fom
Posts:
1,719
Registered:
12/4/12
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Re: Matheology § 255
Posted:
Apr 20, 2013 11:37 AM
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On 4/20/2013 3:16 AM, WM wrote:
> Matheology § 255
>
> Let S = (1), (1, 2), (1, 2, 3), ... be a sequence of all finite
> initial sets s_n = (1, 2, 3, ..., n) of natural numbers n.
>
> Every natural number is in some term of S.
> U s_n = |N
> forall n exists i : n e s_i.
>
> S is constructed by adding s_(i+1) after s_(i). So we have
> (1) forall n, forall i : (n < i <==> ~(n e s_i)) & (n >= i <==> n e
> s_i).
>
> There is no term s_n of S that contains all natural numbers.
Since the s_n are defined in terms of finite sets, this
is true.
> This
> condition requires
> (2) exist j, k, m, n : m e s_j & ~(m e s_k) & ~(n e s_j) & n e s_k.
Notice that (2) is not a consequence of standard mathematics.
Rather, it is a consequence of WM's ultrafinitism. It is
not standard mathematics that derives this self-contradictory
state of affairs; rather, it is WM's peculiar reasoning.
> (2) is in contradiction with (1).
(2) is in contradiction with itself. But this has nothing
to do with standard mathematical practice.
--------------------------------------------------
WM is an unabashed ultrafinitist who refuses to fix
a largest finite number. Each "n" in his description
depends on the subsequence of triangular numbers.
> F(n)=Sum_i(1..n)(i)
>
> 1 :=> 1
> 2 :=> 3
> 3 :=> 6
> 4 :=> 10
>
> and so on
According to Brouwerian intuitionistic reasoning,
when WM's construction reaches the point where
the sequence of triangular numbers exceeds the
ultrafinitist limit, the contradiction nullifies
the construction.
This is WM's model of mathematics:
http://en.wikipedia.org/wiki/Finite_model_property
until he reaches his contradiction and
it vanishes.
=====================================
The triangular numbers correspond with
the number of 'marks' representing numerals
or significant denotations occurring in any
of WM' representations of the form:
1
2, 1
3, 2, 1
...
n, ..., 3, 2, 1
...
-------------------------------------
This number of 'marks' satisfies a structural
feature of the natural numbers called a
directed set:
Defintion
A binary relation >= in a set D is said
to direct D if and only if D is nonempty
and the following three conditions are
satisfied:
DS1)
If a is an element of D, then a>=a
DS2)
If a, b, c are elements of D such
that a>=b and b>=c, then a>=c
DS3)
If a and b are elements of D, then there
exists an element c of D such that c>=a
and c>=b
So, WM's geometric reasoning for any given
n obtains a finite model domain with its
cardinality given by the associated
triangular number. The triangular number
is the "element c" of condition DS3 from
the definition.
-------------------------------------
Finally, Brouwer's explanation for finitary
reasoning is used because WM refuses to
commit to any mathematical statement with
coherent consistent usage.
Brouwer distinguishes between results with
regard to 'endless', 'halted' and
'contradictory' in his explanations
"A set is a law on the basis of
which, if repeated choices of
arbitrary natural numbers are made,
each of these choices either
generates a definite sign series,
with or without termination of the
process, or brings about the
inhibition of the process together
with the definitive annihilation
of its result."
WM cannot be an ultrafinitist and
expect others to not hold him to
task for it. In constrast to
Brouwer, he repeatedly states
that there is absolutely no
completed infinity. Therefore,
there must be a maximal natural
number for his model of
mathematics. Beyond that
number, there is no mathematics.
That is WM's belief as surmised
from his statements and reasonings
as opposed to what he says with
rhetoric.
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