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

On May 17, 3:09 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote:
MoeBlee wrote:
On May 16, 12:28 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote:

I am pointing out that the CLAIM that N = E is an INHERENT PROPERTY of
any infinite set -- i.e., that the belief that IF set X has infinitely
many elements, then that fact BY ITSELF, with no need for additional
axioms, makes it possible for a proper subset of X to have the same
cardinality as X -- is false.

Who made such a claim?

I will try to find time to look it up, but EVERY website, book, etc.,
that I have ever seen states, categorically, that this is a property of
INFINITE SETS!

In the very post of mine to which you are responding, I explain what
it is you're confused about here. Again: (1) To prove "N equinumerous
with E" we do NOT need to invoke "Every infinite set is Dedekind
infinite"; indeed my proof of "N equinumerous with E" makes no use
whatsoever of "Every infinite set is Dedekind infinite". (2) "Every
infinite set is Dedekind infinite" depends on the definition of
'infinite' and/or on some form of the axiom of choice. You will find
many textbooks of set theory that do NOT define 'infinite' as
'equinumerous with a proper subset of itself' and that only show "With
a choice axiom, it follows that every infinite set is Dedekind
infinite". So get this very clear, once and for all, it is NOT the
case that every textbook or set theorist defines 'infinite' as
'equinumerous with a proper subset of itself'.

NO ONE states that there are alternative axioms in which
N = 2E. Are YOU aware of anyone stating that?

(1) First, please stop using '=' that way. What you mean is card(N) =
2*card(E).

(2) It is a THEOREM of Z set theory that card(N) = 2*card(E). We don't
need alternative axioms for that, since we PROVE it FROM the axioms we
ALREADY HAVE.

I have been mentioning all along that we prove N = E from axioms.

Yes, you have, but I have stated all along that Cantor's proof is a
circular argument, which it is, because you know damn well that Cantor
believed that N = E is an unavoidable result of N being INFINITE, when
in fact, it requires an axiom that is NOT necessarily true for every
infinite set

I'm not addressing any proof that Cantor might or might not have given
nor what he might or might not have thought. Rather, I'm talking about
formal Z set theory. In formal Z set theory there is a proof that N is
equinumerous with E.

(Zeno's Dichotomy being the most comprehensible exception I
know of). I asked for a modern proof of N = E, and you gave one, no
doubt, and it uses an axiom which is both (1) logically equivalent to N
= E,

INCORRECT. There is no axiom of Z set theory that is logically
equivalent to "N equinumerous with E".

and (2) an accepted axiom in ZFC. Fair enough. But NO ONE, to my
knowledge, has ever stated, anywhere, that this axiom has alternatives
which can ALSO be used with the infinite set N, and which lead to
equally valid conclusions, such as N = 2E. Do YOU know of anyone who has
stated this???

(1) We don't need alternative axioms. In just about any set theory
textbook you will see a proof (or you can prove for yourself from
basics proven in such textbooks) that for any infinite cardinal k, we
have k = 2*k (where '*' stands for cardinal multiplication).

(2) The question of what alternative axioms we might devise is not
even at issue. By saying that Z set theory proves theorem T, of course
I don't claim that T might not be provable in some other theory or
that ~T is not provable in some other theory.

And if not, then is not the CORRECT implication that no
one even realizes that valid alternative axioms exist? And is that not
logically equivalent to the belief that there are no valid alternative
axioms, which in turn WOULD MEAN that N = E is something that DOES
necessarily follow from N being infinite, alone?

I never said, and nothing I've said remotely implies that I mean that
N is equinumerous with E siimply on account of N being infinite. I
simply gave you a proof from the axioms of set theory.

I won't ask you to give
me a reference to a paper that states that alternative axioms, leading
to results like N = 2E, exist, I'll just ask you if you have ever even
heard of one.

Look in nearly any textbook of set theory to see a proof (or see how
to prove) that for any infinite cardinal k, we have k = 2*k.

Be honest, Moe, if the axiom that if n is in N, then so is n + 1, is
just one of the valid axioms that can be used with infinite set N,

I used no such axiom. "If n is on N, then n+1 is in N" is a theorem
and NOT an axiom of Z set theory. You are probably conflating
different theories (some do have "If n is in N, then n+1 is in N" as
an axiom). I am using Z set theory, in which "If n is in N, then n+1
is in N" is a theorem and NOT an axiom.

then
the idea that N = E is like saying that the sum of the angles of a
triangle is 180 degrees, something that is true for one BRANCH of
infinite set mathematics, but which which is not "simply true." I have
NEVER seen anyone claim that N = E for one, but not all, sets of axioms,
and forgive me, but I don't believe for an instant that you have ever
seen that either. When EVERYONE states and believes that N = E, then NO
ONE believes that an alternative set of axioms, which cause the
statement N = 2E to be true, and the statement N = E to be FALSE,
exists. Do you understand? I accept that under your axioms, N = E. But
can you HONESTLY state that you have ever even heard of an alternative
set of axioms in which the claim, N = E is always false? And if the
answer is no, then can you HONESTLY state that anyone, to your
knowledge, believes that when N is an infinite set, and the right set of
ADDITIONAL axioms is used, that N NEVER equals E?

It is not at issue as to what are or are not theorems of various
alternative theories that can be stated. Rather, I simply gave you a
proof in Z set theory and claimed nothing more than, on the basis of
that proof, it is a theorem of Z set theory that N is equinumerous
with E.

Also, that every infinite set (in the sense of not being equinumeruous
with a natural number) is equinumerous with a proper subset of itself
(i.e. that every infinite set is Dedekind infinite) is also proven
from axioms (and the axiom of denumerable choice is used for this in
addition to just the axioms of ZF).

Neither you nor anyone else has proven,
using only the properties that are INHERENT TO ANY infinite set, that
any infinite set has the same cardinality as a proper subset of itself,

No, we prove "Every infinite set is Dedekind infinite" from axioms.

You skipped recognizing this. You need to understand it.

and THAT is what Cantor both tried and claimed to prove!

Cantor did not work in a formal set theory. We've come a long way
since Cantor.

You skipped recognizing this. You need to understand it.

Go ahead! Do
what no one else has done, and prove, WITHOUT using axioms that do NOT
necessarily follow from the mere fact that a set has infintely many
elements, that N = E!

Just to be clear here, a proof of "N = R" such as I mentioned makes no
use of any premise that every infinite set is equinumerous with a
proper subset of itself.

True, given the existence of different transfinite sets, it would have
been better for me to state that, given an infinite set X, that an
infinite proper subset of X exists that is equinumerous with X.

You missed the point again. My ("my" is ridiculous, since the proof is
just rudimentary) proof makes no use WHATSOEVER of any principle that
every infinite set is Dedekind infinite.

On the other hand, "N = E" entails that there
is at least one infinite set that is equinumerous with a proper subset
of itself; but "N = E" doesn't entail that every infinite set is a
proper subset of itself. Anyway, the proof I mentioned of "N = E" is
from axioms and those axioms do not include "Every infinite set is
equinumerous with a proper subset of itself" and those axioms do not
even entail "Every infinite set is equinumerous with a proper subset
of itself".

If one of those axioms is logically equivalent to "Every infinite set is
equinumerous with a proper subset of itself," then those proofs DO
include such an axiom,

And NONE of those axioms is equivalent to "Every infinite set is
equinumerous with a proper subset of itself". Moreover, we can't even
prove "Every infinite set is equinumerous with a proper subset of
itself" from the axioms I used. I ALREADY said that to prove "Every
infinite set is equinumerous with a proper subset of itself", where
'infinite' is defined 'as not equinumerous with a natural number', we
have to adopt a choice axiom; and my proof makes no use whatsoever of
any choice axiom.

YOU'RE NOT EVEN READING WHAT I WROTE.

Again, in my proof, there is no defintion of 'infinte' as
'equinumerous with a proper subset of itself' and there is no axiom
'every infinite set is equinumerous with a proper subset of itself'
and, in Z set theory with the definition of 'infnite' as 'not
equinumerous with a natural number' we can NOT even prove as a theorem
that every infinite set is equinumerous with itself.

and the axiom "if n is in N, then n + 1 is in N,"

NO SUCH AXIOM in Z set theory.

No, what we prove as a THEOREM from AXIOMS is that if n is a member of
the set of natural numbers then 2n is a member of the set of natural
numbers. Not just any set infinte set X, since we also prove that
there ARE infinite sets X such that n is in X but 2n is NOT in X.

Um, have I stated this badly, and misled you as to what I mean? I'm
trying to say that if there are n elements from 1 thru n, then there are
2n elements from 1 thru 2n. I am NOT trying to say that in all infinite
sets of numbers, that if an element with a VALUE of n is in a set, then
an element with a VALUE of 2n is also in the set. Does that help?

Yes, but it is but a drop in the bucket from you torrential downpours
of confusions.

Again, hopefully this is clear now. I was trying to say just what you
said, that if n is in N, then 2n is in N. That can either be an axiom,
or follow from some other axiom, probably that if n is in N, then n + 1
is in N. However, if an alternative to this axiom leads to the result
that N is NOT = E, then we can no longer claim that a property of any
infinite set X is that it is equinumerous with a proper subset of
itself, because that is then true ONLY if we choose a certain axiom set,
just as it is true that triangles have 180 degrees ONLY in one branch of
geometry.

No such axiom as "If n in N then n+1 in N" is in Z set theory. Rather,
"If n in N then n+1 in N" is a THEOREM proven FROM the axioms of Z set
theory.

And the proof that any infinite set X is equinumerous with a proper
subset of itself is a separate matter and is proven from axioms that
include some weak or full form of the axiom of choice.

But that would
mean that we could also choose an alternative axiom, whereupon this is
NOT true, and no one would be stupid enough to claim that N = E
But that would
mean that we could also choose an alternative axiom, whereupon this is
NOT true, and no one would be stupid enough to claim that N = E follows
simply because N is infinite.

It is consistent with ZF to have an axiom "It is not the case that
every infinite set is equinumerous with a proper subset of itself".
But it is INconsistent with Z (or with ZF) to have an axiom "No
infinite set is equinumerous with a proper subset of itself". And,
anyway, the proof of "N = E" is not very much related to that, since
"N = E" is provable in Z set theory alone, and the question of whether
every infinite set is Dedekind infinite does not affect the proof of
"N = E".

You are EXTREMELY mixed up about all of this. The only hope for you
have of being able talk coherently about this subject is to actually
study it while paying close attention to the PRECISE formulations in
those studies.

Can't resist, can you? EVERY point I made is "simply the result of
EXTREME confusion."

You are RIDICULOUS. Set theory and mathematics are TECHNICAL. There
are very precise, sometimes detailed and intricate, formulations that
need to be handled ACCURATELY. You have virtually no idea about these
technicalities at all yet you are slopping them up all over the place!

But I'm not mentioning that set theory is technical just to put you
off; I'm only mentioning it to alert you that you CAN master the
technicals and speak intelligently about them, but first you have to
STUDY them. But at this juncture, you are in a complete fog about all
of this, and one just can't speak coherently about a technical subject
that one is not familiar with even its rudiments.

I just don't understand what makes you think you can talk meaningfully
about these technical matters when you haven't even studied them at
the most rudimentary level.

MoeBlee

.

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