On 3/18/2013 2:26 PM, WM wrote: > On 18 Mrz., 20:00, fom <fomJ...@nyms.net> wrote: >> On 3/18/2013 1:34 PM, WM wrote: >> >> >> >> >> >>> On 18 Mrz., 18:32, fom <fomJ...@nyms.net> wrote: >>>> On 3/18/2013 6:43 AM, WM wrote: >> >>>>> On 18 Mrz., 07:26, fom <fomJ...@nyms.net> wrote: >> >>>>>> You turn to an outdated strategy directed to >>>>>> a situation that no longer exists rather >>>>>> than do the hard work of grounding your >>>>>> claims. You do this to say that just >>>>>> because you do not believe a particular >>>>>> axiom, >> >>>>> Wrong. I prove that the axiom is nonsense like the axiom that a >>>>> triangle with four edges exists. >> >>>> That would be more forceful if you used the >>>> term 'trilateral'. >> >>> Then it would be trivial. My example requires a little bit deeper >>> thought. >> >>>> Once again. You have *proven* nothing. >> >>> As your foregoing hint shows, you seem to welcome trivialities, but >>> you seem to be not able to understand more difficult ideas. >> >> >> What I do not understand is how you consider >> your refusal to provide a similar framework >> to have merit.- > > There is no framework necessary to show in ZFC that every word belongs > to a countable set.
A set you hold to be interpretable as countable in relation to a theory of time.
There is a reason Brouwer's ideal mathematician is pre-linguistic.
And that is a question of mine from another post for which you have yet to provide an answer.
So, let me explain it to you.
A 'word' constitutes a limitation in the sense of Kant's 5th remark just as a geometric point (*represented* arithmetically in terms of Dedekind cuts) on a line represents a temporal "now" when the continuity of a line is used to represent time in kinematics.
And Kant's 3rd remark explains the error of "putting the cart before the horse."
> 1) > Time is not an empirical concept > that is derived from experience. > [...] > > 2) > Time is a necessary representation > that underlies all intuitions > [...] > > 3) > The possibility of apodeictic principles > concerning the relations of time, or > of axioms of time in general is > grounded upon this a priori necessity. > [...] We should only be able to > say that common experience teaches > that this is so; not that it must be > so. These principles are valid as > rules under which alone experiences > are possible; and they instruct us > in regard to experiences, not be > means of them. > > 4) > Time is not a discursive, or what is > called a general concept, but a form > of pure sensible intuition. > > 5) > The infinitude of time signifies > nothing more than that every determinate > magnitude of time is possible only > through limitations of one single > time that underlies it."
One need not base one's understanding of mathematics on a theory of time. However, that it is possible is to be found in Aristotle's discussion of priority:
"One thing is said to be prior to another in four ways. First and most fully, in time, when one thing is said to be older and more ancient than another; for it is because the time is longer that it is said to be older or more ancient. Second, what does not reciprocate in implication of being. One, for instance, is prior to two; for if there are two, it follows immediately that there is one, whereas if there is one, it is not necessary that there are two. So that from one, the implication of the other's being does not hold reciprocally; and the sort of thing that seems to be prior is that from which there is no reciprocal implication of being. Third, a thing is said to be prior in some order, as with sciences or speeches. For in the demonstrative sciences there is prior and posterior in order, since the elements are prior in order to the diagrams; and in grammar the letters are prior in order to the syllables. And the same is true for speeches, since the introduction is prior in order to the exposition."
Kant's critical philosophy introduces a notion of time in which Brouwer's ideal mathematician is necessarily pre-linguistic. Experience does not instruct the theory.
So the fact that you can count the marks made by your crayons proves nothing because they cannot instruct with regard to what is presumably unsayable and unwritable.
Finally, let me remind you of what has now been identified as the narrowing requirement one must accept to even consider your statements. It can be seen in Weyl where one is asked to forgo one's logic and one's definition just long enough to make his reasoning "true"
In another post I observed the following:
> Note the explicit rejection of logic > and definition in his statement, > > "Therefore, how two sets (in contrast to > properties) are defined (on the basis of > the primitive properties and relations > and individual objects exhibited by means > of the principles of section 2) does not > determine their identity. Rather, an > objective fact which is not decipherable > from the definition in a purely logical > way is decisive; namely, whether each > element of the one set is an element > of the other, and conversely. [...]" > > > So, as a reader of this statement, I > am first expected to reject prior > definitions and to reject logical > relations. Then, I am expected to > understand the discursive assertion > explaining what it is that cannot > be explained.
You are more than welcome to begin again in ways that do not violate the reasoning of rational human beings. But failing to identify principles, failing to identify a logic, and failing to prove your statements is not how one goes about the practice of mathematics.