Re: Cantor's circular "proof" that evens = integers

In article <1sto331ke4sq6f8k1jjb26su865pq7vi9s@xxxxxxx>,
G. Frege <nomail@invalid> wrote:

On Sat, 05 May 2007 09:11:59 GMT, Phil <toob-headman@xxxxxxxxxxxxx>
wrote:


Now, if anyone can PROVE that whenever x is in N, then 2*x is in N,
then you have a true PROOF that N and E are equinumerous.

If you weren't the idiot you are, you might notice that * is an
operation which can be (and is) defined in set theory, such that
n*m in N for all n,m e N. Since 2 e N this implies that 2*n in N
for all n e N.

If you are curious, * is defined the following way:

(a) n * 0 = 0
(b) n * m' = n * m + n

Or if one insists that N begins with 1, as Phil seems to want to do,
one first defines addition, (+) inductively as
(a) n + 1 = succ(n)
(b) n + succ(m) = succ(n + m)
the one gets multiplication (*) indiuctively by
(c) n*1 = n
(d) n*succ(m) = n*m + n.
It is then trivial to prove, again inductively, that both (+) and (*)
are closed operations , i.e., that all sums and products of naturals are
again naturals.


However, simply claiming that "this is obvious" is not adequate [...]

The detailed proofs are trivial, but tedious. If you want them, get a
text on foundations, and do a little of the work on your own.

No-one (other than you) suggests that the claim "this is obvious"
is an adequate replacement for an explicit mathematical proof.

That there are many texts on foundations of mathematics which contain
all these proofs, and more besides, makes it obvious that anyone who
wants such proofs can get them.

If you want tutoring, are you willing to pay the going rates for it?



Actually, WE all know that

"Unproven statements carry little weight in the world
of mathematics." (Amir D. Aczel)

Those statements are proven in many places. That Phil does not have
access to those proofs is largely his problem.
.

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