Is $a$ bigger than $0$ or not?

Question

Consider Goldbach's original conjecture (no need worry, because we don't talk about "Goldbach" itself):

every integer $n> 2$ could be detached as a sum of three primes. (In Goldbach's age, $1$ is regarded as a prime.)

For any $n>2$, we denote the statement above as $p(n)$. Then define

$$a_n:=\begin{cases} 0, &\text{if $p(n)$ is true},\\ 1, &\text{otherwise}. \end{cases}$$

Note that $a(n)$ is definite, since every $p(n)$ could be verified, at least by machine, as long $n$ is given. However, consider the decimal number $$a:=0.00a_3a_4a_5\cdots.$$

What's $a$? Is it a real number?

If $a$ is a real number, then we can compare the magnitude of $a$ and $0$, since the real number field is ordered.

But we can't in fact. If $a=0$, then "Goldbach" is proven, and if not, then "Goldbach" is overturned. However, it's in suspense hitherto whether Goldbach is true or not.

Probably, we may say $a$ is indefinite, that's to say, we do not know what $a$ is yet. But a new question is coming about:

Why we know any digit but do not know the number itself ? In order to know the number, what else we need ?

Further, let's compare $a$ with another classic number $$\pi=3.1415926\cdots.$$

Indeed, we do not know all the digits of $\pi$ untill we compute every one. But we can safely say it's $\pi$. What's the exact difference between $a$ and $\pi$ on earth? Why we think we know $\pi$ but do not know $a$? In another word, what does "knowing a number" mean exactly? And in what sense can we say we know a number?

Answer 1

What you have defined is a real number. Indeed, from the definition given it is also a computable number. Does $a=0$? Well, this is something we simply do not know. Of course, there is nothing wrong with not knowing a number from a definition. For example, the Ramsey number $R(6,6)$ is bounded between $102$ and $165$. However, we have no idea what this number is. Another example, define the number $x$ to be $0$ if the Riemann hypothesis is true and $1$ if it is false. This is a perfect definition, but I have no idea what $x$ is and if you can tell me you will get a million dollars.

To reiterate: One can write a perfectly sound mathematical definition and not know what that number actually is. This is not a paradox, but simply something that can arise naturally.

Answer 2

$a$ is a real number, defined by the definition you have given above.

It is computable, in the sense that we can calculate it to arbitrary numerical precision via an algorithm.

However, it is not currently known whether $a=0$.

None of these statements are in conflict. In fact, there are many such cases in mathematics where there is some perfectly well-defined answer to a certain question, and where we could get successively better computational bounds that decide more and more of the digits of the number, but where we don't yet know whether those bounds will converge to a certain conjectural value.

The word "know" is doing a lot of work in this supposed paradox, and if you specify any particular notion of what it means to "know" what a number is you'll find that it is no longer an apparent contradiction.

Answer 3

You are confusing "we can verify any number of them" with "we can verify every one of them". yes, we can verify any of the ones we choose to but we can't verify any of the ones we don't choose to, and as there are infinitely many of them there will always be an infinitely number of them we don't choose.

So although we can know any $a_i$, we can't know every $a_i$.

Imagine I asked you. Can you read all of Moby Dick in 5 minutes. You say no. I ask how much of Moby Dick can you read in 5 minutes. You say one page, I guess. And I ask which page can you read. And you say well, any page. And I say which pages can't you read. And you answer well I could read any page. so I say if you can read every page and there aren't any page you can't read then you CAN read all of Moby Dick in 5 minutes.

Is that a paradox?

Answer 4

We can know any digit of $a$ (after a possibly very long but finite computation), but we do not currently know every digit. There is a difference: if you wanted to know the $n$th digit of $a$, give me some long but finite time, and I can tell you. But that is not the same as saying that I can tell you, currently, off the back of my mind, every single digit of $a$, for arbitrary $n$.

There is no paradox, you just need to be precise about the meaning of "know". You could say, for instance, that we can know any digit of $a$, but do not currently know every digit.