as it would in the real number system. Meaning that ".999... - 1 =0" in the hyper-real system. Adding 1 to both sides ".999... = 1" in the hyper-real system.
One could also argue that because the statement ".999... = 1" is true in the real number system, and the hyper-reals are a superset of the reals. Then ".999... = 1" is true in the hyper-reals.
Ah, I see your point. I was trying to address the more general "intuition" that people have about this, that 0.9999... falls short of 1.0, and that there's some sort of infinitesimal between them. You can use that to talk about the hyper-reals, but, as you say, it still doesn't "solve" the "problem" that people perceive.
So you're right, and I was answering a different (although related) question.
And .9999==1 in the hyper-reals
>>If it were, then any operation involving only real numbers >>would behave identically to the real number system. >Why is that a problem? Seems to me that that's desirable.
Consider the expression ".999... - 1" in the real number system. As has been established earlier, this simplifies to 0.
Consider the same expression ".999... - 1" in the hyper-real number system. As all the numbers are real, this expression simplifies just as it would in the real number system. Meaning that ".999... - 1 =0" in the hyper-real system. Adding 1 to both sides ".999... = 1" in the hyper-real system.
One could also argue that because the statement ".999... = 1" is true in the real number system, and the hyper-reals are a superset of the reals. Then ".999... = 1" is true in the hyper-reals.