Date: Apr 24, 2013 9:42 PM Author: fom Subject: Re: Matheology § 246 On 4/24/2013 11:46 AM, WM wrote:

>

> 1

> 1, 2

> 1, 2, 3

> ...

>

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.