Re: The modern mathematical concept of infinity is ...

lwalke3@xxxxxxxxx wrote:

On Feb 15, 12:13 pm, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
Ralf Bader wrote:
In a certain sense, this is wrong (see that paper, or Nelsons's
"Predicative Arithmetic")...
I really do not see how you can claim this is wrong. If n and m are
natural numbers, n^m is a natural number. Proof by induction
(whatever) is completely trivial.

That's just it -- proof by _induction_. Nelson's paper
deals with the theory PA-Induction Schema. So in that
paper, a theorem isn't considered proved unless it can
be done so using the first four axioms of PA alone. And
according to Nelson, one can't prove that the naturals
(which he calls "counting numbers" in this very paper)
are closed under exponentiation from the first four
axioms of PA alone. But one can prove that N is closed
under addition and multiplication. Nelson explains how
one difference between these two and exponentiation
is that the latter isn't commutative.

Again, this has nothing to do with WM. His problem is not that he has
some odd ideas, so much as that couldn't distinguish an idea from a
spare bicycle wheel.

So even after WM has appealed to another mathematician,
Isles, to validate his theory, the defenders of ZFC
state that WM is still wrong to do so.

In other words, a straight ultrafinitistic point of view, although
problem-ridden, is much more reasonable than Mueckenheim's "philosophy"
of mathematics, which is a promising competitor for the title of "Most
Stupid Philosophy Of Mathematics Possible To Conceive".
Uh, no, I don't think WM's is that good.

One can only imagine how many more mathematicians
WM must find having similar or even identical
ideas as himself, before Bader, Chandler, and the
others finally accept that WM can have a theory
that isn't the "Most Stupid Philosophy" ever.

That is now getting completely ridiculous. As far as I recall, it was *me*
who for the first time mentioned Isles' paper in a discussion with set
theory cranks in de.sci.mathematik, in
Message-ID: <dq1hga$icb$1@xxxxxxxxxxxxxxxxxx>
on Wed, 11 Jan 2006. However an exact reconstruction is not possible as
Google's groups search has now reached a state of total unusability.
The point is that Isles' or Nelson's work has mathematical content, and
Mueckenheim's has not. And it is totally irrelevant how many more
mathematicians Mueckenheim can find (how about Van Bendegem, to mention him
another time?), this will not improve the total lack of quality in
Mueckenheim's "work".
.

Relevant Pages

  • Re: Skolems Paradox
    ... Whatever philosophy of mathematics you espouse, it remains a rather trivial mathematical fact that and are not equivalent. ... In this case it's a theorem of ZFC that implies, but only for the rather uninteresting reason that both of these are provable in ZFC and all provable statements imply each other. ... One could take the view that there is one and only cumulative hierarchy, the class of all sets, and hence CH is either true or false in the ordinary mathematical sense, but hold that our ideas about the cumulative hierarchy are not detailed enough to determine whether CH or ~CH holds. ... If someone were to ask me whether there exists a set, I would give the natural answer: "sure, for example, there's the empty set, the set of naturals, the set of countable ordinals and so forth". ...
    (sci.logic)
  • Re: Orlow cardinality question
    ... > 1's between naturals violates by being finite in size. ... elements in an infinite set. ... > source of much dispute up to this day. ... mathematics as applied to the real world. ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > construction when things talked of are supposedly infinite objects? ... we define, for all naturals i, f= i; and, for all naturals j, g ... > think this was one of the steps of reasoning which Dr. Zenkin ... anyone else here seems to be talking about - mathematics. ...
    (comp.theory)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > construction when things talked of are supposedly infinite objects? ... we define, for all naturals i, f= i; and, for all naturals j, g ... > think this was one of the steps of reasoning which Dr. Zenkin ... anyone else here seems to be talking about - mathematics. ...
    (sci.math)
  • Re: You Dont Have to Be Nuts to Be a Mathematician ...
    ... unjustified arbitrariness somewhere in between mathematics and physics. ... plus infinity, this implied the lack of a reference point to lean on. ... symmetry between positive and negative numbers. ... I imagine instructions leading from naturals to reals pointing to spots ...
    (sci.math)
Quantcast