Muxdemux: Of course I only proved (1) for natural numbers - this is all one can do with induction. The problem arises when you take the limit and the inequality results in an absurdity.
Bob Bundy: There are no personal attacks. Why the message? And of course I wrote the article. I never plagiarize the work of others and most people do not think or agree with me, which makes me feel pretty good. It's always nice to know that I am correct and the majority are in error. Either way, it does not really matter too much. People who read stuff on the internet (or anywhere else) should not take things personally. If your name is not in the perceived put-down or insult, then why take offence? Kind of silly, isn't it? The world does not revolve around any one individual. Please do not delete my post. I am sharing my unique knowledge with you on your forum. Whether you agree or not is irrelevant.
And one more thing: I don't like it that it says "novice" under my name. This is a major put-down for me. Please change it to something like Expert or Master. Thank you in advance!
Bob bundy: "To say that numbers have no existence at all except as mathematical concepts" is false.
They "don't just get defined" as you say. Numbers have a very real existence as abstract objects when they are well-defined.
The examples you gave, 0!=1, sqrt(a), etc, all have very sound explanations.
A number is a magnitude that has been completely measured.
There is nothing dreamy or "just defined" about the previous statement.
Finally, it's not a case of liking or disliking, but rather using objects that are well-defined.
Muxdemux: Didn't you only prove (1) for the natural numbers?
Of course. Are there any other numbers that you know of which I don't? Please don't tell me infinity is a number. It is a meaningless, non-real concept.
TheTick: .9999.... NEVER equals 1 and it has nothing to do with tolerance.
At no point in the partial sums of 0.999... are you "rounding up". In other words, a carry is impossible. The problem with 0.999... is that it's not a number unless considered as an approximation. A non-terminating (repeating or non-repeating) decimal is an ill-defined concept.