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)
Given any FISON, there exists n such that n is not an element of FISON(a),
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
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
is added to the Peano axioms.
> By construction of the sequence of FISONs it is a > contradiction.
But WM seems to only be contradicting himself.
He is the one whose logic seems to be based on
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 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 statements and reasoning as opposed to what he says with rhetoric.