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

MoeBlee wrote:


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".

Very interesting! Although I am certain that either the axiom of choice does have alternatives, or it is being used, unconsciously, in ways that are equivalent to adding additional (unconsciously) axioms, because all the axioms I have seen that lead to an infinite set being Dedekind infinite ARE logically equivalent -- meaning, if axiom A, then B, and if axiom B, then A -- to N ~ E. Now, there might be an axiom from which it follows that N ~ E, but which is so encompassing that the reverse is not true, but I haven't seen it yet (although I have not studied "choice" from this perspective). BUt the fact remains that if you state that "if n is in N, then so is n + 1," then you DO HAVE an axiom that is logically equivalent to N ~ E, because by assuming the latter, I can now deduce the former.

> 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'.

I never believed everyone DEFINED infinity this way, I "merely" believe that no one I have seen uses, or is even aware of, an axiom set in which N is NEVER equinumerous with E (I'll call that "N |~ E," although I'm sure there's a better way to abbreviate that).


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).

Okay.


(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.

No, I mean an axiom set in which card(N) > card(E), and card(N) is NEVER, under any circumstances, = card(E). If, when n e N, (n + 1) is NOT necessarily in N (again, see the Dichotomy), then N |~ E. That is an example of such an alternative axiom.


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.

Well, I AM addressing what Cantor thought, and believed, and for that matter, what everyone today believes, whether they realize it or not. I realize how that sounds -- how can I know what others think? -- but I do have a specific sequence of thoughts on this matter, and so far, it appears to me that I have failed to communicate them to anyone. PERHAPS that is merely because there is an error in my thinking! Perhaps not. I will try to clarify things so that the answer will become clear to all.


(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".

As far as I can see, you are wrong on this point. Of course, there HAS to an "axiom X" from which N ~ E follows (well, a set of axioms, in which there is probably one key axiom), the only question being whether, given N ~ E, that axiom X then follows. Certainly, in the proof you gave me, you assumed that if n e N, then 2n e N, which IS logically equivalent to N ~ 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).

I definitely misspoke here. I did NOT want to claim that we COULD state that N ~ 2E, but rather that with a different BUT VALID AND LOGICALLY COMPATIBLE set of axioms, that N is NOT equinumerous with E, ever.

(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.

Here is where the subtle distinction comes into play. The phrase (triangles ~ 180) is my shorthand for, "the sum of the angles of a triangle = 180 degrees." We can say (triangles ~ 180) and add that

(1) "This is a property of Euclidean geometry, and is FALSE under non-Euclidean geometry," or we can state that

(2) "this is an inherent property of triangles, period."

Given the Euclidean postulate, it is true that (triangles ~ 180), but the second statement is false, because it is NOT an inherent property of triangles, but rather the consequence of an arbitrary axiom which has viable alternatives. Even amateurs like myself know that if anyone claims to have a "proof" that (triangles ~ 180) IN GENERAL, that they actually have a circular argument, one which merely ADDS a logical equivalent of (triangles ~ 180) to the other axioms, either through error, or in some subtle, hidden assumption. In contrast, an INHERENT property of triangles is that they have 3 sides, something that is true under ANY geometry. To summarize, although (triangles ~ 180) and triangles have 3 sides under Euclidean geometry, we must DISTINGUISH between those characteristics that are inherent properties of triangles, and those properties that are NOT inherent to triangles, but are instead merely inherent to some particular branch of geometry. Any "proof" that (triangles ~ 180) ASSUMES, somewhere, the axioms of Euclidean geometry, and it is imperative that we understand that this proof provides information about the axioms of Euclidean geometry, and NOT about the inherent properties of triangles. In other words, NO ONE claims that (triangles ~ 180) anymore, because we know better. Instead, we observe that the sum of the angles of a triangle does not have one specific answer, because the answer depends on which set of axioms we CHOOSE to use.

Equivalently, we CANNOT state that there exists a proper subset of infinite set X that is equinumerous with set X UNLESS this is truly an inherent property of infinite sets, otherwise we are taking a property that properly belongs to an arbitrary axiom set, and mistakenly attributing it to infinite sets in general. We can state that under Z set theory (ZF? ZFC?), which has some axiom that allows you to state that "if n e N, then 2n and (n + 1) are e N," that N ~ E, but that is then a property of Z, not infinite sets. UNLESS, of course, N ~ E REALLY IS an inherent property of infinite sets, but I have yet to see the key assumption in the proofs you have presented to me, namely that "if n e N, then 2n e N," is something which NECESSARILY FOLLOWS from the mere fact that N is infinite. And until someone DOES present such a proof, any claim that N ~ E is something that is true BECAUSE N is infinite is, in fact, just a circular argument, and an error of reasoning.


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.

Granted, but no one has ever, as far as I know, stated that there are alternative axiom sets in which N |~ E, and since N ~ E WOULD be an inherent property of infinite sets if no such alternative axiom set exists, and virtually everyone else I have read categorically states that it IS a property of infinite sets that N ~ E, I hope you can understand why I would assume that you are NOT the only other person in existence who thinks differently. I just got done looking up several sources, and they ALL state that IF a set is infinite, THEN it is equinumerous with a proper subset of itself. That is consistent with the belief that this is an inherent property of infinite sets, and it is inconsistent with the claim that N ~ E is merely a property of the axiom set we happen to be using with N. That is, i admit, a subtle distinction, but it IS a very important one, and I am NOT claiming that merely to win an argument! Look back at the equivalent situation for triangles and geometry, and you will have proof of how important that distinction really is.


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.

Very interesting! If it doesn't take too long, can you tell me which axiom in Z set theory leads to the conclusion that "if n e N, then (n + 1) e N." I think this should also cause 2n e N.


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.

I hope you understand now why that is irrelevant to the issues I am raising. I always knew it was proved from axioms. My claim was that it was NOT proved from the fact that set N is infinite, but rather was an axiom that was merely ADDED to the other axioms. Well, it did already exist -- at least, AFTER Cantor's proofs on the matter -- you're correct about that, but it did NOT follow from N being infinite.


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.

Again, I hope you understand why this is not, in fact, significant. The error I have in mind is NOT that there is no axiom leading to "if n e N, then 2n e N," but rather that there are ALTERNATIVES to that axiom, and because there are alternatives, the almost universal claim that N ~ E is a result of N being infinite -- and check if you like, but EVERYONE does state that -- is in fact an error of logic and reasoning, just as claiming that (triangles ~ 180) is an INHERENT property of triangles is an error.


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.

I think I addressed this point above, but for the record, your proof DOES make use of something that is logically equivalent to N ~ E.


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.

Now wait a minute! How did you prove that N ~ E if you had NOTHING that is logically equivalent to N ~ E! I've already noted that assuming that 2n e N whenever n e N IS LOGICALLY EQUIVALENT to assuming that there is a proper subset of N that is equinumerous with N. Maybe you used a choice axiom without realizing it?

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.

MAYBE I'm not understanding what you wrote. But I did read it, VERY carefully.

Again, in my proof, there is no defintion of 'infinte' as
'equinumerous with a proper subset of itself'

Well, I should hope not, since that's what you're proving! :-)

and there is no axiom
'every infinite set is equinumerous with a proper subset of itself'

Ditto.

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.

Well you obviously used SOME axiom that allowed you to conclude that if n e N, then 2n e N. Again, I hope you let me know what it is.


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.

Oh Lordy, my "torrential downpour" 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!

A nice general criticism without supporting examples -- but perhaps you are referring to the examples above, although again, I hope you now realize that we had some communication snafus that caused you to believe I was saying several things that I at least had no intention of saying!

>
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.

It's quite simple; the inherent logic is actually VERY simple, and I am, in fact, capable of seeing logical errors that DO EXIST in these extremely simply, at least by my standards, logical structures. The problem I am having is in understanding the terms and symbols, not the extremely simple (if often flawed) logical structures that make use of these terms and symbols.

To give what I feel is a very basic example, if we arrange the set of all (countable) infinite, finite, and infinitesimal numbers like so (where every infinitesimal number is defined as being equal to some finite number divided by a countably infinite number).

[infinite numbers]
....
[finite numbers]
....
[infinitesimal numbers]

I find it utterly obvious that, given an infinite number w, countably infinite and therefore a member of the top class, that it is an INHERENT property of the word "infinite" that:

(1) w times ANY finite number x is infinite (this can include finite fractions)

(2) ANY INFINITE COMBINATION, meaning "w many," of finite x's is infinite

(3) Any finite number x divided by w is infinitesimal

This follows from the only possible workable definition of "infinite." Invoking infinity, by definition, causes a TRANSITION from one class to another. Any number, quantity, whatever x that does NOT cause such a transition cannot possibly be infinite, since given finite a, we would have finite b such that x * a < b, and that would mean that x falls under the Archimedean postulate, and is therefore a finite number. And yet, you guys ACTUALLY BELIEVE that there are infinite sums of FINITE numbers which can be FINITE, thereby violating the above diagram, rules, and fundamental definition of "infinite." There is no finite number which can be added to itself infinitely many times, nor any sum of different finite numbers that can contain infinitely many numbers, without producing an infinite sum! And yet, this ABSOLUTE FACT goes right over everyone's head. That's why I KNOW FOR A FACT that I am qualified to discuss "these technical matters," even though I honestly don't know much about the SUBJECTS; I understand thinking, and in SOME ways -- not in all ways, but in some ways -- you guys don't. You have the brainpower, but you were never taught certain essential tools for making USE of your brainpower.

Phil


MoeBlee



.

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