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<mueckenh@xxxxxxxxxxxxxxxxx> wrote in message
news:1159610608.002214.150810@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> William Hughes schrieb:
>
>
>> For every real number x, there exists a list, L_x such
>> that x is a member of L_x
>
> Where do they exist? It is impossible for most real numbers even to
> name them. You cannot name or construct more than aleph_0 real numbers.
> Nevertheless you insist, that also the other ones, which have no names
> and no other identification properties, should have complete lists?
This is a complete red herring. There is no question that the Real generated
by Cantor's proof is computable (r. e,) if the original list is, as he gives
an explicit construction.
>>
>> There exists a list L, such that every real number x is
>> a member of L.
>>
>> The first is true, the second is false. There is no way to put
>> all the L_x together to get a "countable set of entries"
>> (the list L).
>
> But if they existed, then we could put them together. Why can't we
> connect in our thoughts all these thought lists such that there is only
> one thought list, i.e., the thought list of all thought lists? At least
> a square of all thought lists should be possible.
We can connect them. We know that aleph_0 squared is still aleph_0, because
we pair the elements off as follows:
1 <-> (1,1)
2 <-> (1,2)
3 <-> (2,1)
4 <-> (2,2)
5 <-> (3,1)
6 <-> (3,2)
etc.
The same will work for aleph_0 to any power; it is not until we get to
aleph_0 to the aleph_0 power (which is the same as 2 to the aleph_0 power)
that it changes.
>
> But if you like, you can schematically consider all real numbers by the
> infinite binary tree which contains them all represented by a countable
> set of nodes and edges. I have shown by a rational relation that the
> set of branches (corresponding to real numbers) is not larger than the
> set of edges.
>
Then the set of edges isn't countable either.
> Regards, WM
>
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