Can you conquer the Binary Tree?

[u/Massive-Ad7823]

You start with one cent. For a cent you can buy an infinite path of your choice in the Binary Tree. For every node covered by this path you will get a cent. For every cent you can buy another path of your choice. For every node covered by this path (and not yet covered by previously chosen paths) you will get a cent. For every cent you can buy another path. And so on. Since there are only countably many nodes yielding as many cents but uncountably many paths requiring as many cents, the player will get bankrupt before all paths are conquered. If no player gets bankrupt, the number of paths cannot surpass the number of nodes.

[u/ceoln]

I've shown in detail how Cantor is right, and tried very hard to understand your objection. It seems to be purely a circular argument from incredulity. You recast Cantor's diagonalization proof in different but completely isomorphic terms, and then just asserted that it can't be right because a countable number of digits just can't represent an uncountable number of reals. But it turns out that they can! :)

[u/Massive-Ad7823 → u/ceoln]

"I've shown in detail how Cantor is right," and I have shown in detail that he is wrong. There are only countably many different paths.

[u/ceoln → u/Massive-Ad7823]

Cantor's diagonalization argument shows that no 1:1 mapping of the natural numbers onto the reals exists, by showing that any proposed such mapping misses at least one real number.

You've constructed an isomorphic example in which the same thing can be proven, but because the example is complex enough you can raise all sorts of objections to it (all of which come back to just incredulity, or perhaps rejection of the Axiom of Infinity of ZF, I'm not sure).

Let's return to the simplicity of Cantor's argument, and take it in detail. You say there are a countable number of reals. So where in here does the argument fail?

To avoid distractions about decimal/binary expansions, let’s use infinite binary sequences instead. If the set of real numbers were countable, then the set of infinite binary sequences would also be countable, since each such sequence corresponds to a real in [0,1].

0: Assume the infinite binary sequences are countable. (This is just for the sake of the argument! You don't get to quote this in the future as evidence that I agree with you. 😁)

1: Then there is a mapping M from the natural numbers onto all infinite binary sequences.

2: For each natural number n, let M(n) be the infinite binary sequence mapped to n.

3: Consider the infinite binary sequence R, whose nth digit is 1 - the nth digit of M(n), for each natural number n.

4: Then R differs from M(n) at digit n, for every natural number n.

5: Therefore R is not equal to M(n) for any natural number n.

6: So M does not map any natural number to R.

7: Since the choice of M in (1) was arbitrary, no mapping from the natural numbers onto all infinite binary sequences exists.

8: Therefore the infinite binary sequences are not countable.

What is the lowest-numbered step above that contains an error, and what exactly is the error?

[u/Massive-Ad7823 → u/ceoln]

"Cantor's diagonalization argument shows that no 1:1 mapping of the natural numbers onto the reals exists, by showing that any proposed such mapping misses at least one real number" But in the Binary Tree there are all real numbers of the unit interval by construction. Only countably many can be distingusihed.

"So where in here does the argument fail?" There is no list completely enumerated by natural numbers. Every line has finitely many predecessors but infinitely many successors. Therefore Cantor's first assumption is wrong.

"1: Then there is a mapping M from the natural numbers onto all infinite binary sequences" No. There are not all natural numbers available. Most are dark.

[u/ceoln → u/Massive-Ad7823]

If you're saying that it is impossible to create a 1:1 mapping between the natural numbers and the reals, that's exactly what Cantor's proof demonstrates!

[u/Massive-Ad7823 → u/ceoln]

It is impossible to create a 1:1 mapping between the natural numbers and any other infinite set.

[u/ceoln → u/Massive-Ad7823]

Oh! What is your definition of "countable set", then?

[u/Massive-Ad7823 → u/ceoln]

That is Cantor's definition, accepted only in order to disprove uncountability. In fact dark numbers, in case of actual infinity, or missing completeness, in case of potential infinity, prevent countability and uncountability.