Suppose you’re looking at a 0.999~cm high domino from a certain distance. You measure it and you get a result like 0.9cm. Of course, this is not true, but that’s what you can see from that distance; you can’t see the remainder, it’s too small. You decide to come closer, just to be sure, and now you measure the domino to be 0.99cm high. You keep coming closer, and each time you come closer, you get a result that is closer to, but not quite equal to, 1.00cm. Now, if you kept doing this for eternity (assuming you had an eternal life to live) and got a sequence of measurements that goes something like 0.9cm, 0.99cm, 0.999cm, 0.9999cm and so on, that would be a domino that is 0.999~cm high. If, on the other hand, you got a sequence of measurements such as 0.99cm, 0.999cm, 1.000cm, 1.000cm, 1.000cm and so on (basically stablizing at 1.00cm high at some point) that would be a domino that is 1.00cm high. Clearly, they are not the same height.
There are objects with a height that is not equal to a finite number of equally-sized units (such as centimeters) which means there is no combination of an integer and a unit that can represent them exactly. Finite decimals are more expressive but they have their own limitations. Most importantly, they can only represent finite quantities. Hence, infinite decimals.
Infinity might be a difficult concept to grasp for some. Mostly, it’s the kind of infinity that Aristotle called “actual (or completed) infinity” that they have problem with. It’s difficult to understand that height, width, depth, length and distance can be infinite.
Regarding the bolded part, one must ask: is that true? Let’s not be bound by our number system. There are number systems other than base-10 and these other number systems can represent numbers that our number system cannot, such as numbers that are larger than 0.999~ but smaller than 1. Consider base-16 number system a.k.a. hexadecimal number system. In hexadecimal number system, 0.999~ is represented the same way, as 0.999~. But since we’re dealing with hexadecimals, which work with more than 10 digits, there are numbers higher than it but lower than 1 e.g. 0.AAA~, 0.BBB~, 0.CCC~ and so on. Of course, the basic idea of the quoted part of your post is correct, but I think it’s important to note that 0.999~ is not the largest number that is lower than 1. Rather, it is the largest number lower than 1 in decimal number system. This means it’s possible for your hand to be closer to the table than 1cm away from it and still not knock the shorter domino over.
The point is that there are numbers larger than 0.999~ but smaller than 1. For example, 0.FFF~ (in hexadecimal system) is larger than 0.999~ but smaller than 1. In base-20 system, we have 0.JJJ~ which is larger than both hexadecimal 0.FFF~ and decimal 0.999~.
This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago
When I was reading through this, I noticed the following that seems to bring it all to a salient conclusion.
It makes sense that “0.999…” is just an expression signifying that some ratio cant be expressed by a fixed number of digits (unlike 1). It isn’t actually a number. And neither are all of those other expressions that end with “…”. And apparently that is why the Wikipedia proofs are misleading.
I am not sure why you think it’s not a number i.e. a symbol representing some quantity. It appears to me that it clearly is. It has many properties that numbers have e.g. it’s greater than some numbers and less than others.
What it isn’t is a finite quantity, that’s for sure, and that’s why it can’t be 1.