How can we say there are Reals that CAN'T be "injected" to a natural number?

Question

When it comes to real numbers like $\pi$ it is easy for us giving an algorithm that can compute it (even if the algorithm never halts, we know it will converge so basically we will find new digits and hence we can find the nth digit).

Each algorithm can be encoded to a natural number (Turing used powers of prime numbers to do that), so as long as we give a real number that can be written using an algorithm (Even if the algorithm never halts) that can approximate it as good as long as it is left running, we have a Injection from Real Numbers to Natural Numbers.

However when someone say there are also Reals that can't be represented with an algorithm I have really have some doubt about that fact.

The argument used by people who claim that (if I understood correctly), is to provide an algorithm that if written, would require a tape of infinite lenght, however: to actually prove that this new algorithm is infinite, they should define it.. providing anyway an algorithm.

If I use a TM that writes this "infinite definition" and simulate/execute is as long as it is written, I can create anyway an algorithm, that can be encoded again as natural number.

So actually I think that it is wrong when someone say that there are Real numbers that can't be mapped to natural numbers, but I have anyway the suspect that I'm wrong. Why?

The reason for which Reals are uncountable to me is clear (no bijection with natural numbers), However I'm not convinced that there are real numbers that can't be mapped to a Natural number.

EDIT:

Clarification: I'm not speaking of uncomputable numbers, but to show me a Real number (Even one that we don't know how is value exactly is) that can't be encoded as natural number. This is not the boring and ever asked question about computable numbers ^^.

EDIT2:

I assume real numbers are a subclass of algorithms (and hence turing machines), I assume that because I think it is essentially true. When we give transcendent numbers we are actually defining them through an algorithm.

From a comment below:

$\pi$ is a number, and there are various definitions of real numbers in ZFC. Yes, it is computable, which means there exists an algorithm which computes it. Note also that there are many algorithm who compute it, not just onelockquote

How can we construct a number that is not an algorithm? We can't, because to actually show that a number is not in an algorithm set, we have to show the number, actually constructing it (using a algorithm, that regardless if that is computable, we can write, and hence encode as natural number)! So, we give $\pi$ but in reality we defined it through an algorithm.

EDIT3:

I also assume that Cantor just showed there's no bijection between natural numbers and reals, he didn't showed in anyway that there are more Reals than Naturals. From the comments below I think to understand that most people is convinced that Real numbers are more than natural numbers, I'm not convinced of that.

Infact every algorithm can be encoded as natural number (skipping some natural numbers), and every natural number can be encoded as a real number (skipping many reals).

Answer 1

It is possible to prove (see, for example, Theorem 7.1.5 in the book How to Prove It by Daniel Vellemsn) that the following are equivalent:

1. A set A is countable (finite or countably infinite).

2. There is an injection from the set A into the set of positive integers.

3. There is a surjection from the positive integers onto A.

Since there is no bijection from the reals to the positive integers (i.e. the set of real numbers are not countable) there is no injection from the set of real numbers to the set of positive integers.

Answer 2

Who says there are reals that can't be encoded to a natural number?

Here is a way to encode any real number $x$ to a natural number: if $x$ is a natural number, encode it as itself; otherwise encode $x$ as $1$, encode $1$ as $2$, encode $2$ as $3$, etc.

So I am confident that no competent mathematician says deliberately that there are real numbers that can't be encoded to a natural number.

What mathematicians do say is that there is no single encoding of real numbers to natural numbers such that the domain of that single encoding contains all real numbers. Therefore, once you choose a particular encoding, some real numbers (an uncountable number of them, actually) will be left out.