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?

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    $\begingroup$ What exactly do you mean by "we can know every digit of a"? $\endgroup$ – Yuriy S Jan 9 at 7:21
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    $\begingroup$ There is no paradox here. You correctly observe that we can have a very well-defined - even explicitly computable - real number $a$, yet be unable to prove whether or not $a=0$. Why should this be problematic, though? $\endgroup$ – Noah Schweber Jan 9 at 7:24
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    $\begingroup$ "Note that a(n) is definite, since every p(n) could be verified, at least by machine, as long n is given" . there are infinitely many $n$. We can only verify a finite many of them. We can potentially know any $n$ we want but we will only know the ones that we actually do verify and potentially being able to verify any of them is not the same as actually verifying everyone of them.... Consider that you could read any page, but only one page of Moby Dick. Since each one of every page can be read does that mean you can read every page. No, it does not. $\endgroup$ – fleablood Jan 9 at 8:05
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    $\begingroup$ Just because we can verify any $a_k$ doesn't mean we will and we won't know what an $a_k$ is until we verify it. And we are NOT going to verify all off them. we only have enough time in the history of the universe and the lifetime of every human being to verify maybe $10$ trillion of them. And it's true we can verify any $10$ trillion but whichever ten trillion we do verify there will by infinitely more that we dont get around to verifying. $\endgroup$ – fleablood Jan 9 at 8:20
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    $\begingroup$ It may help to distinguish between (1) properties that are satisfied by exactly one real number, and (2) properties that a number might satisfy. Those of type (1) can be used to define a real number, and those of type (2) are questions we could ask about a number. As an example, one can prove by calculus methods (rates of increase and intermediate value theorem) that "$x^2 + 1 = 2^x$ and $4 < x < 5$" is satisfied by exactly one real number. Let's give this number a name, say $B.$ We can now ask things like is $B$ rational, what is the $10$th digit to the right of the decimal point of $B,$ etc. $\endgroup$ – Dave L. Renfro Jan 9 at 11:24

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.

  • $\begingroup$ Why we know every digit and do not know the number itself? In order to know the number, what else we need ? $\endgroup$ – mengdie1982 Jan 9 at 7:37
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    $\begingroup$ Do you know every digit of $\pi$? No? But defining $\pi$ as the ratio of the circumference to the diameter is a perfectly reasonable definition. We can find as many finite digits as we like but we can never know an infinite number of digits unless something else is proved. Same with your number $a$. $\endgroup$ – QC_QAOA Jan 9 at 7:55
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    $\begingroup$ I say your Riemann number is 0, and my spouse says it's 1. Please send 1 million dollars to our joint account. ;-) $\endgroup$ – Joffan Jan 11 at 14:30

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?

  • $\begingroup$ Sir, please compare $a$ with $\sqrt{2}$. What's the difference between them, on the point " know" ? Why we know $\sqrt{2}$ but do not know $a$? $\endgroup$ – mengdie1982 Jan 9 at 11:01
  • $\begingroup$ @mengdie1982: One can prove with significant machinery (Intermediate Value Theorem) that there is a unique positive real number whose square is $=2$. That number is usually called $\sqrt 2$, and we can in principle compute any digit of it. Everything "extra" we know about it is that by definition, it is a number whose square is $2$. Often when dealing with it, it is actually only that information that one uses (and e.g. its negative in $\mathbb Q(\sqrt 2)$ can play the same role, or one of the two numbers in the $3$-adics $\mathbb Q_3$ whose square is $2$ ...). $\endgroup$ – Torsten Schoeneberg Jan 9 at 16:19
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    $\begingroup$ [continued] Exercise: Define a real number $x$ where deciding whether the real number $\sqrt 2 \stackrel{?}= x$ or not is equivalent to some open conjecture. $\endgroup$ – Torsten Schoeneberg Jan 9 at 16:22
  • $\begingroup$ We don't know all the digits of $\sqrt{2}$. Why did you think we did? We have methods to calculate them to whatever accuracy we want but we certainly don't know what they all are. $\endgroup$ – fleablood Jan 9 at 16:25
  • $\begingroup$ Maybe more strikingly, mengdie1982, how well do you think we "know" $\sqrt 2$? Do you know if the digit $7$ occurs infinitely often in its decimal expansion? I don't, and I think it is not known. (Cf. the same question about $\pi$: math.stackexchange.com/q/2791/96384. If that is known about $\sqrt 2$, it would probably by some intricate argument via continued fractions. What really is not known, according to Wikipedia, is whether $0, ..., 9$ all have the same asymptotic density in $\sqrt 2$'s decimal expansion: en.wikipedia.org/wiki/Normal_number) $\endgroup$ – Torsten Schoeneberg Jan 9 at 16:51

$a$ 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.

  • $\begingroup$ Sir, my question is the last sentence. $\endgroup$ – mengdie1982 Jan 9 at 7:32
  • $\begingroup$ Why we know every digit and do not know the number itself? In order to know the number, what else we need ? $\endgroup$ – mengdie1982 Jan 9 at 7:34
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    $\begingroup$ We don't know every digit! Every digit is knowable, in the sense that a computer program could in theory compute it. But we do not actually know, say, the $n$th digit where $n=\lfloor e^{e^{e^{1000}}}\rfloor$. $\endgroup$ – RavenclawPrefect Jan 9 at 7:39

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.


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