“Number”
If memory serves, this is the first time you’ve suggested otherwise. But that’s sort of my point, (L) can’t be a real number for all the reasons I’ve pointed out, so if it’s not a reason then we agree there. I just think that’s a big problem for your argument, because you’re treating (L) as though it’s a real number.
I’m not ignoring it, I’m pointing out that it’s vague. (L) is clearly not a number like the ones we usually work with, and so I’m not making assumptions about what you mean by “number”. You haven’t clarified what happens when we subtract 1 from (L), what kind of number do we get? If it’s a real number, than that number plus 1 must be (L), in which case (L) is the sum of two real numbers. If it’s some number less than (L) but of the same type, then does (L-1 = L)? Is (L) hyperreal?
Failing to say explicitly what you mean lets you play calvinball around (L): no conclusion that could defeat your claim can be argued for, because you offer attributes of (L) only if and when they become necessary to undermine some argument.
Here is a great example: just confirm or deny the inequality!
I have yet to see a coherent definition of the “the largest number”, and this question makes me think whatever you have in mind has some problems. But I guess we’ll see when you tell me what you have in mind.
This here is treating (\infty) like a number, and it isn’t a number. I would say that that equation is meaningless. I’ve probably been sloppy in my language around this, and I apologize. When I say that two infinities are equal, I mean that every element of one set can be mapped to exactly one element of the other set. So, for example, (.\dot9) and (9.\dot9) have the same number of decimal places, in the sense that each decimal place in one corresponds to exactly one decimal place in the other.
Somewhat as an aside: in your system, (.\dot9) is ambiguous, since if I just say (.\dot9) we don’t know if it’s just the simple repeating decimal, or some different version of the repeating decimal obtained by multiplication by a power of 10 and subtracting the integer portion. This strikes me as a larger problem than you seem to acknowledge (how might 1 fewer zeros on the end of an integer’s decimal expansion change its value?).