Cantor diagonalisation

Consider decimal numbers between 0 and 1 in binary.

Here's how I am going to synthesize this set.

Step 1: Take non-decimal binary numbers and consider them to be padded with an infinite of zeros at the left.

...0000000

...0000001

...0000010

...0000011

...0000100

...0000101

...0000110

..........

Do we agree that this will contain all the non-decimal non-negative binary numbers?

In particular, is the following number in the above set? ...111111 (all ones, not zero-padding on the left). If not, why not. And if not, this seems to be a matter of definition to me. If yes, move on.

Step 2: Place a decimal at the end of each line above and flip left and right.

0.0000000...

0.1000000...

0.0100000...

0.1100000...

0.0010000...

............

Do we agree that this contains all the decimal positive binary numbers between zero and one: [0, 1).

Let's now apply diagonalisation on this.

It says that the number 0.11111111... will not be present in the above set.

Perhaps someone can see my confusion and enlighten me. :-) Thanks!

I am fully aware of the constructionist approach to mathematics.

But your argument goes somewhat like this:

The reals are real

In computer programs however, we can only do so much

Every computer program represents a real number

For every bounded amount of bits, there is only a countable set of computer programs

Therefore, the reals are countable.

This is not how math works. You extrapolate from a bounded set that is countable to the collection of all those bounded sets - and then claim because every one set is bounded, the collection must be bounded as well - and therefore countable.

Let me make a similar argument just to show how silly your argument is. Here we go: 1. Every set of integers with only one element is finite. Call it a trivial set, for example: {1}

2. Every finite union of such trivial sets is finite

3. Every finite union of such a finite union of trivial sets is finite

4. and so on - after many iterations, your finite set will look pretty big to the human eye - almost as if it was the set of integers.

5. Therefore, the set of integers must be finite.

Sounds pretty silly, right? Thats what you've done for a different property of a different set.

That is nonsense. This is why arguing with illiterates about math is so annoying. They think they can write non-stringent arguments and then claim they have "somewhat proof". In math, there is no such thing as "somewhat proof". You are either right or you are wrong. The reals are the reals by definition. You can not make them countable.

What you can make countable is sets of numbers that you work with. Sure. You wont ever create a computer that has access to all the reals, PRECISELY because it operates from a position of countable-ness (even quantum computers). But thats not what I care about when I talk math.

The real numbers are not accessible to in a "reality" way. They lie fully in the realm of arcane mathematics. Like many other things do. You would never claim that the hyper-exponentiation operator had anything to do with reality, and it doesn't. Neither do the reals. You're just confused about them because they appear so prominently in middle school mathematics.

Accept it and move on.

Mathematics is the realm of absolute truth. Everything we have proven is de facto true. You can make an argument that we are doing the "wrong kind of mathematics". And that may actually be true. But it doesn't change anything about our discipline. If you want to get philosophical and create a new Mathematics discipline based on different arbitrary rules (axioms), you are free to do so. But you will have to prove a lot of very mundane things and go through centuries of early day fuckups and its very likely that you will soon seek to merge with the already established mathematics.

Call it math2.0. I'll play with it. If its interesting, why not. But don't expect your new math to be more interesting than the one we've got.

Pi looks somewhat like this: 3.1415... "and so on". Let's ditch the 3.

0.1415... and so on. flipped over equals:

...5141

Easy enough, right? But pi had infinite decimals. Infinite decimals on the left side means what? You tell me :)

Infinity is not an element in the real numbers. All the irrationals, when flipped over, get absorbed into infinity - and infinity itself has no decimal representation. For people who are not in math, saying "and so on" is fine. But you can't do math on "and so on". "And so on" is a handwaving way of saying that you lack the mathematical language to describe whats going on. Which is fine, for casuals.

The definition of the number 2 is by axiom. The peano axioms state this:

1. There is a number. We call it 0

2. For every number n, there is a successor S(n). The successor of 0 is called 1

3. Let m and n be numbers. m=n is equivalent to S(n)=S(m). This means that if two numbers are the same, they have the same successor

4. For every number n, S(n) is not 0. This means that 0 is the first such number

This completely describes the natural numbers. If you want to say that 0 is not part of the natural numbers, just write down the same things and say that 1 is the first number, and construct 0 later. For some funky reason, this doesn't actually matter. Although you'd think that difference between 0 and 1 is pretty big.

Nowhere in this does infinity appear. Infinity does, in fact, not belong to the natural numbers. Infinity as a NUMBER is only defined much later in mathematical literature. As a concept, it just means "big". When we talk about infinity in terms of the reals it just means "if we construct a series of numbers that get succeedingly bigger and they are not bounded, then they 'tend toward infinity'". But infinity is not a thing. Its just a name for "this shit gets big brutha".

The problem is that I can't give you a formal definition for "and so on", because it can mean many things.

I will give you a formal definition for a number instead.

Pretend again that we look at the number .111111111...

and now we flip it around. We look at the number ...111111.0

What this means is the following. This number is the sum over all the exponents of 10. The number is obviously identical to 1 + 10 + 100 + ... = 10^0+10^1+10^2+...

We still get the and so on. Now we need to find language that seeks to eliminate the notion of "and so on". Keep in mind that this number is not part of the real numbers, because it would be identical to infinity.

We prove this in the following way:

Our number is obviously bigger than 1, because it is 1+10+...

It is obviously bigger than 10, because it is 1+10+...

Now, we perform something devious. An argument by induction. This happens in the first week of math education and it is precisely when we leave the realm of "all the things we can write down".

We prove that for every n, the number ...1111.0 is bigger than 10^n.

1. We already know this for n=1, because 1+10+... is bigger than 10^1 = 10.

2. For n>1, 10^(n-1)+10^(n) is clearly bigger than just 10^(n), and our number contains all the exponents of 10.

(^Up there is a tautological proof. You already know this, I just write it down "semi formally". Proving a fact about natural numbers by induction is always somewhat tautological because the mechanism that makes induction possible IS the natural numbers themselves.)

Now we know that for every exponent of ten, our number is bigger than that. This means that for every natural number that we can possibly think of, our number is bigger than that. Therefore, the number we are looking at is not part of the natural numbers and by extension, not part of the real numbers.

This is because every natural number has a successor and by definition, the successor is bigger than the number itself. But our number is bigger than all the numbers already. In order for it to be a natural number, it would have to have a successor, which would be bigger than itself. But our number is bigger than all the numbers. This sounds a bit clunky. I could write it down mathematically, but that would just confuse you, probably.

Now I'm just being wordy. The proof was over long ago.

edit: Writing this down mathematically goes like this: Let b=...1111 and n be an arbitrary natural number. b>n => n is not successor of b => b is not in N. (You see that I had to reduce the argument to something I already knew - the successor axiom - in order to wrap it up).

I've shown that your number is bigger than all the possible number on the real line and if its not on the real line, it is not part of the real numbers.

Now you would ordinarily say that it is infinite. And that's kind of true, but you can only say that once you understand that infinity is just a concept, not a number in the way you understand numbers.

tl;dr: Infinity is not part of any of the ordinary sets of numbers (natural numbers N, integers Z, rationals Q, reals R, complex C, ... - yes, there are more of these)