- #1

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- Thread starter deimors
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- #1

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- #2

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

- #3

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- #4

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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

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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

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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.

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- #8

honestrosewater

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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.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).

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

1) the

present the same problem as your original expression, only more directly? Under some

Perhaps even more straightforward is

2) any

The referent of (2) is the set of all objects that aren't referents. So (2) seems to require, where

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

If you let

4) [tex]\exists I \exists f \ [I(f) \in I(f) \ \leftrightarrow \neg(I(f) = I(f))[/tex]

If you further identify singletons with their member (so that

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

Hm, right? Does anyone see a mistake?

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