The Richard Paradox

Philosophy Forums Science, Technology, and Math
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The following is a construction of the Richard Paradox found in the book "Godel"s Proof by Ernest Nagel and James R. Newman with editing and forward by Douglas Hofstadter.

I am posting this paradox for two reasons.

  1. This paradox is the lynch pin for Godel’s Incompleteness Theorem, which dealt huge blows to Russell’s and Hilbert’s drive for Formalism in the construction of an axiomatic theory of arithmetic.

  2. The concept of mapping true sets of sequences of sentences to the counting numbers was a new concept to me. And this is one of the great joys in my life. I hope it will be that way for you.

I should also say that there is a logical flaw in this paradox. Maybe you will find some joy in trying to find it.

Consider a language e.g. English in which the purely arithmetical properties of the cardinal numbers can be formulated and defined. Let us examine the definitions that can be stated in the language. It is clear that, on pain of circularity or infinite regress, some terms referring to arithmetical properties can not be defined explicitly - for we can not define everything and must start somewhere- though they can be understood in some other way. For our purposes it does not matter which are the undefined or “primitive” terms; we may assume, for example, that we understand what is meant by “an integer is divisible by another”, “an integer is the product of two integers” and so on. The property of being a prime number may then be defined by “not divisible by any number other than 1 and itself”; the property of being a perfect square maybe defined by “being the product of some square and itself”; and so on.

We can readily see that each such definition will contain only a finite number of words, and therefore a finite number of letters of the alphabet. This being the case, the definitions can be placed in a serial order: a definition will proceed another if the number of letters in the first is smaller than the number of letters in the second; and, if two definitions have the same number of letters, one of them will precede the other on the basis of alphabetical order of the letters in each. On the basis of this order, a unique integer will correspond to each definition and will represent the number of the place that the definition occupies in the series. For example, the definition with the smallest number of letters will correspond to the number 1, the next definition in the series will correspond to the number 2, and so on.

Since each definition is associated with a unique integer, it may turn out in certain cases that an integer will possess the very property designated by the definition with which the integer is correlated. Suppose, for instance, the defining expression “not divisible by any integer other than 1 and itself” happens to be correlated with the number 17; obviously 17 itself has the property designated by that expression. On the other hand, suppose the defining expression “being the product of some integer by itself” were correlated with the order number 15; 15 clearly does not have the property designated by the expression. We shall describe the state of affairs in the second example by saying that the number 15 has the property of being Richardian ; and in the first example, by saying that the number 17 does not have the property of being Richardian. More generally, we define “x is Richardian” as a shorthand way of saying “x does not have the property designated by the defining expression with which x is correlated in the serially ordered set of definitions”.

We come now to a curious but characteristic turn in the statement of the Richard Paradox. The defining expression for the property of being Richardian ostensibly describes a numerical property of integers. The expression itself therefore belongs to the series of definitions proposed above. It follows that the expression is correlated with a position-fixing integer or number. Suppose this number is n. Now we pose the question, reminiscent of Russell’s antimony: Is n Richardian? The reader can doubtless anticipate the fatal contradiction that now threatens. For n is Richardian if, and only if, n does not have the property designated by the defining expression with which n is correlated (i.e., it does not have the property of being Richardian). In short n is Richardian if, and only if n is not Richardian; so that the statement n is Richardian is both true and false.

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A definition can be stated in various different but equivalent ways. A different number of words and letters will give it a different place in the series. Hence it is possible for the number associated with a specific definition to be Richardian with one representation of that definition but not with another (but equivalent) representation. For instance, a prime can be defined as “a number that can be divided only by 1 and itself” or by “a number which, when divided by another number, will only give an integer as the outcome when the numerator and the denominator are equal OR when the denominator is 1”. The second definition is longer and will be placed lower in the list than the first one, even though both are equivalent. If the short definition is correlated to a prime, then the second need not be. Hence the same number can be both Richardian and non-Richardian, as the outcome can be “manipulated” by carefully tuning the length of the corresponding definition.

Quantum mechanics and relativity are peppered with paradoxes (real or perceived) like the Richard paradox. I never gave them much attention, partly because it is beyond my intellect to resolve puzzles like these and partly because they take up too much time.

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what a good workout. we need more threads like this.

is it that in order to create a sequence of numbers to designate your laws, you need to make laws that say what order the numbers go in?

1: one comes after zero
2: two comes after two

in this case, there are no examples of a rule that does not describe its arbitrary number… wait no i dont think thats it

i got it! being a perfect square is a property of integers. this property goes into the list of rules. being richardian is a property of a rule, not a property of integers. so therefore the rule “a richardian integer is blah blah” would not be a part of the list to begin with because it does not belong in a list of integer qualities or rules, it belongs in a list of rule qualities.

which does not belong?

1: a number minus itself equals zero
2: a number times itself equals a square
3: a number times another number equals a product
4: a rule on this list that has a certain property is called richardian
5: a number thats less than zero is negative

actually, if the number designating its place in the sequence wasnt a number, but a rule, then this could work. if you used the rules like numbers in some arbitrary order to designate the place in the sequence that the richardian rules lies, and the rule designating the richard rule happens to be richardian in your other list containing all the rules corresponding with numbers then i think you still have the paradox.

i sprained my brain.

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future man you have not got it. being stated in english it belongs in the list, because the list orders english statements (which its why its called formal, it doesnt care about meaning just form)

ed there is no logical flaw, the fact that a defined system of symbols will reference more problems than solutions is in fact true, as it is true that any formal language will be able to express at least one unverifiable property of an object [in that language].

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You say a definition with 17 letters that applies to the number 17 implies that 17 is not richardian, right? so a definition with 17 letters that did NOT apply to 17 would mean 17 was richardian? since i’m sure both exist (with some creative wording), then you have two cases, one where 17 is richardian, one where it isn’t. Obviously, the richardianism is a property of the definition (specifically a property of the number of letters in the definition), not of the number 17 itself.

SO…Future man is right…does this help the paradox?

As stated: is the definition of richardianism with n letters imply n is richardian or not?

since the property of being richardian is not a fundamental property of the number n (or of any number), then the definition of richardianism is richardian.

Trust me, it’s ironclad…

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you mean he’s listing all random combinations of letters that there are?

and if the property of being richardian did not ostensibly describe integers, it would therefore not belong to the series of definitions proposed above? i think it describes the property of an item in that list. not integers, list items.

the number 5 does not have the property of being richardian without the existence of the list. ‘richardian’ requires both in order to make sense.

there is no property being designated at number n. define “property”. when it was just a list of things like 'not divisible by any integer other than 1 and itself’ then a ‘property’ is a characteristic of numbers.

‘x does not have the [characteristic that some numbers happen to have] designated by the defining expression with which x is correlated in the serially ordered set of definitions’

you could say that a number doesnt have the property it designates, but you cant say that a number doesnt have the property of not having the property it lists if its not listing any other properties besides that one.

well i guess you could say that. but it would not mean the same thing as ‘the number five doesnt have the property of being a square’ because in order to figure that out, you do math. in order to figure out that the number 5 doesnt have the property of being richardian, you dont do math, you just look and see that there are no number characteristics being listed at position 5.

4: square - do math, 4 does have the property that is listed; not richardian
6: prime - do math, 6 does not have the property that is listed; richardian
5: richardian - look and see that there is no number property being listed and therefore “5 does not have its listed characteristic” but that is so for a different reason than the previous two. it doesnt have the numerical property listed because there is no numerical property listed.

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hah, beat you

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Tmminionman2 and Future Man are correct.

In Future Man’s example we have:

1: a number minus itself equals zero
2: a number times itself equals a square
3: a number times another number equals a product
4: a rule on this list that has a certain property is called richardian
5: a number that’s less than zero is negative

Statement 4 does not belong in the list. (However, the list is out of order, assuming that it was intended to be ordered per the instructions specified.)

On a personal note, when I was reading this article, I made the same observation as To Wander, and still believe that under the prevailing rules the text is flawed in this regard. I simply assumed that there was an unstated contracting rule that did away with this problem. In addition, I had forgotten about the paradox problems in Quantum Mechanics & Relativity and it was nice to have my memory jogged. Anyway, I identify strongly with his post.

zeno leapt ahead to the generalized conclusion. His conclusion is true in all systems as complex as arithmetic, with adding and multiplying; and assuming finitisic rules of transformation e.g. rules like adding a number to each side of the equation still results in an equation and various rules of logic.

There are interesting proofs of consistency, lead principally by Gerhard Gentzen, but this proof requires a special rule of inference called ‘the principal transfinite induction.’ In addition this rule lies outside the formal theory.

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The interesting thing about some of these paradoxes is, that they are thought experiments that cannot be performed other than in your brain. Yet these thought experiments yielded answers that were “true” or “false” in the Platonic sense and the most succesful scientific theories ever constructed by Man rely on them.

The Twin paradox is a very famous thought experiment from special relativity. It says: of two twin brothers, one undertakes a long space journey while the other remains on Earth. When the traveller finally returns to Earth, it is observed that he is younger than the twin who stayed put. The paradox arises if one takes the position of the travelling twin: from his perspective, his brother on Earth is moving away quickly, and eventually comes close again. So the traveller can regard his brother on Earth to be a “moving clock” which should experience time dilation. Special relativity says that all observers are equivalent, and no particular frame of reference is privileged. Hence, the travelling twin, upon return to Earth, would expect to find his brother to be younger than himself, contrary to that brother’s expectations. Which twin is correct?

(If you want a detailed analysis of the Twin paradox, then click here.)

The EPR paradox is much more complex.

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ed3, you left out a condition requiring that the list only contain arithmetic properties. what you stated was " Consider a language e.g. English in which the purely arithmetical properties of the cardinal numbers can be formulated and defined". this does not logically imply non arithmetical properties are not acceptable, it just implies all arithmetical properties are.

if we take that at face value, the property of being richardian, although not arithmetic, can be regarded as acceptable in the list. since the property of being richardian relies in its definition only on already accepted items, such as logic, arithmetic properties and their derivatives a strong case exists for the property being implicitly acceptable.

if you want to limit it, you should explicitly say so.

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Hi to zeno & To Wander:

To Wander: I very much appreciated the link on the twin paradox, its bothered me for almost 30 years. Who could believe 2 new ways to see things in just 2 weeks (I thought that at a constant velocity in an inertial system time would always contract).

Zeno: I assumed, when reading the text, that the descriptive ‘purely’ would sufficiently restrict the domain.

By the way, I would have enjoyed playing bridge, but I do not have any time left in my day to do so. I am curious if you are a New Yorker, from some of your posts I thought this might be a possibility. If so I thought that there might be a possibility that you play go. Just curious.

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i played some go years ago, so you will probably crush me. which, ofcourse, makes it worth my time. wanna try ?

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zeno:

Yes, I would like to :smiley: . It has been years for me too, except for a few rusty games. Also, I do not know the logistics as far as how to setup an electronic game.

If I can prevail upon you to walk me through this I would appreciate it.