gib wrote:If L is the largest number, then it is a number. If it's a number, it represents a certain quantity of things. So if you had a set consisting of that many things, there's no reason to say you cannot add one more thing to the set. If you add one more thing to it, you will have L + 1 things (this was the argument you so adamantly insisted was true about adding to infinity). But if L is the largest number, you cannot have a larger number L + 1. Therefore, adding one more item to a set of items whose quantity is L will not give you L + 1 items. <-- A Contradiction.

This is based on the erroneous assumption that every number has a number greater than it. This holds true for naturals, integers, rationals and standard reals but it does not hold true for all numbers. Indeed, it holds true for all numbers EXCEPT for one: the largest number.

Of course I have. There's no other possibility. Unfortunately, you refuse to explain to me what your concept of number is. The best I can do, therefore, is to respond within the context of the standard definition of number.

Well, you could try, oh I don't know, explaining what you mean by "number"?

Anything that is more or less than something else is a number. The largest number is a number because it's something that is greater than every other number. The same applies to infinity: it's a number because it's something that is greater than every integer (or standard real, if you will.)

Yeah, well, when the only example you can dig up for the definition of infinity has to be described as "closer" to the definition you have in mind, you know you're reaching. On the other hand, if you type "definition of infinity" into google, you get a whole ream of exact definitions, and none of them mention integers.

Why do I have to use the same exact words as other people do? Back when I was in school, it was highly desirable for students to use their own words for the simple reason that by doing so they prove they aren't parrots.

"Greater than every integer" and "Greater than every standard real" are two different (and only slightly so) expressions of one and the same thing. There's no difference. Nothing "new and exotic" about the former.

Wikipedia wrote:Infinity (often denoted by the symbol \(\infty\) or Unicode ∞) represents something that is boundless or endless or else something that is larger than any real or natural number.

The Wiki says "Greater than every natural number". Still not "Greater than every integer", the "new and exotic" definition that I'm putting forward.

That you're better off defining infinity in terms of reals rather than integers. You stated earlier: "Infinity is a number greater than every integer, so it's greater than 1,000 as well as 1,000,000. And yes, infinity (if it refers to a specific quantity greater than every integer, and not merely to any such quantity) has a place on the number line." <-- If you mean that infinity falls on the hyperreal section of the number line, you could state this more clearly by saying infinity is greater than all reals. Otherwise, it leads one to wonder whether you think there are real numbers not only greater than infinity but all integers (and given some of the ideas you've defended in this thread, I wouldn't put it passed you).

If something is greater than all integers, it's also greater than all standard reals.

The problem is that you refuse to acknowledge you're talking about hyperreals.

That's not true.