Cantor was wrong.

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I think you're supposed to be looking at the numbers with leading 0. If you drop this, then 1.11111... isn't found in your scheme.

>>7988413
This is a strawman. Cantor's diagonilization proves that the set of all infinite sequences of binary digits cannot be listed. It is then shown that there is a 1 to 1 correspondence between T and a subset of R.

>>7988562
Why strawman?
OP thinks he found a loophole in how one may get all numbers into a list, that's the point.
I'm still waiting for a counterargument for 1.111 not being listed, though.

>>7988583
The strawman is in stating that the diagonilization attempts to list the real numbers in base 2. It doesn't. It attempts to list the infinite binary sequences and then shows that since that set is uncountable, the reals are also uncountable. If we wanted to use this argument on the reals directly we would have to take care not to list the same number twice, which OP does.

>>7988413
>>7988607
>>7988413
>>7988607

Right, OP's list contains infinitely many equivalent pairs, you could pick any one of the pairs and say "oh look, even though I used diagonalization I ended up with a number that was still in the list".

Diagonalization is guaranteed to give a digit expansion that is not in the list, to use it to generate a real that is not in the list you'd have to provide a bijection between the reals and digit expansions in the list.

>>7988413
Wouldn't both set be infinite?

Obviously I don't know shit about math but I'm curious.

>>7988413
Not bad, OP Two things I wanna say.

1) Just because you didn't obtain a contradiction doesn't mean there isn't one. I could take any proof by contradiction, erase the last line, and say "no contradiction to see here".

2) Repeat the argument on a different diagonal. You probably won't end up with something that ends with an infinite string of ones.

Now, here's the question I pose to /sci/. Where does this argument fail for the rational numbers?

>>7988493
>>7988583
It doesn't matter that 1.111... isn't on the list. Adding a single element to an infinite set won't change the cardinality of that set.

>>7989333
Just to elaborate on 2), the reason OP doesn't obtain a contradiction is because he ends up with a number that has two different representations. The motive behind the diagonalization argument is "well, if I change every number, then surely I'll with something that's different from everything else on the list". But clearly that isn't the case here.

But, it's definitely not true that if you do this on every diagonal you'll always end up with something that ends in an infinite string of ones.

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>>7989368
>But, it's definitely not true that if you do this on every diagonal you'll always end up with something that ends in an infinite string of ones.
So then are the reals only sometimes countable, like quantum mechanics but with infinite sets, and how the cardinality of the set collapses into either a countable or uncountable infinity depending on how you measure it via the ordering?

>>7988961
The naturals are countable infinite.
The reals are uncountably infinite.

A Countable set is one where there's a bijection (a one to one function for both the domain to range and range to domain) between the naturals and that set. Uncountable means such a function doesn't exist.
Similarly a finite set implies there's a bijection between some bounded subset of naturals (like all naturals less than N) and the set. Infinite means there is no set and function

Math majors feel free to correct me.

I might be retarded, but how is S defined here? it doesn't seem clear to me.

>>7989845
By that definition finite sets are uncountable.

>>7989874
Define it in anyway so that the main diagonal will end up being 1.000...

>>7989881
Thanks.
A denumerable set is a set with a bijection between the naturals and that set.
A countable set is either denumerable or finite. An uncountable set is infinite and not denumerable.

Fix'd?

>>7988413
No, he wasn't. See Rudin's proof. He shows that every subset of the reals in base 2 is a proper subset, and the reals are therefore uncountably infinite.

>>7988413
The only real numbers with two binary representations are fractions with a power of two in the denominator

Those are "small" in R, roughly speaking, and removing them doesn't affect the cardinality.

In fact, Cantor's actual proof (not whatever the fuck you wrote) does remove them. It uses a separate mapping for those rationals and the binary strings that don't represent a unique number.

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Reminder that classical set theory does not allow repetition of elements in any set.

this is how plebby classical mathematicians are.

>>7989343
hmmm ill need to see a proof of this to believe it

>>7989343
Adding an uncountably infinite set to it does though

>>7989834
wat. No.

Not obtaining a contradiction in a proof != There is no contradiction to be found.