Re: Cantor Confusion

David Marcus wrote:
Han.deBruijn@xxxxxxxxxxxxxx wrote:
stephen@xxxxxxxxxx schreef:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

stephen@xxxxxxxxxx wrote:
But everything can be modelled as a set.
Define "everything" and prove that claim.
By "everything", I meant everything mathematical. Of course that is not 100% precise.
And no, I cannot prove it. But so far all the various objects of mathematics can be
modelled using set theory. That is what is meant by set theory being a foundation
for mathematics. If someone were to invent something "mathematical" (whatever that may
mean exactly) that could not be described in terms of set theory, then set theory would
no longer serve as a foundation. But given that the basics such as the real numbers,
functions, limits, calculus, etc. all can be founded in set theory, it would have to
be something strange indeed. Not that there is anything wrong with strange, but you
probably would like it less than set theory.
Correction. By "everything" you probably mean "everything according to
nowadays mainstream mathematics", which _is_, of course, "mathematics",
according to your probably rather limited view. But since you can not
really prove anything of the kind, I will rest my case.
It's not much of a case. You have not presented any evidence that there exists
any sort of mathematics not describable by set theory. Until such evidence
exists, the hypothesis that mathematics can be modelled with set theory has
not been falsified. And don't bother presenting something that uses limits,
functions, etc. as all of those things can be modelled with set theory.
Ah, now you are trying to put the burden on me. But that is false play,
of course. You said something like "all heat is phlogiston". I do not
have to argue that this is not so. The burden remains yours. I didn't
even say that set theory is useless within mathematics. I've only said
that there's more to mathematics than set theory.

All the mathematics I've ever seen in a math class or read in a math book or journal or done myself can be done in ZFC. Admittedly, there are large areas of mathematics that I know little or nothing about. Still, that would seem to be quite a bit of evidence right there for the statement that ZFC can be used as a foundation for mathematics. So, the burden is now on you to show some mathematics that can't be done in ZFC.
<snip>

Hi David -

Is it sufficient to show that there are conclusions derived from application of set theory that may not be mathematically correct in all senses? If a conclusion based on premises of set theory does not match the conclusion based on other mathematical methods, then is there not a contradiction between the premises, and therefore premises which are not subsumed under set theory?

The infinite staircase comes to mind, where point set topology considers the limit of the staircase from (0,0) to (1,1), as the number of steps increases without bound, to be the same object as the diagonal line from (0,0) to (1,1), since the locations of the corresponding points become arbitrarily close. This produces a contradiction in measure, the object being of length 2 for all staircases, but of length sqrt(2) for the diagonal line. While the locations of the points in each set approach each other with no lower limit, the directions of the corresponding sub-segments of the two objects are always at a 45 degree angle to each other, producing the error of sqrt(2)/2, the cosine of that angle. So, what we have are a diagonal line of length sqrt(2) and a fractal "line" or curve of length 2. In other words, characterizing the objects as sets of points misses the distinction between the objects in terms of measure, whereas characterizing them as sequences of segments preserves the distinction in terms of direction and overall length.

Now, sequences may be said to derive from ordered sets, but sets are said to be determined solely by membership, with order unimportant. So, the notion of a sequence derives really from an inductive definition such as Peano's, and not from the one primitive in set theory, membership, alone. The notion of order is not captured by "is an element of". Do you disagree?

Tony
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... you claim that set theory is ... theory in which to express virtually all of mathematics. ... S (call this function 'omega pre S'). ...
    (sci.math)
  • Re: Cantorian pseudomathematics
    ... What I mean is set theory was not born just by the great thought of Cantor. ... of mathematics behind, which help greatly, and which even where necessary for the theory to be imagined. ... formalization, at a level up, for meta-theory. ... scientists and engineers doesn't have to be used by engineers. ...
    (sci.math)
  • Re: Cantor Confusion
    ... > If actual infinity is assumed to exist, then the second case is valid. ... of limits in mathematics are formulated in such a way that the limit ... Set theory is more basic than analysis. ... sequences of sets are in general not defined. ...
    (sci.math)
  • Re: Bedeviled by the .999~ = 1 "debate"
    ... of people who understand mathematics disagreeing. ... and infinite sequences, and people who don't want to learn ... Obviously those who learned about limits started from ...
    (sci.math)
  • Re: Cantorian pseudomathematics
    ... What I mean is set theory was not ... >> prove only the mathematics we set out to prove and not statements about ... >> formalization that does not compromise the other criteria. ... >> or show a formalization that does not compromise the other criteria. ...
    (sci.math)