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

On Apr 30, 5:56 pm, Phil <toob-head...@xxxxxxxxxxxxx> wrote:

The reason I constructed my own proof the way I did
was in order to avoid having to DIRECTLY
state that that if n is in N,
then so is 2n.

I retorted,
Why in the hell would anybody want to AVOID stating
such an OBVIOUS truth??

Well, let's see ... this is mathematics,
so we do not ASSUME ANYTHING???

WHY are you STILL such an ignorant insulting ***?
WE are the ones who TOLD YOU that it was necessary
to state your assumptions! WE TOLD YOU to STATE
SOME AXIOMS! IN MATH, OF COURSE you have to
assume things! The part you haven't figured out yet
is that BEFORE we begin, WE STATE those assumptions.

In geometry,

Oh, SHUT UP. YOU DON'T KNOW *** about the history
of axiomatizations of geometry. Logic as we know it wasn't
even invented until the 1880s and it took that long for a
coherent axiomatization of geometry to emerge as well
(Google "Pasch" with "betweenness" and "1882").

given the "obvious" truth that the fundamental
characteristics of a figure do not change if we
merely scale the figure

That is NOT an obvious truth; it is not even a TRUE
truth, UNLESS YOUR AXIOMS PUT YOU in a UNIVERSE
where that truth is PROVABLE! Sometimes OTHER
things are possible, ESPECIALLY if you have NOT
stated your axioms. THE DIFFERENCE between what
is going on here and YOUR LAME analogies to past
innovations that YOU DO NOT UNDERSTAND is that
WE KNOW what the axiomatic method is and WE HAVE
stated our axioms!

up or down in size, it became possible
to prove the Euclideam parallel
postulate from the other 4 postulates,
plus this "obvious" truth!

THAT DID NOT become possible until AFTER This
obvious truth WAS ADDED AS AN AXIOM!
THAT IS THE WHOLE POINT! You INTENTIONALLY
make your axioms your obvious truths!

The problem, as I'm sure you know,

No, I don't know, and you don't either.

was that it turned out that this obvious ability
to scale without causing internal changes is
LOGICALLY EQUIVALENT to the Euclidean parallel postulate,

SO WHAT?!?!? HOW OR WHY is THAT a PROBLEM?!?!?
If the two postulates are logically equivalent then YOU CAN
JUST USE EITHER or the other, and it NEVER MATTERS
which!

and for that matter, to the "fact" that the sum
of the angles of a triangle is 180 degrees.

None of this IS ANY kind of PROBLEM!
These "facts" you are talking about DID NOT HAVE
*proofs* from AXIOMS *except* in the context where
the parallel postulate, or the scaling, or the angle-sum,
WERE TAKEN as axioms! Non-Euclidean geometry
(violating ALL this crap) was discovered all the way
back in 1837 by Lobachevsky! The fact that it violated
intuition WAS JUST IRRELEVANT! IT WAS STILL
*logically* consistent and NOBODY Had ANY disproofs
of its results! NONE OF THIS HAS ANYTHING TO DO
with the situation in which YOU now find yourself,
in which WE DO have axioms and proofs!


George, these are INFINITE sets we're talking
about here. The very idea
that there are as many even numbers
as even plus odd number is
"obviously" wrong.

So FUCKING what!?!?
ANYBODY who WANTS to postulate a theory to that
effect is PERFECTLY FREE to do so! GO AHEAD!
MAKE MY DAY!

In other words, why in the hell would anybody with a
PhD in philosophy have to ASK why we need to
avoid simply CLAIMING that x is true because it's
OBVIOUSLY true?

I do NOT HAVE a PhD in Philosophy, dumbass.
Read the address line. I have an M.S. in Computer
Science from UNC. And YOU ARE ASKING A STUPID
QUESTION, you stupid ***. Just because SOMEbody
SOMEtime didn't get their axioms right does NOT mean
WE are having any trouble with knowing that sum or product
or exponentiation of 2 natural numbers is natural!
YOU DO NOT doubt that! YOU HAVE NEVER SEEN
a theory where it MIGHT be otherwise! DO YOU REALLY
SUSPECT that this might NOT be PROVABLE?????!???

Why did I want to avoid stating that if n in in N, then so is 2n?

Because you're an idiot, that's why.

Because I didn't have an ironclad
PROOF that it was true,

PLEASE go *** yourself.
YOU don't have an ironclad proof OF ***
because YOU have NO CLUE what a proof *IS*!
YOU actually thought that if I had axiom A, and proved
something of it, that this proof might somehow become
invalid if I then added axiom B to the mix. You began
this whole discussion by REFUSING to state your axioms,
a refusal that you have MAINTAINED to this day, while
still claiming that you could "spot flaws" in other people's
reasoning! *SHUT*UP!*

and I wanted to have the strongest proof that N
and E are equinumerous that I could get.

Idiot: there is no such thing as a stronger or a weaker
proof. A proof is just string-matching. EIther the patterns
match or they don't. How strong or weak your proof
of the fact that there are as many even naturals as
there of naturals in toto DOES NOT MATTER.
This is a SIMPLE fact that GALILEO proved 350
years ago. IT DOES NOT NEED infinitary axiom-schemata
from ZFC and careful attention to being "strong enough".
You might as well be worrying whether you were strong
enough to take candy from a baby. And that goes SQUARED
for proving that if n is natural, then n+n is too. That can
hardly even COME UP, it's so basic. Theories where
it canNOT come up ARE BETTER for that reason!


that 1+1=2, do you want to DOUBT that 2 is natural??
Why is it a DIFFERENT question when it is 69+69=138?
Do you REALLY think "the proof that 138 is natural" COULD
POSSIBLY be relevant to this issue??? OF COURSE the
double of a natural number is a natural number!
OF COURSE
the square of a natural number is a natural number! Why?
Because DEFINITIONALLY, the SUCCESSOR of a natural
number is a natural number, and, AGAIN, DEFINITIONALLY,
natural numbers consist of NOTHING BUT more applications
of successor!

Gee, and all this time people
thought that the subject of INFINITE sets
was inherently more difficult than finite sets.

Of course it is, but that has NOT A DAMN thing to do with
the fact that for ONE (not infinitely many, for ONE) natural,
that ONE natural is OBVIOUSLY going to force anything
you can do to it with addition, multiplication, and other naturals,
a natural number of times, to ALSO be natural.

If only George had been
there to help everyone see the light.

It wasn't ME, fool. It was the inventor of THE INFERENCE RULE
OF UNIVERSAL GENERALIZATION. This is the rule
that says you can prove that something is true for ALL
members of a class, simply by CHOOSING AN ARBITRARY
(new) NAME to denote ONE member of the class, and
proving the something ABOUT THAT ONE MEMBER.
BECAUSE the name is new, because it was not used anywhere
else prior, it follows that the exact same proof WOULD work
on any OTHER term or entity you could've put in place of
the arbitrary name. SO YES, YOU CAN show SOME things
about infinitely many things JUST AS EASILY as you could've
shown ONE thing. The people who know something about
logic KNEW THAT ALREADY, Phool. YOU ARE THE ONLY
ONE who hasn't read the relevant INTRODUCTORY textbook.


.

Quantcast