Then I’m confused. If the question is, “Is .999… = 1 true in standard math?” the answer is yes without a shred of doubt. I could point you to a hundred books on calculus and real analysis. If we’re talking about standard math, how could anyone hold a different opinion?
But yes! In which case .999… = 1 in standard math and there is no question or dispute, other than to clarify for people what the notation means and why it’s true within standard math. It’s a theorem in ZF set theory. It’s a convergent geometric series in freshman calculus. It’s even true in nonstandard analysis, which some people aren’t aware of. There’s just no question about the matter.
So you really have me puzzled, Magnus. If you agree we’re talking about standard math, what is the basis of your disagreement?
Ok. So you admit that you are NOT talking about standard math, but rather about your private nonstandard use of mathematical symbols. In which case you can define .999… = 47 and I would have no objection. If that’s one of the rules in your game, I am fine with it; just as I learned to accept that the knight can hop over other pieces in standard chess.
You have already said that you are talking about standard math AND that you are talking about your own private nonstandard math. It’s not hard to misunderstand you!
Now let me talk delicately about infA. When I came to this forum several years ago, James was already a prolific poster, an ILP Legend to beat all ILP legends. I am reluctant to criticize him since he is not here to defend himself. He has far more mindshare on this forum than I do. I respect his prolific output, if not always its content.
That said, the concept if infA is confused and wrong in the extreme. The idea seems to be some sort of mishmash of the ordinal numbers, in which we do “continue counting” after all the natural numbers are exhausted; and nonstandard analysis, in which there are true infinite and infinitesimal numbers. The infA concept borrows misunderstood elements from each of these ideas and simply makes a mess.
One really valuable thing I got from this thread a few years ago is that James caused me to go deeply into nonstandard analysis, to the point where I understand its technical aspects. For that I appreciate James. But the infA concept is just bullpucky, I don’t know what else to say.
It means exactly what I’ve described to @Ed3. It’s a particular map from the positive integers to the set of decimal digits; a constant map, in fact, in which f(n) = ‘9’ for all inputs n. We then interpret this symbol as a real number as in the theory of geometric series, in which it’s proved rigorously that .999… = 1.
Again I agree that if you choose to make up a new system in which .999… has some other meaning, you are perfect within your rights. After all there do happen to be many variants of chess; played on infinite boards, or with a new piece called the Archbishop, etc. If someone enjoys playing alternate versions of standard games it’s ok by me.
Yes ok. Then we are done!
I fail to follow that. Did you learn geometric series at one point? The definition of a limit? I can’t tell where you’re coming from.
Actually the limit is defined as the sum. That’s what a limit is. It’s a clever finessing of the idea of “the point at the end” or whatever. You are adding your own faulty intuition. If you would consult a book on real analysis you would find that the limit of a geometric series is defined as the limit of the sequence of partial sums; and that the limit of a sequence is defined as a number that the sequence gets arbitrarily close to. I explained all this to @Phyllo the other day. That’s the textbook definition. You’re just wrong in your impression, either because you had a bad calculus class (as most students do) or none at all. It’s not till Real Analysis, a class taken primarily by math majors, that one sees the formal definition and comes to understand that the sum IS defined as the limit of the sequence of partial sums. That cleverly avoids the confusion you’ve confused yourself with.
Ah, the evil cabal of mathematicians. I will fully agree with you that most math TEACHERS form an evil cabal. One doesn’t get all this stuff sorted out properly till one sees the formal definitions; at which point, one learns that the limit of a sequence is defined by arbitrary closeness. The belief you have is a bad intuition that mathematical training is designed to clarify. It’s sad that we don’t show this to people unless they’re math majors, and you can rest assured that when I am in charge of the public school math curriculum the teaching of the real numbers will be a lot better.
Till then, I apologize on behalf of the math community that you weren’t taught better. But limits are very rigorously defined and your idea is just wrong.
Again I hope I’m not coming on too strong, I’m criticizing your ideas and not you. I know you are sincere. Except for the part where you say you’re talking about standard math AND that you’re not. That point confused me.