Re: On the diagonal argument again (2)

[|-| E R C]

On May 21, 8:31 am, Jim Burns <burns...@osu.edu> wrote:
> On 5/20/2012 5:03 PM, LudovicoVan wrote:
>
>
> > "LudovicoVan" <ju...@diegidio.name> wrote in
> > messagenews:jpbk5l$i8b$1...@speranza.aioe.org...
> >> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
> >>news:55b27f39-f8f2-46e9...@b1g2000vbb.googlegroups.com...
> >>> I do not yet understand why should a diagonal s[y] that is not on the
> >>> list have an ordinal y that eneumerates the entries of the list?
>
> >> Yes, I guess I agree: "we cannot meaningfully substitute y for x".
>
> > More specifically, if the non-finite ordinal in question is simply
> > denoted by w (omega), we have:
>
> > s[w] is a string such that s[w][w]=/=s[w][w]
>
> > There the problem is not really in stating 's[w]', we have in fact just
> > defined what s[w] is, the problem is that there is no w-th digit. (And,
> > if there were an w-th digit, i.e. in an extended diagonal argument over
> > N*, we'd still be at square one, with an impossible string that has to
> > differ from itself.)
>
> I am curious if you have any objections to the general diagonal
> argument.
>
> Let A be a set, P(A) be its powerset, and f: A -> P(A).
> Then there is at least one element D of P(A) (D subset A)
> such that, for all x in A, f(x) =/= D.
>
> Proof: Consider D = { y in A | y not in f(y) }.


In a countable sets UoD this is d = { n | n ~e n }

See: http://tinyurl.com/PureSetTheory


Herc

[LudovicoVan]

[Cross-posted to sci.logic.]

"Virgil" <vir...@ligriv.com> wrote in message
news:virgil-84FD7F....@bignews.usenetmonster.com...
> In article <jpc315$lra$1...@speranza.aioe.org>,
> "LudovicoVan" <ju...@diegidio.name> wrote:
<snip>

>> This string is like it is saying "I am not listable", similar to "I am
>> not
>> provable". Is that a proof of anything? Are flying unicorns a proof
>> that
>> zoology is incomplete?
>
> Actually, the constructed string is saying "I was not listed in that
> particular list" since it is perfectly possible to put that string in a
> different list.

Right: not listable by that list, not provable by that theory, not
computable by that system, since...

-LV