As an apt comment on this and other, similar, threads, I recommend (01)
On Jan 29, 2010, at 6:20 PM, Christopher Menzel wrote: (03)
> On Sat, 2010-01-30 at 12:28 +1300, Rob Freeman wrote:
>> On Fri, Jan 29, 2010 at 3:16 PM, Christopher Menzel
>>> ... On reflection, I will qualify my original claim that Russell's
>>> paradox has *nothing* to do with undecidability. For there is in
>>> fact a similarity between the usual argument given in presentations
>>> of Russell's paradox and the method of diagonalization in
>>> computability theory -- a method often used in proofs of
>>> undecidability. Russell's argument can be used more generally to
>>> show that, for any given set A, the set of all non-self-membered
>>> members of A cannot itself be a member of A and the proof is a clear
>>> instance of diagonalization. The paradox itself ensues from this
>>> theorem if one also assumes (or one is able to prove) that there is
>>> a universal set.
>>> So there is a similarity between the method of proof in Russell's
>>> paradox and a common method of proof in computability theory. But,
>>> conceptually, the paradox itself has nothing to do with
>>> undecidability/incompleteness/computability per se.
>> I'm not sure what you mean by
>> "undecidability/incompleteness/computability per se."
> It means the paradox has nothing to do with the concepts themselves.
> Russell's paradox is a paradox of set theory; it was discovered
> before computability theory was even formulated and none of the
> of computability theory are presupposed by the paradox in any way.
> does it have any implications for them. The only connection, as I
> pointed out, is that the *structure* of the usual proof of Russell's
> paradox (more accurately, the usual proof of a proposition that, with
> the assumption of a universal set, entails Russell's paradox) is
> to the structure of some proofs of undecidability.
>> I'm happy with your agreement that "there is in fact a similarity
>> between the usual argument given in presentations of Russell's
>> and the method of diagonalization in computability theory -- a method
>> often used in proofs of undecidability."
> My agreement?? Again you are being entirely disingenuous. For X to
> agree with Y about A implies that Y has asserted A; and it typically
> presupposes that Y also understands A. So to characterize my
> observation about diagonalization in the proof of Russell's paradox as
> something I have *agreed* with you about suggests that it is something
> that you yourself have asserted and that you understand. You've
> certainly not asserted it, and the numerous confusions and errors
> logic and computability in your own posts lead one, at the least, to
> suspicious of whether you understand it.
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