# Cardinality of the set of binary-expressed real numbers

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1. Dec 3, 2015

### PengKuan

Cardinality of the set of binary-expressed real numbers
This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis.
1. Counting the fractional binary numbers
2. Fractional binary numbers on the real line
3. Countability of BF
4. Set of all binary numbers, B
5. On Cantor's diagonal argument
6. On Cantor's theorem
7. On infinite digital expansion of irrational number
8. On the continuum hypothesis

Cardinality of the set of binary-expressed real numbers
http://pengkuanonmaths.blogspot.com/2015/12/cardinality-of-set-of-binary-expressed.html
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2. Dec 3, 2015

### micromass

Staff Emeritus
Sorry, it's wrong. You say there are $2^n$ "fractional numbers" with $n$ decimals. Correct so far. But then you say "let $n$ go to infinity and you obtain $2^{\aleph_0}$". This is an incorrect reasoning. Cardinalities do not work like this.

There are other very obvious errors in your paper. I suggest to pick up the excellent book "Introduction to set theory" by Hrbacek and Jech and work through it carefully.

Also, if you claim to have found an error in Cantor's diagonal argument, then it is not ok to make some kind of analogy with the Hilbert grand Hotel. Instead, you need to specify where exactly the error is. In particular, here is the complete rigorous proof of "the reals are uncountable" http://us.metamath.org/mpegif/ruc.html Every single logical step is detailed from the axioms. You need to say exactly which step is not allowed and why. Otherwise, nobody will take you seriously.

3. Dec 3, 2015

4. Dec 3, 2015

### micromass

Staff Emeritus
Well, your reasoning is wrong in any case and shows you do not really grasp set theory well. In particular, the proof by contradiction.

5. Dec 3, 2015

### Staff: Mentor

I might be restating what micromass already said, but in your example of the Grand Hotel, to find a room for one more guest, all that needs to happen is that each current guest gets moved from room N to room N + 1 (where $N \ge 1$), thereby freeing up room 1. The new guest can be assigned that room (room 1). The argument is a lot simpler than what you have described, and has nothing to do with Cantor's diagonal argument.

A similar argument can be used to accommodate any finite number of guests, simply by shifting each current occupant from room N to room N + k, where k is the number of new arrivals. The hotel can also accommodate a (countably) infinite number of new guests, by shifting each current occupant from room N to room 2N, thereby freeing up rooms 1, 3, 5, ..., 2N + 1, ... for the new arrivals.