A google search seems to prefer the name "Richard's paradox". --Brion 23:49 Oct 1, 2002 (UTC)

I don't believe that anyone not previously familiar with this paradox would be able to follow this article. I sure can't. For example, this

One example sentence would be "That positive real number whose square is two."

The sentence quoted is not a sentence. Ortolan88 Probably the one who wrote above isn't a native English speaker (neither am I, anyway) and got confused between sentence and phrase.

I think the article use the word "sentence" in the technical sense...e.g. the phrase "red car" is a sentence because its "logical form" is "there exists and x such that x is red and x is a car" Philosophy.dude (talk) 05:57, 21 July 2008 (UTC)

The paradox is implying that the set of these phrases (let's call it P) is countable. If it is, then the "paradox" is quite obvious, as R is an uncountable set. If P isn't countable, then it makes no sense to say "the nth phrase" because doing so requires a mapping between N and P. And it is by no means explained why couldn't P be uncountable.

I`m absolutely not mathematical, and maybe I`m just being dumb, but I don`t understand this:-

"Thus, if a number is Richardian, then the definition corresponding to that number is a property that the number itself has. (More formally, "x is Richardian" is equivalent to "x does not have the property designated by the defining expression with which x is correlated in the serially ordered set of definitions".)"

You're right. Thanks for the catch. Typos usually aren't that bad, but they tend to be really nasty when you forget to use the word "not". Heh. Eric Herboso 01:05, 23 Jan 2005 (UTC)
Also, the reason why the set P is definitely countable is because every element in P is a string of characters, and we order the set by putting each element with a lower amount of characters before any element with a higher amount; and for any two elements with the same number of characters, we order according to standard alphabetical order (including extra characters in the order as well). If this isn't sufficiently clear in the article, feel free to more specifically address this issue there. Hope this helps. Eric Herboso 01:13, 23 Jan 2005 (UTC)
That's true if we explicitly require to use finitely long phrases. If we use infinitely long phrases, we can express any real number, for example the one for pi begins "the real number whose floor is three, whose first decimal digit is one, whose second decimal digit is four..." And in the article, nowhere is it stated that definition must be finite. --Army1987 22:06, 9 August 2005 (UTC)

A substantial portion of this article appears to be taken from the book Gödel's Proof by Nagel & Newman (link goes to Amazon). While there aren't any direct quotes, phrases such as "an essential but tacit assumption" appear in both, and are unlikely to have been developed independently. I don't think it quite warrants blanking the article, but a serious rewrite and/or citations are needed. I don't have time to work on it, though. Benandorsqueaks 23:55, 27 March 2006 (UTC)

I was the one who wrote that substantial portion of the article. I did use Nagel & Newman's book as a reference when rewriting this article, so this might explain my particular choice of phrases being similar to Nagel and Newman. Indeed, the flow of the argument is also taken from Nagel and Newman, since I felt their order of presentation was very good for their discussion of Richard's paradox. But I do not feel that there has been a copyright violation here, as the source material was merely used as a reference when rewriting the article due to its ideas on how to present Richard's paradox most efficiently and readably. If such a use of reference were considered a copy vio, then when writing an article on calculus, you wouldn't be able to reference a calculus book's presentation flow, or when writing an article on history, you wouldn't be able to make the same points that some recent book you'd read had talked about. Because I merely took the ideas from Nagel and Newman and used them to write the article, rather than quote from the text, I feel that this does not constitute a copyright violation. Of course, I am no lawyer, so if someone who knows more on this disagrees with me, then they should say so on my talk page, and I will rewrite the appropriate sections. — Eric Herboso 20:42, 3 April 2006 (UTC)

## Disputed

Hang on... in the article it says "Explaining away Richard's Paradox is as easy as being careful to distinguish between statements within arithmetic (which make no reference to any system of notation) and statements about some system of notation in which arithmetic is codified." Half the trick of Godels Incompleteness Theorem is the invoking of Godel numbering of notation symbols so that mathematical notation becomes equivalent to arithmetic. I get the feeling that Richard's Paradox is not supported by expert logic mathematicians. User:rem120 5th April, 2005

Gödel's proof is a formalization of Richard's paradox, in which metamathematics (about statements in the formal system) is represented by arithmetic (on the Gödel numbers of those statements). Gödel proves that this is valid, and that the arithmetical statements are provable within the formal system. The key is that we're representing metamathematics, not discussing it directly. Benandorsqueaks 20:26, 2 April 2006 (UTC)

So are you saying that the article is incorrect in stating that Richard's paradox is fallacious? Remy B 05:46, 3 April 2006 (UTC)
Richard's paradox is fallacious, but this is because of how it is set up. The initial conditions specified that we number statements concerning (A): the arithmetical properties of integers. Then we set up the property of Richardian concerning (B): statements concerning the arithmetical properties of integers. The important point is that these two things are different; the list we set up in the beginning is a list of statements concerning (A), while the property of being Richardian does not concern (A); it just concerns (B). Therefore, the property of being Richardian was not enumerated in that list.

[So what happens if you change the initial specification so it refers to (B) instead of (A)? E.g. replace Consider a language (such as English) in which the arithmetical properties of integers are defined to Consider a language (such as English) in which the arithmetical properties of integers and the properties of statements concerning the arithmetical properties of integers are defined. (Certainly it would not be the first time that English had been applied to define some properties of English.) Then does the paradox come back to life? If not, why not? It appears to me that the last section about explaining away the paradox really only ring-fences the paradox off from purely arithmetic properties. Is this correct? The paradox is still alive and kicking in a different place.]

The reason why it is fallacious is because we set up our initial list such that the property of being Richardian is not in the list. But a very similar argument was taken by Gödel where he took a system that could codify self-referential statements. The Richardian paradox is fallacious, but Gödel's proof is not. — Eric Herboso 20:57, 3 April 2006 (UTC)
In the article it says that you could consider "the first natural number" and "not divisible by any integer other than 1 and itself" as arithmetical properties. Why cant you consider "is Richardian" as one too? What is it about being Richardian that isnt an arithmetical property? Remy B 09:27, 4 April 2006 (UTC)
The statements of arithmetical properties apply to numbers and you can see them as functions, like e.g. f(x) where f is the property and x is some number or set of numbers. Now the property of "being Richardian" is not a property of a number or set of numbers but of a pair (x,y) where x is a statement about numbers and y is the number assigned to it. Hence, to be RIchardian is a property of the type f(x,y). It is hence not on the same level as the other definitions and statemnts, because it predicates someting about them. When trying to apply it to itself, you get a paradox not unlike the Liar's paradox. Therefore we have to distinguish mathematical statements from statements about mathematical statements, just as we have to distinguish sentences and assertions about the truth-value of sentences. See also Alfred Tarski and the distinction between language and metalanguage. Cat 12:09, 27 April 2006 (UTC)
Forgive my persistence, but I'm still not convinced from that line of reasoning. Why cant f(x) be a function that returns 0 or 1 (or whatever results are desired) whether the number x is Richardian or not? That is formulated in the same way as, say, how an f(x) might be a function that returns 0 or 1 (or whatever) whether the number x is "the first natural number", or perhaps "not divisible by any integer other than 1 and itself". I dont see why it has to be a double-parameter function when you are dealing only with the Richardian property. (Just to clarify, I do actually believe that the logic is false, purely because prominent mathematicians state that it is false, and I believe their abilities far above mine. I just dont see the truth in the reasoning given in the article, and on this talk page, and that makes me think there is something I am missing). Remy B 12:49, 27 April 2006 (UTC)
Why cant f(x) be a function that returns 0 or 1 (or whatever results are desired) whether the number x is Richardian or not? Carefully consider the input neede for such a function. It would need 1) the statement or definition about numbers as input and 2) the number assigned to that very statement in order to decide whether it is Richardian or not. Consider the other statements and definitions: "1 is the first natural number", "a number divisible by 2 is even", independently from the number assigned to them they are applicable to numbers. The property of being RIchardian depends on the the pair of number and statement. Look at the definition itself: "x is Richardian" is equivalent to "x does not have the property designated by the defining expression with which x is correlated in the serially ordered set of definitions" Contrast this with "x is even" is equivalent to "x is divisible by two". In the latter statement no reference is being made to the "serially ordered set of definitions" or the numbers correlated to them. That is why we would get a paradox if we would add "Richardian" to the set of definitions. The fact that we would get a paradox indicates that "being Richardian" is a different kind of property that being even or odd. Cat 13:16, 27 April 2006 (UTC)
I'm not sure why it is a paradox to include a statement which contains "serially ordered set of definitons" to the set of definitions. If it is well-defined, it has a definition, which means it has a unique number which can be included in the order (eg. taking the Godel number of its algorithm). Also, why do you have to input the statement "Is Richardian" (or its equivalent) to the function? You do not need to pass the statement "is the first natural number" to a function f(x) which determines whether or not x is the first natural number. Remy B 13:50, 27 April 2006 (UTC)
It generates a paradox because it predicates something about itself and then you get an infinite loop at each stage of which the truth value flips. If you consider "being Richardian" a statement of the form f(x), then when evaluating the property of "being Richardian" of itself as part of a list of statements, you effectively plug it into itself thus f(f(x)). Thereby, if it applies, it renders itself untrue, but at the next iteration f(f(f(x))) it would be true again, in virtue of the fact of being untrue of itself at the earlier stage ... and so on ... this is a paradox. We can "solve" the paradox by looking at the nature of the statement being made. A statement like "S is P" (Subject is Predicate, like "snow is white") assigns a property to an object, e.g. whiteness to snow, which is similar to f(x) -> White(Snow) and FirstNN(1). A statement like the property of being Richardian is different. Why? This property is formulated in a conditional way: IF the statement does not apply to its own number THEN the number is Richardian. This is not anymore an "S is P" case, but rather an "IF S is not P THEN S is R". Now the paradox occurs when we take "being Richardian" itself as a possible candidate for "being Richardian". So in the statement "IF S is not P THEN S is R", where R=being Richardian, we plug in "being Richardian" also as P, hence getting the form "IF S is not R THEN S is R" which is a paradox. Seeing as we have obtained an untruth, we have to revise our premises: Clearly there is something fishy going on with "being Richardian". The property R should not be applicable to itself, like a function for one kind of object is not applicable to objects of other domains. So if R can be applied to a structure like "S is P" by saying "If S is not P THEN S is R", we must reformulate this into some less vulnerable form. For instance we can say that "R" is not a property of S, but of statements like "S is P", and it is true exactly when "S is P" is not true (when the statement does not apply to its own assigned number). Then R is not a property of an object, but of a statement, and hence it is a higher order function: R(f(x)), which cannot be applied to itself, but depends on the outcome of the function f(x). Then you also see again that we would get an infinite loop alternating between true and false if we would apply it to itself, as the truth value would come to depend solely from R applied to itself and not from R applied to f(x). R applied to itself does not have a definite value, but f(x) does in every case, and hence R(f(x)) does too. The error lies in considering R a statement at the same level of f(x), while it is not. Cat 08:20, 28 April 2006 (UTC)

I don't know, I think Remy is right. We can number the one-place predicates of Peano arithmetic, using arithmetic. The idea that the system Gödel took is somehow different from this system doesn't seem right. All he needed was addition, multiplication, and first-order logic. The article mentions "not divisible except by 1 and itself", which seems to imply that the system does have all this. Therefore no matter what convention we use, "is Richardian" is, I'm thinking, also an arithmetical property. -Dan 14:22, 24 May 2006 (UTC)

I suppose the point is that this is being done in natural language, not in a formal system, which is not arithmetical. My mistake. Followup question: Any recursive predicate can be expressed in arithmetic. Why doesn't this prove people are not algorithms? -Dan 15:07, 24 May 2006 (UTC)
With humans, at some point, the recursiveness stops. We may be able to describe recursiveness with a single statement, but certainly you'll admit this is quite different from actually going through the recursion. Humans can comprehend four levels of recursiveness, or 200 or maybe even 5000 levels. But saying it in shorthand is not enough. You have to be able to work it through. I'll admit it certainly seems that humans can handle any level of recursiveness, if they are given enough time to work through it. But even then, there's only so much time available to spend on it, even if you only count the age of the universe. And it is only assumed that humans can handle 5000 levels of recursiveness; I doubt no one has in actuality truly tried to comprehend that deep a level, though I suppose it's possible.
In short, the ability to conceive of certain levels of recursiveness proves people are (at the least) algorithms of a certain level of sophistication. But using shorthand to refer to recursiveness proves nothing; even algorithms of a basic type can utilize shorthand in place of actual recursion in order to talk about recursion. Using short hand proves nothing about the sophistication of the algorithm. — Eric Herboso 16:59, 27 April 2007 (UTC)

## last Resolving the Paradox paragraph is very weird

The last paragraph in the Resolving the Paradox section seems very strange; it seems to be trying to make some point (by putting words like resolving and paradox in quotes, over and over again). What is the point of this paragraph? Would some references be appropriate here? Right now it seems sort of vague and philosophical (perhaps original research?). Cispa 02:22, 18 March 2007 (UTC)

Okay, after looking around for more information on this for a while, I'm going to remove this, as it seems inappropriate for the article. If someone wants to put it back, they should of course fully reference it, but also, give multiple reliable references to establish the notability of this discussion in relation to Richard's paradox.Cispa 16:09, 18 March 2007 (UTC)

For reference, I am copying the deleted text to here. When I get more time, I'll manually go in and see if I can clean it up and reference it so it can be put back in. — Eric Herboso 19:11, 18 March 2007 (UTC)
Richard himself explained the 'paradox' to Poincare, and the discussion is in A. Garciadiego, Bertrand Russell and the origina of the set-theoretic 'paradozes'. He simply said that the domain E was infinite, and, therefore, undefined. Thus, it is not necessary to look for false steps within the 'paradox' in order to demonstrate that it has no logical content.
Criticisms of Richard's paradox place other paradoxes into doubt, as Garciadiego demonstrates. This has further implications for the three schools of mathematics generated in response to the set-theoretic 'paradoxes', all designed to 'resolve' or 'avoid' them. Godel's theorems are also designed to 'avoid' such 'paradoxes'. The three responses are known as "natural" mathematics and had a huge influence on scientists who did not know the avant garde of mathematics in their day. They were influenced by Poincare's Science and Hypothesis, which also had as its mission of 'avoiding" the 'paradoxes'. Kimura in biology, Sraffa in economics, and Einstein in physics (who was an enthusiastic fan of 'Science and Hypothesis'), were all influenced by "natural" mathematics.
I don't like this idea. Certainly, we should make the article readable to a general audience. But to not describe the paradox is, I think, out of the question. We should preserve content on the wikipedia whenever possible. Perhaps we cancreate tw major sections, one that glosses over details, but is highly readable, and a second that goes into detail on the paradox? — Eric Herboso 17:02, 27 April 2007 (UTC)

For the original paradox by Jules Richard compare:

• J. Itard: Richard, Jules Antoine, Dictionary of Scientific Biography, 11, Charles Scribner's Sons, New York (1980) 413-414.
• Les principes des mathématiques et le problème des ensembles, Revue générale des sciences pures et appliquées 16 (1905) 541-543.
• The principles of mathematics and the problem of sets (1905), englische Übersetzung in Jean van Heijenoort, "From Frege to Gödel - A Source Book in Mathematical Logic", 1879-1931. Harvard Univ. Press, 1967, p. 142-144.
• Lettre à Monsieur le rédacteur de la Revue Générale des Sciences, Acta Math. 30 (1906) 295-296.
• Jules Richard

217.94.215.52 07:26, 1 April 2007 (UTC)Regards, WM

I'm afraid I'm not following you. Are you saying that the explanation of his paradox is not the same in this article as the way he originally explained it? If so, I agree with you. But the explanation given here is far easier to understand, and is functionally equivalent to his original. Furthermore, it is the version that is most popular in texts that are attempting to delve into the workings of the paradox (see Nagel & Newman's Godel's Proof). A quicker version, of course, is the enigmatic: the smallest natural number without any interesting properties, which, after a moment's thought, is actually equivalent to Richard's paradox. — Eric Herboso 17:10, 27 April 2007 (UTC)

The paradox, as translated by van Heijenoort, goes as follows (quotations are Richard's words from van H p. 143-144):

• Define a certain set of numbers E as follows:
• (1) Alphabetize alll permutations of the twenty-six letters of the "French alphabet" starting with letters "two at a time", then three at a time, and so forth. The permutations can contain repetitions of the same letters.
[I would have encouraged Richard to include the "space" " ", "=", "(", ")", and the period "."]
[Thus E is {aa, ab, ..., az, ba, bb, ..., zz, aaa, aba, aca, ...}. Had he included "(", ")", "=", "1", used Peano's primitive "s" or successor, and defined his numbers only in conventional unary, this enumeration does not take long to produce E = { ..., zero=( ), s( )=1, ... ,s(1)=11,... s(11)=111, etc. }
• (2) Eventually, for any integer p, any permutation of the twenty-six letters taken p at a time will be in the table. Thus, eventually "everything that can be written" in the French alphabet of 26 letters will be written in this table.
• (3) Some of these strings of symbols will be definitions of numbers; most will not be:
[ e.g. E = { ... THE NUMBER NAMED FOUR IS THE SUCCESSOR OF THE NUMBER NAMED FIVE, ... , THE NUMBER NAMED THREE IS THE SUCCESSOR OF THE NUMBER NAMED TWO, ..., THE NUMBER NAMED TWO IS THE SUCCESSOR OF THE NUMBER NAMED THREE... } ]
• (4) Cross out all those strings of symbols that are not definitions of numbers.
[ e.g. E = { ... THE NUMBER NAMED FOUR IS THE SUCCESSOR OF THE NUMBER NAMED FIVE, ... , THE NUMBER NAMED THREE IS THE SUCCESSOR OF THE NUMBER NAMED TWO, ..., THE NUMBER NAMED TWO IS THE SUCCESSOR OF THE NUMBER NAMED THREE... } ]
[Nowadays a construction such as this implies to us the existence of (i) a pre-defined "dictionary", an (ii) "active agent" who can look up "meanings" in said "dictionary", (iii) some kind of pre-defined syntax/parsing mechanism/definition to allow the "agent" to determine if a sentence is well formed. For example: what is a "SUCCESSOR", what is the meaning of "THE NUMBER NAMED"? Is the combination of symbols THREE (TROIS in French) a valid name for "a number" (and what just what number does TROIS identify? and just what is a "number", anyway?) Apparently the agent here is a (trained, French- or English-speaking) human called a mathematician or logician or philospher.]
• (5) Now identify (that is: label with a number in sequence, map onto, "enumerate", "count") each remaining valid sentence/definition, in the order presented. "Thus we have, written in a definite order, all with numbers that are defined by finitely many words. ¶ Therefore, these numbers that can be defined by finitely many words form a denumerably infinite set. ¶ Now, here comes the contradiction. We can form a number not belonging to this set."
[Does this beg the question? It is very close to begging. What does "enumerate" mean? How does the active agent know how to do this?]
• (6) Here comes the trick: we have in our listing, as well as the simple example sentences above, the following sentence that invokes Cantor's diagonal argument. It creates, in the typical diagonalization manner, a number N that will not be in the listing-set E:
"Let p be the digit in the nth decimal place of the nth number of the set E; let us form a number having 0 for its integral [integer] part and, in its nth decimal place, p+1 if p is not 8 or 9, and 1 otherwise."
[Woops ... Richard's alphabet does not include a "+" symbol, nor a "0", "1", "8", or "9". But this is what is written by Richard and presented by van Heijenoort. So we have to assume that the symbol "+" is written out "PLUS", "0" is written out "ZERO" etc.]
• (7) As is always the case with a diagonalization (finite or infinite) the number N defined above does not belong to the set E. "If it were the nth number of the set E, the digit in its nth decimal place would be same as the one in the nth decimal place of that number, which is not the case."
[Example of a diagonalization of just two symbols 1 and 0 (ignore the "."). Note that a diagonalization always involves an "enumeration" (putting into some order and then labelling the rows with "numerals" and the columns with the same sequence of "numerals"):
c: 1234
1 .1000
2 .0100
3 .1100
4 .0010
d .1100
~ .0011
In the above, row "d" denotes the diagonal number a11 a22 a33 a44 as shown in bold-face, and "~" is the "anti-diagonal" number defined by 1-a11 etc. Observe that the anti-number ".0011" does not appear in the collection {.1000, .0100, .1100, .0010} ]
• (8) Denote by G the collection of letters between quotation marks (the one that appears between (6) and (7)). The [diagonal] number N is defined by the words of collection G, and this is finitely many words. Hence it should belong to "a certain set of numbers E", but it does not.
[This seems to work for finite sets. But is this diagonal-number N, as defined by the letters of G, truly finite? Richard says the numbers in set E "form a finitely denumberable set". How can a process that writes symbols ad infinitum write only "finite" descriptions? We can see that if we were to restrict the use of only the "successor" and "unary" symbolism then s(1111...1)=111...11 goes on ad infinitum, contrary to Richard's assertion. If we specify binary symbolism for the numbers then the power set is appearing, and [confusion is rampant]. More importantly, in a "system" with a highly restricted syntax the long sentence that defines N could not exist. So is N ultimately infinite? Does this infinite number N really belong in the same set as all the finite numbers in E?]
[As the tiny finite example shows, for any number p there will indeed be an anti-diagonal number N made up of p symbols, but not in the set E. As p continues to grow the number is still finite. The problem seems to occur when we end up at the "completed infinite".]
• (9) "Such is the contradiction."

In the last two paragraphs of his letter Richard offered a way out of his paradox, but van Heijenoort reports that Peano rejected his explanation. van H reports that "the paradox is generally considered solved by the distinction between language levels." Do all commentators agree? I will research. The above commentary needs work.

The struck-out paragraph above seems to be invoking the Finsler (1926) argument, "What we now have is a sequence, namely the antidiagonal sequence, that is definable but not formally definable" (van H p. 438). Finsler constructs an interesting argument around this -- he demonstrates that he can state a false sentence -- known to be false "outside the system" -- but undecidable within the system.

Both Finsler (1926) and Godel (1931) invoke the Richard paradox. Godel claims not to have known of Finsler's work. And Finsler was recognized by Godel as follows:

"... As for work done earlier about the question of (formal) decidability of mathematical propositions, I know only a paper by Finsler published a few years before mine ... He also applies a diagonal procedure in order to construct undecidable propositions. However, he (Finsler) omits exactly the main point which makes a proof possible, namely restriction to a definite (some well^defined) formal system in which the proposition is undecidable. For, | he had the nonsensical aim of proving formal undecidability in an absolute sense. This leads to the nonsensical definition given in the first two parags. . . . and this leads to the flagrant inconsistency that he decides the <"formally> undecidable<"> proposition by an argument <<sect. 11, p. 681>> which, according to his own <definition in> the two passages just cited, is a formal proof while on the other hand he asserts on p. 681 it/s formal undecidability. If Finsler had confined himself to some well^defined formal system S, his proof (by changing replacing the nonsensical section 11 with a proof that the <proposition in question> is expressible in S) could be made correct and applicable to any formal system. I myself did not know his paper when I wrote mine, and other mathematicians or logicians probably disregarded it because it contains the obvious nonsense just mentioned." (strike-outs in the original letter to Yossuf Balas (1970); Godel never sent the letter. Footnotes and inserts etc in this draft letter are fascinating. From Kurt Gödel: Collected Works, Volume IV, 1990, ISBN 0198500734.) More to come. wvbaileyWvbailey 02:52, 21 August 2007 (UTC)

## Arithmetical properties ?

It's not clear in the description of the paradox what is an "arithmetical property". I also don't understand why do we speak about "arithmetical properties" and not just "properties" (which would make the paradox much harder to be solved).--Pokipsy76 20:21, 25 September 2007 (UTC)

## Changes

Basically I interchanged the info in the articles about Richard and the paradox. The description of the paradox in the article about Richard was much the better than the description previously given here. I didn´t modify the text at all, just swapt the articles.Alterationx10 (talk) 19:41, 28 June 2008 (UTC)

## "Fits" in Variation

I'm not sure why this section introduces the concept of a number "fitting" its definition. It's not used.--Antendren (talk) 04:37, 27 August 2011 (UTC)

## The paradox as described by M.J. Richard himself in 1905

It helps to read the original. If you do, you'll realize that the article and this talk page reflect a number of misunderstandings.