The French mathematician Jules Richard came up with a paradox in 1905 involving sets and definitions of numbers. Using these, he constructed what appears to be a number that is both definable and undefinable, as well as a definition for that number that is both clear and ambiguous. Let’s get into how such a number is to be constructed, using my own exposition. I name some of the propositions that follow for easy ordering, using ‘P’ along with a numeral, and conclusions from these propositions using ‘C’ and a numeral.
P1: Some expressions of English define real numbers unambiguously.
For example, the definition “The integer between 2 and 4” defines a real number unambiguously because there is no other number besides 3 that fulfills this definition. Note that real numbers are integers (i.e., whole numbers less than zero, zero, and whole numbers greater than zero), rationals (i.e., any numbers that can be written as fractions, where the numerator and denominator are integers—that is, written as ratios) and irrationals i.e., those numbers, such as pi or the square root of two, that cannot be written as fractions where the numerator and denominator are integers).
P2: Some expressions of English do not define real numbers unambiguously.
For example, the vague-at-best definition, “The integer that is almost half of three but not quite, especially as approximated a few minute after last call at the bar.” This definition does not define any particular number. At best, it defines an arbitrary list of numbers according to the drunken state of individuals at a certain moments, which may change at different attempts at the approximating calculation.
P3: Arranging all possible expressions of English in order is possible by arranging all combinations of single English characters (i.e., letters, numerals, spaces, special characters, etc.) lexicographically (i.e., ‘a’ – ‘z’, then ‘1’ – ‘0’ as it is along the top of the keyboard, as well as making up a specific order for the special characters) and according to length (i.e, all single characters come first, and then all pairs of characters, then triples, and so on).
(Note that a list just is some arrangement of things in a particular order. And so, I will use terms such as ‘arranging in order’ or ‘ordered’ or ‘orderings’ interchangeably with ‘listing’ or ‘listed’ or ‘lists’, respectively.)
To explain, the first ordered combination of English characters would be the single characters ‘a’ then ‘b’, then ‘c’ and so on; and once all the single characters are ordered according to our lexicographical standard, next would come all the ordered pairs of English characters, ‘aa’ and ‘ab’ and so on; from there would come all the combinations larger than pairs, likewise by order of size and our lexicographic standard. Recall that English characters include spaces and numerals and even special characters, such as commas, periods, and parentheses. Given an ordering of all such combinations of English characters, it follows that most of these combinations of strings of characters are cat-keyboarding nonsense; however, some of them will include expressions such as “It is raining” and “The number that is 1 plus 1.” What’s more, it follows that such a list is infinite. What could I mean by this? Take the string of characters ‘aa’. Well, for this string, another string may be constructed that is the concatenation of ‘aa’ and ‘a’, ‘aaa’. With this string, however there is another string that may be constructed; namely, the concatenation of it and ‘a’. And this step may be repeated over the constructed string indefinitely, yielding larger and larger strings of ‘a’ characters. However, all such constructions are just strings, or combinations, of English characters. Thus, such strings of ‘a’ characters that could be constructed indefinitely are also in any arrangement of all combinations of English characters. Thus, any arrangement of all combinations of English characters is infinite.
A final note before proceeding: in what follows I will use proper capitalization within quoted strings of characters, but I do this so sentences do not look grammatically improper. For our purposes, there will be no meaningful difference between “it is raining” and “It is raining”. That is, capital letters can be considered used just as a matter of style, and they can be considered to be the same character as their lower-case equivalents (akin to differences in font). If you prefer to include them as separate characters, however, you may think of capital letters as being special characters.
P4: (Usually implied in other expositions) any definition of a real number in English is also a combination of characters of English.
That is, one English definition of 3—e.g., “The integer between 2 and 4”—whatever else it may be, is also a combination of English characters.
P5: (Usually unstated, but some version of this proposition will fill in a gap toward the conclusion, we will see) For any English definition, the definition defines the same thing if the period at the end is replaced with the string of characters ‘ and ‘, and then the string of characters of the original definition is repeated there after.
For example, since “The integer between 2 and 4” is a definition, “The integer between 2 and 4 and the integer between 2 and 4” is also a definition, albeit for the same number as the former definition. This latter definition perhaps sounds clumsy, but it would still define 3 unambiguously. It is no matter that the same info for this integer is repeated twice; the same integer gets defined nonetheless.
C1: From the infinite list of all combinations of English characters, an infinite list of English expressions of finite length will define real numbers unambiguously.
How so? We have already established that unambiguous sentences of English of finite length exist that also define real numbers, and that they are finite characters of English. Therefore, arranging all English characters into an ordered list of all of the finite combinations of those characters will include plenty of definitions for real numbers (although again most combinations will be nonsensical cat-keyboarding). Next, we also made explicit the fact that any such unambiguous definition of finite length can be made into another, larger, unambiguous definition of finite length that defines the same number, as outlined by P5. And what P5 expresses can be repeated indefinitely, so that it repeats a definition twice or four times or eight times, and so on. Again, since all such definitions are also a finite string of English characters, then our list of all combinations of strings of English characters will also include in it an infinite list of English expressions of finite length that define real numbers unambiguously. To have a convenient name for this list of unambiguous definitions of real numbers, let’s call this list ‘D’.
With C1 serving as an additional premise, we can get the following conclusion as well:
C2: An infinite list of real numbers corresponds to the definitions of the list referenced in C1, or list D, by being those very numbers defined.
How so? Consider that the real numbers also includes the integers. No doubt, in the list of definitions, one will say “The number 1 added to 1”, and another “The number 1 added to 2”, and another “The number 1 added to 3”, and so on. That is, all combinations of numerals of finite length will eventually follow the space after ‘to’ and so form their own definitions. And since there are infinitely many combinations of numerals of finite length (recall that numerals are just names for numbers; specifically, in English, they are the characters ‘1’ – ‘0’ as they appear along a row near the top of a keyboard, and then room on the right side for concatenating more such numerals, according to their combinations), there are infinitely many real numbers defined.
We now have two different lists of infinite size. One is a list of unambiguous definitions in English of real numbers, which we call ‘D’, while the other is a list of numbers which are so defined in the former list (let’s call this list ‘N’). So the things referenced by each list are very different, and this difference is reflected by our names for these lists. These lists are perhaps best thought of, and often are thought of, as a kind of ordered set, where the things in that set are all and only those things that would get listed, and ordered as they would appear on the list. Lists and ordered sets are basically the same entities, and may be used interchangeably with slight adjustments to terms (e.g., lists typically reference things, while sets have things as members). I will continue speaking in terms of lists.
Another caveat concerns how we are to order the list N. I initially thought we might order it by numerical ordering (this likely shows how much of a n00b I am at mathematical logic). So the least number would come first, followed by the second least and so on. But this is untenable. First of all, any definition we take to define the least number might be concatenated with ” and subtracted by one” to plausibly result in a lower number. Even if we defined the least number as a sort of limit toward negative infinity, such a number does not seem to be unambiguously one real number or another. Also, assuming we could unambiguously define such a least number, further difficulties arise in deciding what the second least number would be. Maybe it is the least number times 1.5. But why not times 1.05—or 1.005? Maybe we’d multiply it by some 1 plus some infinitesimal. But if this is possible, then where would we find, say, the number 3 in our list, which would have to be there somewhere for being defined in D? Multiplying the least number by 1 plus some infinitesimal would indicate that there are an infinite number of ordered numbers before 3 on the list. But then we could never scroll down such a list (even given unlimited space and time) and get to 3. There is no ordered position n on the list such that n is a positive integer and 3 is in that position.
So obviously we need a better alternative. The most straightforward way I can see, and how we will proceed, is to order the numbers in N by the way their corresponding definitions in D are ordered. Let’s use ‘r1‘, ‘r2‘ and ‘r3‘ and so on with ‘rn‘ for every (numeral of) positive integer n (replacing the subscripted ‘n’ in ‘rn‘), as names for the first entry, second entry, and third entry respectively, and so on, in list N. And so, the way we will order list N is as follows: ‘r1‘ names the first number defined in our ordered list of definitions, ‘r2‘ the next number defined in our ordered list of definitions, and so on for every number defined in our ordered list of definitions, D. If a definition defines more than one number (e.g. “The integers between 2 and 5”), then we will just say that the numbers are added to the list from least to greatest (this may give rise to the difficulties I previously alluded to in ordering our list from least to greatest to begin with; I suppose alternatively we could just ignore definitions that define more than one number; deciding one way or another shouldn’t affect what follows). And if a definition defines a number that has already been defined by a previous definition, then the number is just listed again. Now we are finally fully prepared for our paradoxical definition.
We will define our paradoxical number p as follows: the integer part is 1, and its decimals are as follows: if the nth decimal place of rn is 1, then the nth decimal place of p is 0. On the other hand, if the nth decimal place of rn is not 1 (e.g., is 3), then the nth decimal place of p is 1.
To illustrate this definition, let’s list the real numbers as they might be defined by our set of definitions, D. List N might go as follows:
r1 = 1
r2 = 2.1333333333333 . . . .
r3 = 1.3333333333333 . . . .
By this list, we can see how to construct the number p by how it is defined. First of all, the integer part is 1. Next, we’d see that r1 has nothing in its 1st decimal place (or in its tens place; alternatively, a number with nothing in its decimal place is identical with a number that has a 0 in that place), and so is not a 1, and so must p must have a 1 in its 1st decimal place. Thus p would be 1.1 . . . . Next, for r2, its 2nd decimal place has a 1, and so p‘s 2nd decimal place must be 0. Thus p would be 1.10 . . . . And for r3, the 3rd decimal place is 3, and so p‘s 3rd decimal place would have 1. Thus p would be 1.101 . . . .
So far so good. Such a number would be of infinite length, given that there are infinitely many real numbers listed in our list N. However, this is fine since its definition is finite. And such a definition appears to be a perfectly good one: we intuitively see how to calculate such a number using the definition. There won’t be any numbers that we could come across in list N that would be ambiguous about how to construct p. Or are there?
This “number” is paradoxical: suppose p is non-ambiguously defined, as it appears to be. If so, then its definition is in D. But if that is so, then p is in N. But if that is so, then which number might it be?
In fact, p can be no number of N. Proof:
Suppose it is r1. But, if that is so, then the nth decimal place of rn is 1 if it is 0, and 0 if there is no 1 there. That is, r1 is 1 in its tens place if it is 0 there, and 0 if it is not 1 there. However, since p = r1, this is a contradiction. If
r1 has a 1 there, then p has a 0 there, by its definition, and since p = r1 it has both 1 and 0 there in its tens place; on the other hand, if it is some other number from 1 there, or no number at all, then it is rather both that number, or no number, and 1. If we return to our arbitrary and incompletely filled-out table above, supposing r1 = p amounts to supposing that it is the “number” 1.(1 and 0)01 . . . . But of course that “number” is contradictory, and so ambiguous, particularly about which number is in its tens place.
Finally, supposing that p is rather some other arbitrary number in the table will do us no good, for let’s now suppose that p = ri where ri is just some arbitrary number in list N that is in ordered position number i. If so, then the digit of the ith-place will have to be given some numerical value by the definition of p. But once more, the number given by definition will contradict the one that is already there, such that both will be there. For suppose that the digit of the ith place is 1. If so, then, since p = ri, the digit of the ith place will have to be both 1 and 0. Suppose rather that the digit of the ith place is not 1 (is blank or is 0 or 2 – 9). If so, then it is that very digit that is both not 1, and is 1. Thus, p cannot be ri. But ri was just an arbitrary number of N. Therefore, p cannot be in N.
Because all unambiguous definitions of real numbers in English expressions of finite length must have their number(s) in set N, the above definition for p, despite first appearances, is not in D.
However, if the above definition is not in D, then there is no real number p it defines that is in N. However, supposing that such a number is not in N seems to render the definition perfectly capable of providing the instructions to make a real number, without contradictions, since a contradiction only came about by supposing that p is in N. Without being in N, a number may be built by referencing the digits of all the numbers in N. But if that is so, then such a definition is unambiguous after all. But, if that is so, it is in D. But if that is so, then p is in N. But we just showed that p cannot be in N, since assuming it is any arbitrary number leads to a contradiction, and from here we can go around the whole loop of reasoning once more.
Thus p is both defined ambiguously and unambiguously, and is listed in N and is not listed in N. This is the paradox.
Resolving the paradox.
I think resolving this paradox is fairly straightforward. First let’s list the conditions (necessary and sufficient) for something being in list D and then in list N.
The condition for being listed in list D: something x is listed in list D if and only if x unambiguously defines (in English and in finite length) some real number r.
The condition for being listed in list N: something y is listed in list N if and only if y is defined in D.
Our Paradoxical Definition is Not Unambiguous
We have previously proven that no member of set N is defined by our paradoxical definition above (let’s now call this paradoxical definition ‘P’). But if this is so, then there is no y that is defined by P in D. But if P is not in D, then P is not an unambiguous definition. Let me offer a much more clear proof of this conclusion. What follows is an argument by contradiction from the assumption that P is unambiguous:
Supposition: Suppose P unambiguously defines some real number.
i) By the condition for being listed in list D, P is in list D.
ii) By our Supposition, P unambiguously defines some real number. Let’s call this number p.
iii) By the condition for being in list N, p is in list N.
(Explanation for iii): p is defined by P, as stated in ii), and P is in list D, as stated in i), which, taken together, entails that p is defined in D, and thus, meets the listing condition for N, so that p is in list N .)
iv) But, as we showed in the last section, p cannot be in N.
Conclusion: since iii) and iv) directly contradict one another and are also entailed by our supposition that P unambiguously defines some real number, this supposition must be false and P does not unambiguously defines some real number.
The definition we took to be unambiguous is not really so. Perhaps we initially thought it was, provided we ever did at all, because we did not consider the real number that it was—at least in part—defining as belonging to N. Now, merely taking the expression to unambiguously define a real number does not meet the condition of that supposed number belonging to set N. It must also must be unambiguously defined by some definition in D. That definition we took to be P. But, similarly, taking this definition to be unambiguous does not make it so. It must be unambiguous in fact. But there is no possible condition where P is unambiguous and p is a real number defined by P. To say that it is unambiguous is to put it in list D and the number it defines in N, but this cannot be done except in a contradictory and thus ambiguous way. And failing to put the definition in the one and the real number in the other of these lists while assuming that the definition is unambiguous is simply to violate the necessary and sufficient conditions of being on these lists. In particular, D would no longer be the list of all unambiguous such definition, despite having a condition that entails that it is.
So why not just assert the logically derived conclusion that our paradoxical definition is ambiguous? Specifically, it seems to be a closed case exactly why the definition is ambiguous, since it ascribes some real number with the contradiction of having a particular decimal place based on its supposed position in the list N be either both 1 and some other number or no number, or else both 0 and 1; namely, it is the decimal place that is in the nth place of the nth ordered element in N, where we suppose that p is that very same element (Again, following our proof from two sections ago, this digit is just ith digit of some arbitrary number in N that is in the arbitrary position of i as ordered in the list, and which we also may suppose is identical to p. That is, the number is ri where we suppose that p = ri; I will no longer explain at length what this contradictory digit is, and from now on refer to it by name. To give this contradictory digit a name, let’s call it our “Contradictory Digit”. )
The answer for why definition P seems plausible at first glance, I believe, has to do with that we are not readily cognizant of the implications of the conditions of membership for D and N when reading off the definition of P, and that because of this we think that there is some real number perfectly well-defined so long as this number is outside of set N. But what we forget is that being perfectly well-defined meets the very condition of being in set N.
What Kind of Definition is P?
The answer is no kind. Look at the following “definition”:
“The number that is 1 and 0”
Obviously, this is not a definition for a number. This is because no number is 0 and 1. And it says nothing else about any other number. Thus, it defines no number. And, since it says nothing else, the expression simply fails to be a definition.
I am not going to give a full account of what a definition is, and I am mainly relying on intuition here; however, at the least, a necessary condition for a definition should be that it expresses a property that something or other has. For definitions of numbers specifically, it should express a property that a number has. The reason the above expression is not a definition is because it expresses no property that a number has.
Likewise, supposing that P defines a real number entails that the real number defined is in set N, the entries of which are explicitly referenced in P. However, this entails that P defines a digit that is both 1 and not 1 (i.e. some other number between 2 – 9).
The issue is really one of quantification, because the entries of list N that P references are entries of a list of all definable real numbers. P defines a number by referencing every entry of this list, which must include the number being defined itself. When we assume that P defines no real number, on the other hand, then P seems to define a real number. Why is this?
The reason it may seem P defines a real number is because we readily see that a number can be constructed out of list N in the way expressed by P, with the exception that such a number is not in N. but this is foolishness: such a number would have to be in N. To my mind, this paradox has the same logical form as that of the Barber’s Paradox. There, we can avoid a contradiction by supposing that the Barber’s conditions on whom he will shave do not apply to the Barber himself (that is, such conditions do not quantify over everything). Similarly, we can get the number that P seems to construct (when it is, albeit contradictorily, supposed to not be in list N), by changing P so that it does not quantify over itself as an entry of list N. Or, at least, so that it does not quantify over itself with the same condition. The most straightforward way to define a number of this sort would be to define the number’s digits using all entries before p on the list, and then define no digit using p, and then define the rest of digits using all entries after p, but making sure to define the nth digit when rn = p by “scooting over” by one digit all the entries that follow p in the list. Let’s examine the following definition to make this idea clear:
Define a number p as follows:
The integer part of p is 1.
And, for all rn in N,
If rn comes before p in N then,
If the nth decimal place of rn is 1, then the nth decimal place of p is 0. On the other hand, if the nth decimal place of rn is not 1 (e.g, is 3), then the nth decimal place of p is 1.
If rn is in the same entry as p in N, then do nothing.
If rn comes after p in N then,
If the nth decimal place of rn is 1, then the nth minus 1 decimal place of p is 0. On the other hand, if the nth minus 1 decimal place of rn is not 1 (e.g, is 3), then the nth decimal place of p is 1.
Notice that the clause that follows “If rn comes before p in N then” is just the original definition P. The clause “If rn is in the same entry as p in N, then do nothing” ensures that we do not apply this definition to p itself. Finally, what follows “If rn comes after p in N then” is the same as the original definition P, except for what is in bold. This addition ensures that we do not miss the nth decimal where rn = p. And we don’t miss any subsequent decimal places for p, since every rn that is after p on the list provides the decimal for the nth minus 1 decimal place of p.
Now, if you’re initial thought is anything like mine was when I first made a definition like the one above, you would think that I’m an idiot and that these definitions are ambiguous, because p might very well equal any old element of N, for all the definition has said. Although this seems like it would be the case, recall that we earlier had decided on the way that the numbers of N are ordered—precisely by how their corresponding definitions are ordered in D. Thus, p has a place in list N which is determined by the ordered place of this new definition in D. It will be right after a number defined in the same length but less in the lexicographical ordering, or after one defined in a shorter length. From there, p will be in a determinate place in list N, and p will have a determinate value by its definition. Thus, there is nothing ambiguous about p based on our new definition!
Now this realization that p would actually come in the order in which it is defined in list D potentially warrants a new investigation into what we presumed is defined by our original paradoxical definition P! This is because we are no longer to consider that there is some arbitrary number listed in N that p also might be, but from which definition P determines a contradictory digit for p. Looking further into this demands its own section. However, in the end, P remains contradictory, and thus still cannot be in N, but according to different terms than those used in the previous sections.
Perhaps Our Paradoxical Definition is Not Contradictory?
We ordered the elements of list N based on the ordering of their definitions in D. Therefore, when we take it that p is defined by P, p will have a determinant position in N solely based on the length of its definition and its lexicographical ordering in D. Recall also that we stipulated that any duplicate numbers defined in D will just be added again to the list N. The implication of this is that there is not going to be some other number in place of p in list N. That is, until p is listed in N, there is just no number in that spot.
But now we have a choice. If there is no number in that spot prior to p being defined, then there are no digits either. So this may be as good as simply putting a 1 in place of the nth digit for rn for where rn = p, by our definition P, since what is there is not a 1. On the other hand, their being no digits in p‘s spot prior to being defined may indicate rather that definition P is meaningless, since the supposed digit that the definition is supposed to reference does not exist prior to the definition.
Let’s look into this option that reads the definition P as presupposing that there is a real number already in every spot of list N, from which P can then determine all the digits of p based on the real numbers that are in those spots. Since there is no number in the nth digit place of rn where p goes (when assuming, contra previous proofs, it is defined unambiguously), then definition P is rather meaningless, because the number it presupposes in order to determine p is just not there.
For ease, see again that the definition P defines number p as follows: the integer part is 1, and its decimals are as follows: if the nth decimal place of rn is 1, then the nth decimal place of p is 0. On the other hand, if the nth decimal place of rn is not 1 (e.g., is 3), then the nth decimal place of p is 1.
The two “if” clauses of this definition may here be read as presupposing the existence of a decimal for any rn. Of course, this must be a perfectly sound assumption since every member of N (i.e. any rn) must be a number. However, in assuming p itself is such a number, then, where rn is p, there is no decimal place there because definition P presupposes a decimal place from which to then define it, and without defining it there is no decimal place there. (But again, as we have already proven far above, p cannot be in N. Thus, no rn is p. Unrelatedly but perhaps important to note, if there now is some other real number in the place where p might have gone if P were meaningful, then this is simply because it is defined by some other, meaningful definition. )
For the former reading of P that I introduced where we just put a 1 in the digits place that we considered previously to be contradictory, no contradiction would seem to arise from this, because the value at this digit is just 1 and not also some other value, because no other number is in this position in N, and thus nothing here has any digit from which to be included in addition to 1. And so, thanks to how we decided to order list N, there is no contradiction involving two numbers being in one and the same decimal place to be derived from definition P!
This is a nice result. That said, this is not to say that there is not a blatant kind of contradiction here of another sort (maybe you are smarter than I and see it at a glance, rather than after weeks of study). We will get to this. Because of this choice between reading the definition as presupposing that there is a decimal place for any rn vs. not presupposing any such number and rather handling a lack of a number in that decimal place by determining that a 1 is there (since, having no number there indicates that there is not a 1 there), then we should maybe conclude that definition P is ambiguous after all. Making a non-ambiguous and meaningful version of P seems rather straightforward. Let’s attempt to define a real number p unambiguously as:
The integer part is 1, and its decimals are as follows: if the nth decimal place of rn is 1, then the nth decimal place of p is 0. On the other hand, if the nth decimal place of rn is not 1, or there is nothing there at all, not even a decimal place, then the nth decimal place of p is 1.
This is exactly the paradoxical definition, except for the addition in bold. However, alas, explicitly attempting to make the definition unambiguous makes it clear that there is still a looming contradiction here: specifically, p would both have “nothing there” in the nth decimal place of p where rn = p as well as have a number 1 in that decimal place. To explain, see that, by the definition, having nothing in that decimal place is sufficient for having a 1 there. The definition is using the material conditional (“if . . . then”), and is not merely implying that some creative force makes a new value after there was nothing (numbers themselves don’t change, after all). That is, if the “if” condition is true and there is nothing there, then the consequent is simultaneously true with that “if” condition, and 1 is also there. Thus, it has both something and nothing in that spot. By this updated definition, p has a Contradictory Digit with both “nothing there” and the number 1.
And so, this updated P remains contradictory, and it defines no number.