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
Notice the use of a union here.
What is the index of the union.
Union_i (i e |N) s_i
In other words, this is disallowed in WM's world
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.
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:
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:
If a is an element of D, then a>=a
If a, b, c are elements of D such that a>=b and b>=c, then a>=c
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.