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Topic: Matheology § 246
Replies: 246   Last Post: Apr 26, 2013 2:37 AM

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 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology § 246
Posted: Apr 19, 2013 8:23 AM

On 4/19/2013 4:35 AM, WM wrote:
> On 18 Apr., 09:36, Virgil <vir...@ligriv.com> wrote:
>

Perhaps I misunderstood what WM had been trying
to prove here.

> Wow thousands of matheologians don't see that every natural number in
> {(1),(1, 2), (1, 2, 3), ...} is also in {1, 2, 3, ...} and vice versa.
>

This must be the affirmative claim.

Thus,

(1)=1
(1,2)=2

etc

So,

1=(1)=((1))=(((1)))=...
2=(1,2)=((1),(1,2))=(((1)),((1),(1,2)))=...

etc

Assuming that this might have been a proof by contradiction,
suppose not.

> Every FISON(n) of the sequence is a subset of |N.

Given n, FISON(n) is a subset of |N

But, it is not the case that n=FISON(n) by the assumption.

> There are all n in FISONs, but not all n are in one and the same
> FISON.

Given n, there exists a such that n is an element
of FISON(a)

n<a

Given any FISON, there exists n such
that n is not an element of FISON(a),

a<=n

So, |N is linearly ordered.

>
> Wow thousands of matheologians don't see that this implies the
> existence of at least two natural numbers, m and n, that are
> distributed over at least two FISONs. There exist m, n, a b in |N,
> such that m is in FISON(a) and not in FISON(b) and n is in FISON(b)
> and not in FISON(a).

So, from the fact that |N is a linearly ordered
set for which every element is contained in
some finite set, WM concludes

b<m<a

a<n<b

for some a,b,m,n in |N

under the assumption rejecting impredicative
definition.

>
> This is the only logical conclusion (if all FISONs exist
> simultaneously).

It would seem that rejecting impredicative
definition is equivalent to simultaneous
existence of FISONs.

So, WM seems to be asserting that a completed
infinity is tantamount to reflecting some
finite value into an unusual location.

For example, the claim would be as if

10=10000

is added to the Peano axioms.

> By construction of the sequence of FISONs it is a

But WM seems to only be contradicting himself.

He is the one whose logic seems to be based
on

n=triangular_number(n)

because of his geometric reasoning.

The arithmetical treatment of completable
infinities as transfinite numbers is
based on extending the notion of limit
to discrete sets ordered by a successor
function.

This is not a stipulation of the form
WM seems to portray.

---------------------------------------

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
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

"A set is a law on the basis of
which, if repeated choices of
each of these choices either
generates a definite sign series,
with or without termination of 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 statements and reasoning
as opposed to what he says with
rhetoric.

Date Subject Author
4/12/13 mueckenh@rz.fh-augsburg.de
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4/12/13 dan.ms.chaos@gmail.com
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4/12/13 dan.ms.chaos@gmail.com
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4/13/13 mueckenh@rz.fh-augsburg.de
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4/13/13 dan.ms.chaos@gmail.com
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4/13/13 Virgil
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4/13/13 Virgil
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4/14/13 Tanu R.
4/14/13 Virgil
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4/13/13 fom
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4/13/13 mueckenh@rz.fh-augsburg.de
4/13/13 Virgil
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4/14/13 Tanu R.
4/14/13 Virgil
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