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Re: Matheology § 203
Posted:
Jan 29, 2013 4:36 AM
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On 29 Jan., 10:18, William Hughes <wpihug...@gmail.com> wrote:
> On Jan 29, 10:09 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
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> > On 29 Jan., 09:54, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Jan 29, 9:33 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > "All" and "every" in impredicative statements about infinite sets.
>
> > > > Consider the following statements:
>
> > > > A) For every natural number n, P(n) is true.
> > > > B) There does not exist a natural number n such that P(n) is false.
> > > > C) For all natural numbers P is true.
>
> > > > A implies B but A does not imply C.
>
> > > Which is the point. Even though A
> > > does not imply C we still have
> > > A implies B.
>
> > > Let L be a list
> > > d the antidiagonal of L
> > > P(n), d does not equal the nth line of L
>
> > > We have (A)
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> > > For every natural number n, P(n) is true.
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> > > This implies (B)
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> > > There does not exist a natural number n
> > > such that P(n) is false.
>
> > > In other words, there is no line of L that
> > > is equal to d.
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> > And how can C be correct nevertheless? Because "For all" is
> > contradictory.
>
> B: There is no line of L that is equal to d
>
> does not imply
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> C: For all n, line n is not equal to d.
>
> B correct does not mean "C correct nevertheless"-
But we know of cases where C is correct nevertheless. I quoted four of
them in the § 203. Or do you disagree to one of them?
In case you have forgotten the old discussion concerning the
configurations of the Binary Tree construction, here it is repeated:
The complete infinite binary tree is the limit of the sequence of its
initial segments B_k:
____________________
B_0 =
a0.
____________________
B_1 =
a0.
/
a1
____________________
B_2 =
a0.
/ \
a1 a2
____________________
...
____________________
B_k =
a0.
/ \
a1 a2
/ \ / \
...
....a_k
___________________
...
The structure of the Binary Tree excludes that there are any two
initial segments, B_k and B_(k+1), such that B_(k+1) contains two
complete infinite paths both of which are not contained in B_k.
Nevertheless the limit of all B_k is the complete binary tree
including all (uncountably many) infinite paths. Contradiction. There
cannot exist more than countably many infinite paths.
Alternative consideration: Obviously every B_k is finite. None does
contain any infinite path. The infinite paths come into the play only
after all B_k with k in N (by some unknown mechanism). If that is
possible, however, this mechanism can also act in Cantor's diagonal
proof such that the anti-diagonal enters the list only after all lines
at finite positions.
Regards, WM
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