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Alan Smaill wrote:
> "John Jones" <jonescardiff@xxxxxxx> writes:
>
> > John Jones wrote:
> >> Alan Smaill wrote:
> >> > "John Jones" <jonescardiff@xxxxxxx> writes:
> >> >
> >> > > If every sentence had a truly unique representation as a string, a
> >> > > string would be a sentence. We are laying down sentence and string
> >> > > together even though there are differences.
> >> >
> >> > Yes, there are differences.
> >> > But why should there be a problem with having a unique representation?
> >> > the representation is not the entity, just as the numeral is not the
> >> > number.
> >> >
> >> > What is a number has has no corresponding numeral?
> >> >
> >> > --
> >> > Alan Smaill
> >>
> >> If a numeral was unique to a number, the numeral would be the number.
> >
> > For a numeral to be unique to a number in any circumstance, the numeral
> > would have to reflect each and every property of the number. i.e. be a
> > number itself.
>
> You still do not provide a number with no or many corresponding
> numerals, I note.
>
> It looks as though there is confusion about the meaning of "unique" here.
>
> No-one is suggesting that there is some inherently iconic representation
> that carries its referent around as a metaphysical doppelganger.
> There are many representation systems for numbers -- if I see "11"
> as a representation, I need to know if it's a decimal or binary
> representation, there are other characters in different languages,
> French and English have different names for natural numbers,
> we could have used "3" as the numeral for the number four, etc etc.
>
> What is claimed is that, given an agreed convention of a shared
> language/representation system, it may be that for every number there
> is one and only one corresponding representation, and for every
> representation there is one and only one number. I thought you were
> claiming that this is impossible -- maybe you are?
>
> Let's go back to playing cards. Suppose we take conventional playing
> cards and some Tarot cards, the same number of each, and we want to
> play a game involving one set of cards with the "wrong" set of cards.
> We can agree a correlation between the cards of one sort and the cards
> of the other sort such that for each card of one sort there is one and
> only one (ie unique) corresponding card of the other pack (and vice
> versa). Now we can play the game -- we can say that the card of one
> pack represents the card from the other pack.
>
> Can you imagine playing such a game?
Yes. I am still lost.
> If every sentence had a truly unique representation as a string, a
> string would be a sentence.
That's the original formulation. Godel is claiming, and everyone else,
that a sentence has a unique string and that the string represents or
corresponds to the sentence. But in that case, either
1) the string is simply a sign (and not a string) representing the
sentence. This cannot correlate sentence and string as 2) and 3) can,
2) or the sentence is identified by occupying a position on the page
laid out in ink in the same place as the string (in which case it is
also not a string but a position),
3) or string and sentence collaborate in implicit self-reference ("this
sentence is a string"), invoking obscure identity relationship,
4) or the string is unique to the sentence, in which case there is one
sentence for a particular length and character depicting string. In
this case the string is the sentence - otherwise it is not a string but
a referencing device, and these are arbitrary and not unique (see1)).
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