On 4/20/2013 11:32 AM, WM wrote: > On 20 Apr., 17:37, fom <fomJ...@nyms.net> wrote: >> 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 >>> forall n exists i : n e s_i. >> >>> S is constructed by adding s_(i+1) after s_(i). So we have >>> (1) forall n, forall i : (n < i <==> ~(n e s_i)) & (n >= i <==> n e >>> s_i). >> >>> There is no term s_n of S that contains all natural numbers. >> >> Since the s_n are defined in terms of finite sets, this >> is true. >> >>> This >>> condition requires >>> (2) exist j, k, m, n : m e s_j & ~(m e s_k) & ~(n e s_j) & n e s_k. >> >> Notice that (2) is not a consequence of standard mathematics. > > No? And this is all counter-argument you have collected in five > posts??? >
There is no arguing with you. The assertions have no basis in standard mathematics. Either you are proving a contradiction within standard mathematics or you are not. You are not. Assertion is not proof. Besides, I already looked at this argument carefully. It is a WMytheology mistake -- not mathematics.
> How do you remove the problem that all n are in S, that all s_i are in > S, that nothing else is in S, but that not all n are in one s_i?
S is a sequence of sequences. It is not |N.
The s_i in S are there by definition.
No s_i is defined to have all n in |N.
> In > "standard mathematics" they usually say that every guy dances with a > girl,
> but that no guy dances with all girls.
That depends on definitions.
For example, given any fraction in lowest terms, there is a representation for each n,
p/q :=> np/nq
There is a "guy" for each "girl" in |N
But one lucky "guy" gets all the "girls" because that "guy" is the standard form.
> That means there is Joe > who doesn't dance with Nancy and there is Karl who does not dance with > Mary. But that is impossible in case of Steve who simultaneously > dances with all girls he ever has met. Any other proposals?
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.