On 4/20/2013 12:14 PM, WM wrote: > On 20 Apr., 17:15, fom <fomJ...@nyms.net> wrote: >> On 4/20/2013 2:01 AM, WM wrote: >> >>> On 19 Apr., 21:41, Virgil <vir...@ligriv.com> wrote: >> >>>>> You have not understood the relativity of mathematics. There is no >>>>> fixed largest number in mathematics. >> >>>> Because for every positive number, whether natural, integral, rational >>>> or real, there is another twice as large >> >>> and finite and belonging to a finite set. >> >>>> As soon as any positive number has been identified, so has its double. >> >>> Not as soon! Every calculation, even the easiest, requires some time. >> >> But there is no calculation involved. Virgil's statement gives >> no specific numerical value to which to apply the axioms. > > No problem. Every numerical value is finite. The axioms are to be > applied to finite naturals only, because only a finite natural changes > its value when 1 is added. And the result is again a finite value, > counting the elements of a finite set. This does never change. >
Sadly, this is the kind of statement which makes apparent your failure to understand.
Constants and closed terms do not change values.
Succession defines how natural numbers are situated with respect to one another.
One of Frege's contributions had been to address the confusion over constants and variables in the practice of nineteenth century mathematics. However, no one really paid attention.
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.