1. Jul 23, 2007

### deimors

What is 'the least number which cannot be described in less than nineteen syllables'? Is this not a description of it, only 18 syllables long?

2. Jul 24, 2007

### Niv

I think it is the same number that cannot be described in 20 syllables :p

3. Jul 24, 2007

### deimors

Not entirely, as 'the same number that cannot be described in 20 syllables' is 16 syllables long, and so this isn't a contradictory description of the number being described. Although, this could turn out true if one held the premises that a) all descriptions which describe nothing (physical / metaphysical (/ otherwise?)) describe the same thing (are descriptions of the same nothing), and b) every number can be described in any countable number of syllables (or, in the less general, any number can be described in 20 syllables).

4. Jul 25, 2007

### Niv

I think that any number that can be described as 'the least number which cannot be described in less than N syllables' can also be described in 18/20/100 syllables.
Just in the way u did.

If we have a very long number that adds syllables, like N=10^(10^10), we can describe it in less syllables. If the description is still N>18 we can give the last one a shorter one also, till we r short enough for N=18 syllables.

* N is any finite number of syllables.

If it is b), so we have an infinity of numbers. If we have infinity so we don't have a least number I think.
So I think it is a)

5. Jul 27, 2007

### deimors

The problem I see with with b) is that it seems at least plausible to consider that there's a linear relation between a number n and the number of syllables in the smallest possible description of n (say, S(n)). Further, one must consider what can be used as a description of a number. Obviously using proper names won't work, as I could say that a description of 10^10^10 is 'Joe', yet this is a description that doesn't really convey information about the number and is perhaps only understandable by me. However, would this exclude common vernacular, such as 10^100 = 'googol'? One could say that a description of a number must be constructed according to simple rules, such as a context-free grammar containing the symbols 'one' and 'plus', and a rule saying a description is comprised of either the terminal symbol 'one' or a composition of the form 'one plus X', where X is a valid construction in our little language (ie: X is either 'one' or another description of the form 'one plus X'). Thus, in this language, every number n is described in exactly 2n-1 syllables, giving us a nice linear relation. However, when we say 'description', are we necessarily referring to a description constructed in a limited subset of our language, or could a valid description be made using the entirety of english (or any other language, for that matter)?

Regarding a), holding this premise would entail 'the present king of france' and 'the round square' are co-referential (refer to the same nothing). This is problematic, however, as it is possible that 'the present king of france' refers to something (being that, uttered a few centuries ago, it would refer to something), whereas 'the round square' necessarily refers to nothing. If we were to hold that the two are logically equivalent, then they must be logically equivalent independent of time, which is obviously not the case.

6. Jul 28, 2007

### Niv

I think that the paradox is when using the whole language.

I see the problem with a). I think that for this case we can say that any description that is mathematically impossible under certain axioms is logically equivalent.

7. Jul 30, 2007

### deimors

The problem appears to be that, for any language sufficient enough to describe all the propositions we'd want to create, there does not exist a set of axioms which could prove every proposition (Gödel's incompleteness theorem). By restricting the descriptive power of a language in order to render a given proposition true or false, we lose the ability to talk about all the things we'd like to talk about. While the results may be to our liking once we've eliminated all the paradoxes, there still exists the original language riddled with contradictory propositions.

8. Aug 6, 2007

### honestrosewater

What about an inconsistent set of axioms? The set of all propositions is a set of axioms, so there you have it already, though a set of axioms technically does not prove anything, as axioms are just propositions. You need inference rules in order to prove anything.

Why would you want to be able to prove every proposition anyway? It seems to me that the whole point of studying logics and reasoning systems is to partition your set of propositions in some way, e.g., into ones that are true and ones that are false or into ones that are logically connected to some given set and ones that are not.

The "problem" with your expression doesn't seem to necessarily have anything to do with numbers or well-orderings. The problem seems to me to be that your expression is talking about its own interpretation and you identify a singleton referent with its own member.

Say that you have a set E of nominal expressions (a certain type of string of some formal language), a set O of objects (objects not necessarily being atomic individuals but possibly sets), and a map I from E to the power set of O, Power(O). Doesn't

1) the o in O that I does not assign as the value of this expression

present the same problem as your original expression, only more directly? Under some I, e.g., the standard English one, Ien, (1) is making a claim about its own interpretation. Under Ien, (1) is putting a condition on Ien, a condition that Ien cannot satisfy, and this seems to me to be the problem with your original expression.

Perhaps even more straightforward is

2) any o in O that I does not assign as the value of any e in E

The referent of (2) is the set of all objects that aren't referents. So (2) seems to require, where I(e) denotes the set assigned to e by I (i.e., a referent and member of Power(O)) and f is in E,

3) $$\exists I \exists f \forall e \forall o \ [o \in I(f) \ \leftrightarrow \ \neg(o = I(e))]$$

If you let I(f) be a member of O, then, when o = I(f) and e = f, (3) becomes

4) $$\exists I \exists f \ [I(f) \in I(f) \ \leftrightarrow \neg(I(f) = I(f))$$

If you further identify singletons with their member (so that p is a singleton and q is in p implies p = q) and assume that I(f) is a singleton (changing (2) to the o in O...), then (4) becomes

5) $$\exists I \exists f \ [I(f) = I(f) \ \leftrightarrow \neg(I(f) = I(f))]$$

Hm, right? Does anyone see a mistake?

Last edited: Aug 6, 2007