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I thought about this question for a longer time. There is a standard proof by contradiction that there is no smallest positive rational/real number $r$ by considering $r/2$.

Now what irritates me about that proof is that it relies on the assumption that such an $r$ or the algorithm for constructing such an $r$ could be given explicitly if it existed. Maybe it just can not but it still can exist. Maybe it can only be addressed by introducing a new way for the description of infinitely small numbers or, equivalently, the numbers that "lie between arbitrary large numbers and infinity". If we do not have a way of talking about such a regime of numbers, then how can we comprehend a set like the natural numbers as it is one object with infinitely many numbered elements but not containing infinity? Normally, infinity is not included in the natural numbers because one would argue that if it was a number, then one could find the number just before infinity but that would not be possible. But what if we introduced a way of talking about such a number? A way of talking about a smooth transition from infinity to numbers that we can write down? Maybe the name or algorithm for the smallest positive decimal number would just take an infinite amount of time to be written down. What if there are more numbers than there are names?

exchange
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    Was there some "popular thing" about this? This is like the fourth question about this matter in the past couple of weeks. – Asaf Karagila Mar 24 '17 at 10:12
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    Maybe look up hyperreals or non-standard reals. – Prince M Mar 24 '17 at 10:13
  • Also, the question itself does not make a lot of sense from a mathematical point of view. – Asaf Karagila Mar 24 '17 at 10:13
  • @AsafKaragila Really? No, I did not read anything popular about it. But you are welcome to send me the related questions. Why does it not make a lot of sense please? – exchange Mar 24 '17 at 10:13
  • if you make infinity a number, it seems to suffer from the same defect as the smallest number, infinity + 1 is a new larger number - so infinity isn't a number. – Cato Mar 24 '17 at 10:17
  • @Cato: Well, infinity could be the number to which you can not add more numbers. Then the "usual" definition of numbers would only apply to final numbers. I thought about introducing infinity into a new system with additional rules to handle it to escape the contradiction that an infinite set with naturally numbered elements does not contain infinity. – exchange Mar 24 '17 at 10:21
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    The set of names (even if we allow arbitary long names) is countable, the set of real numbers however is uncountable. So, there must remain reals that have no name, no matter how we try to name the real numbers. – Peter Mar 24 '17 at 10:22
  • You write "it relies on the assumption that..." but where exactly is that assumption used? – Mark Mar 24 '17 at 10:44
  • @Mark I mean that one states: "Assume, for contradiction, that $r$ is the smallest rational number. Then, no matter what $r$ is, one can divide it by 2." But when you say "no matter what $r$ is", what is meant (colloquially) is: "No matter what number or algorithm you give to me, I could give you half of it.." So there is the assumption that you could give it to me, but maybe one can not. What I tried to say is: Just because one can not give such a smallest number, it does not mean that it does not exist. – exchange Mar 24 '17 at 10:49
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    @Peter: Not so fast -- the correspondence between names and the numbers they mean is generally not definable in the formal system we work in (such as ZFC set theory), so just because a surjection does not exist as a single mathematical object in our universe doesn't necessarily mean that the universe contains a number that has no name. – hmakholm left over Monica Mar 24 '17 at 10:52
  • That "standard proof" does *not* rely on "on the assumption that such an $r$ or the algorithm for constructing such an $r$ could be given explicitly if it existed." Rather it relies on the basic fact that *every* number $r$ has a half $r/2$, not just the numbers that can be "given explicitly". – bof Mar 24 '17 at 11:03
  • @HenningMakholm Naming every number can only work if we allow infinite long names (or if we change the names whenever we find a new number, which would be "cheating"). But I would not consider an infinite representation to be a name. So, even a definition outside ZFC does not change the situation. – Peter Mar 24 '17 at 11:15
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    @Peter: No, that is not so. There exists models of ZFC where every set -- in particular every real number -- has a finite definition in the form of a logical formula with one free variable that is true for exactly that set and nothing else. – hmakholm left over Monica Mar 24 '17 at 11:31

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It sounds like you have been confused by the colorful phrasing sometimes used to explain proof steps -- that is, things like, "for every number you can give me ..."

Speaking that way generally tends to make it easier for beginners to understand the structure of proofs, but it is also possible to take it too seriously, which you appear to be doing.

In particular, the actual content of a proof (say, a proof that there is no smallest positive number), does not really depend on anyone physically "giving" numbers to each other. As far as the proof goes, saying "every number you can give me" does not mean anything that's different from "every number" -- the "you can give me" is just a vivid way of reminding ourselves that the proof we're constructing is not allowed to rely on things that are only true for some numbers.

The assertion the proof depends on is simply that every positive number equals two times some other number, which is also positive but smaller than the first number. Note well that in this formulation, the claim does not speak about anyone doing anything -- it just states an eternal fact about numbers, not that we "can divide every number by 2", but that half of each number simply exists, has always done so and will always do so. And this general fact does not depend on whether the numbers have names or whether we can communicate each of the numbers exactly between each other.

Of course, if you desire to, you can start investigating what you can prove if you do insist on only speaking of things you can actually imagine doing -- that is, for example, only speak of numbers that can be communicated using finite description. This leads to constructive mathematics, which is an entirely respectable area of study. You just need to be aware that constructive mathematics is not what ordinary mainstream mathematics aspires to be: everyday proofs do allow things that cannot be justified constructively, and if the default assumption when you speak to mathematicians is that the proofs you speak about are according to the everyday non-constructive rules, however vividly they are phrased.

hmakholm left over Monica
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  • Thanks for the answer! I admit that I thought that when fixing some $r$ for constructing a contradiction, it is an implicit assumption of the possibility to give an algorithm for its construction. I voted your answer up for clarifying this for me. However your answer is still not really what I am looking for because the question is actually meant to be deeper. I may reask how do we know that really every $r$ that we fix is still greater zero when divided by 2? What I am thinking of is to enlarge the vision on numbers to be able to talk about e.g. $\delta:=\lim_{n+1\rightarrow\infty}1/2^n$ – exchange Mar 25 '17 at 13:24
  • such that $\delta>0$ but $\delta/2=0$. I know that we usually do not think of a construction like that because there is no notion of something like $\infty-1$ but my question is exactly what IF we would really develop a precise meaning for the realm of numbers between arbitrary large numbers and infinity. – exchange Mar 25 '17 at 13:26
  • Because if we could do something like that, then we would be able to construct a new way of looking at e.g. the natural numbers including infinity. This would be more intuitive in the sense that a set that is numbered $1,2,3,...$ with infinitely many elements actually contains the infinity. Because if we would really have an INFINITE amount of time to write down the natural numbers, we would have to get there. – exchange Mar 25 '17 at 13:29
  • But the way we handle natural numbers (and the smallest point bigger zero) now is that we just say that these points near and at infinity do not exist, so effectively we are talking about things that we can write down in finite time even though we call it an infinite set. So that is why I thought that there might be more numbers than we can think of in the current approach. So I actually want to do the opposite of a intuitionistic or constructive mathematics approach - I would like to extend the view on numbers to be able to talk about something that in a finite time may be unconstructable. – exchange Mar 25 '17 at 13:37
  • Thank you very much for consideration! I apologize that my arguments are formally not well developed but I hope that the meaning of what I want to say can come across. – exchange Mar 25 '17 at 13:42
  • @exchange: Well, it's mathematics: You're allowed to defined whatever you please. However, when you do that, the burden is on you to show how your definitions are worth caring about -- not because the math police will come after you otherwise, but because if you fail to convince, everyone else will simply keep using the same concept of number we have used all the time, which seems to work quite fine to us already. – hmakholm left over Monica Mar 25 '17 at 14:39
  • For the particular plan you're outlining, you're going to run into the problem that in the arthmetic we already know and love, it is an absolute fact that when $a/b=c$ we also know that $b\cdot c=a$. If you want a $\delta\ne 0$ such that $\delta/2=0$ this cannot be true anymode, because $2\cdot 0$ is and always was $0$, which you insist is not your $\delta$. So you face quite an uphill battle explaining what advantages your new numbers bring with them that are worth sacrificing the defining property of division for. – hmakholm left over Monica Mar 25 '17 at 14:39
  • That's not to say your task is impossible -- you have thought about this plan longer than I have, so perhaps you can see an desirable goal it achieves that does not spring to my mind immediately -- but it is yours to carry out all the same. – hmakholm left over Monica Mar 25 '17 at 14:41
  • Thank you very much for your constructive answers and for looking at the idea in a tolerant way! This helped me a lot! You are probably right that it is at least a difficult task with a lot of trouble. I will save this discussion and think about it. In any case, this question and your answers have given me new ideas and understanding! Thanks! – exchange Mar 28 '17 at 08:45
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If a number can be defined, let its name be its (possibly long) definition. Then any number we can define has a name. (F.i., the name for $\sqrt{2}$ would be "the positive root of $x^2=2$".)

mlc
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    This shows that not every real number can be defined. – Peter Mar 24 '17 at 10:23
  • What Peter said is exactly the point: You may be able to axiomatically define all real numbers (together) but when naming each individual of the uncountable set you defined, then you can not only use one definition. The argument of your answer does only hold if a definition is used to only define a single thing (number) at a time, which is not the case for number systems (in particular not for infinite ones). I also ask about the possibility of enhancing the number system and hope to obtain more information regarding that. But thanks for your answer! – exchange Mar 24 '17 at 10:56
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    You convinced me that my answer is inappropriate (thank you!) and I am going to delete by tonight. (I am leaving it for a while so you get a chance to read this.) – mlc Mar 24 '17 at 11:05
  • @exchange Henning apparantly disagrees ... – Peter Mar 24 '17 at 11:51
  • Thanks for the answer mlc! @Peter Yes Henning says that the correspondence is generally not definable. But even if this is so, the answer above only holds for things that you define individually, right? But I think it can still be left there and does not have to be deleted. It was also interesting for me. – exchange Mar 24 '17 at 12:16
  • @exchange As I understand the comments, Henning claims that it is possible to name every real number by pointing out that there is some model in ZFC allowing this. And since a finite description is a definition, the claim also implies that every real number can be defined. – Peter Mar 25 '17 at 08:31
  • But that necessarily means that we also have uncountable many FINITE definitions, otherwise there would not be a surjective map from the finite definitions on the real numbers. – Peter Mar 25 '17 at 08:37
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    @Peter I have to admit that I can not yet judge the argument of Henning about the possibility of such a map in the special model of the ZFC he points out. Therefore I will not argue about it now but I will save the link and this discussion for later consideration when I obtain more understanding. Thanks again for your answers. – exchange Mar 25 '17 at 13:51
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Set theory has the concept of transfinite numbers (for example the first infinite ordinal $\omega$), so learning about that might answer some of your questions (I'm not sure anything would qualify as a "smooth transition from infinity to numbers that we can write down" -- you would need to describe more rigorously what you mean by that).

As for whether arbitrarily small positive rational numbers can be written down (in principle, given enough time and paper/ink), the answer is clearly yes. Suppose I have written down some rational number $p/q$ where $p$ and $q$ are in decimal digits; well, I just add a zero to the end of $q$ and I have a smaller number. So in that sense every positive rational number has a "name" (digits, followed by "/", followed by more digits) although that "name" may be very long.

As for whether every irrational number can be "defined", it appears to be more complicated (I think it may depend on what model of set theory you are presumed to be working in and what exactly you mean by "define"). This MO post discusses such issues.

Regarding some concept of number where $\delta > 0$ but $\delta / 2 = 0$: I doubt anything exactly like that exists. What is $\delta / 1.1$ in such a system? Or $\delta / 1.0000001$? However, as mentioned by another commenter, the hyper-real numbers are sort of like this.

NikS
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