this doesn't make sense to me, smart people? (infinity)

[daviddem]

Hey, totally grunching the thread here. I read daviddem's first reply that explained cardinality and it made perfect sense. I've done an MA in Econ, which isn't nearly as math-y (we get looked down on by math types) as some disciplines, but however. I hadn't really needed to understand cardinality but have been made familiar with these concepts before. Anyway.

One thing that has always bugged me:

In the second video (the one with the clearer explanation and proof of the infiniteness of the set (0,1) being larger than that of the set of all natural numbers), the proof explicitly excludes the number 9 in the construction of 'x'.

This is justified as a way to avoid ambiguities like 0.499999... = 0.5

I had personally always thought that 0.499999... != 0.5

because we can always choose a sufficiently small number 0.000....01 such that 0.5 minus this number equals 0.49999...

Am I wrong? This has always pissed me off.

This is already discussed at length in some posts in this same thread. I am not doing this again, but essentially, yes, you are wrong.

0.4999... equals 0.5 because 9... represents an infinite number of 9's, not "some finite number of 9's that we can make as large as we want" or "a number of 9's that tends to infinity". Being infinite and tending to infinity are two different things.