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

karl malbrain wrote:
"Phil" <toob-headman@xxxxxxxxxxxxx> wrote in message
news:464AB295.4060803@xxxxxxxxxxxxxxxx

Virgil wrote:

In article <Grc%h.820$SC4.190@xxxxxxxxxxxxxxxxxxxx>,
"Karl Malbrain" <malbrain@xxxxxxxxx> wrote:



"Virgil" <virgil@xxxxxxxxxxx> wrote in message
news:virgil-478616.14250805052007@xxxxxxxxxxxxxxxxxxxxxxxxxxx


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.

I'm afraid Phil has chosen to begin, not with axioms, but with his
properties, and apparently has a "proof system" in mind that is based

upon

them.

karl m


If Phil chooses to reject our axiom systems, like ZF for example, then
he has no right to quarrel with what can be deduced in those systems.

And until he comes up with a system in which all his own assumptions are
made as explicit as in such systems as ZF, we have no reason to accept
his rodomontades as anything but egotistical blustering.

I know this is a waste of time, since you KNOW, with godlike certainty,
that the current beliefs are all perfect and flawless, but "just for the
record..."

I am not rejecting ZF.

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


Yes, I don't think it can be done without the axiom of induction, which is
an axiom that is not implied directly by the set being infinite. What is
your problem with the axiom of induction?

karl m


I have no problem with the axiom of induction, or for that matter the axiom of succession. My "problem" is that everyone, including mathematicians, believes that it necessarily follows SOLELY from the fact that N is infinite, that N = E. That's what Cantor believed, and supposedly proved. Neither ZFC nor PA existed when Cantor published his conclusions about infinite sets, so Cantor was NOT saying that, given the axioms of ZFC (or PA), that N = E. For example, George claims that Galileo "proved" that because N is infinite, that N = sqrt(N), basically the same thing as N = E (although more extreme). That is true ONLY if you accept axioms such as, "if n is in N, then 2n is in N, and n + 1 is DEFINITELY in N." Jesse even accused me (indirectly) of being a complete *** and a liar, saying that I also believed that n + 1 was in N, but when I saw that this "fact" contradicted my arguments, that I then claimed otherwise (I should mention that I made my arguments about Cantor 4 years ago on the sci.physics.relativity group, prior to joining the logic group, so proof exists that Jesse's claim -- at least in this case -- is false). Do you see what I mean? The CLAIM is that once N is defined as containing infinitely many elements, that N = E, period, when in fact, the conclusion that N = E REQUIRES, in addition to N being infinite, an additional axiom that is logically equivalent to N = E, such as "if n is in N, then so is n + 1, and 2n."

Now, if alternative axioms are not compatible with N being infinite, THEN N does indeed equal E, and Cantor was correct. And if anyone has proven this, or does prove this, THEN they did/will prove Cantor correct. When asked what I didn't understand about "if n is a number, then so are n + 1 and 2n," a question which implies that this IS the only possible compatible axiom, I used Zeno's Dichotomy, which uses the same infinite set of halfway points from the middle of the room to the door that Zeno's Achilles uses, but crosses them in reverse, to show that after moving from the door to the 1/4 point, there is only one more halfway point remaining in the infinite set, the one in the middle of the room, after which another point in this infinite set does not exist. I also gave the example where, given the infinite set of points in the segment [0,1], the point corresponding to the position 0.75 is in that set, but 2 * 0.75 is not. These examples seem to indicate (not prove) that alternative axioms are both VALID, and COMPATIBLE with N being infinite. If that is indeed the case, then we have a situation similar to geometry, where it turned out that there were two (main) branches, the old, standard Euclidean geometry, and non-Euclidean geometry. In the case of infinite sets, it may indeed be the case that if we assume one set of VALID, COMPATIBLE axioms, then N = E, but that if we assume a different set of equally valid, equally compatible axioms, then N |= E.

Have YOU ever seen an article or a book that states that if N is infinite, that N may or may not have the same cardinality as E, depending on which additional axioms are used? I never have! Everything I've ever seen states that Cantor proved that infinite sets have very different rules than finite sets, and that E has just as many elements as N. Well, he DIDN'T prove that, because if we are only given that N is infinite, then N may or may NOT = E! If one chooses to also accept the axiom that if n is in N, then so is n + 1, okay, NOW it's true! But if there are sets in which the existence of n does NOT guarantee 2n, or n + 1, then n does NOT = E. I just got back from Hawaii, and haven't had time to read that much, but several posts appear to acknowledge that my use of the Dichotomy is valid, that it is possible to use alternative axioms in which the existence of n does NOT guarantee the existence of either 2n or n + 1, but they then add, "So what? So there's an alternative set of axioms in which N does NOT equal E, why should that matter?"

It matters for three reasons. First, it shows that I am not just a kook, and that I can see flaws in widely accepted claims. Even if some mathematician SOMEWHERE already figured this out, no one here appears to have been aware that Cantor's proof required the use of an arbitrary axiom, meaning an axiom with valid alternatives, similar to the various parallel postulates. I figured this out, and no one else here did. That doesn't mean I'm better than everyone here, but it does mean that people should acknowledge that I am intelligent, that I can think VERY well, and that my arguments on other matters should not be dismissed out of hand "because Phil is JUST a crank." I am NOT just a crank, I can see things that no one else here managed to see, and it's time people figured that out.

Second, it may well be the case that NO ONE, ANYWHERE, has figured out that Cantor's proof is actually just a circular argument, and in that case, I need to publish this, as it really is similar to the realization that a non-Euclidean geometry exists, and people here should be helping me to do so, not insulting me. I have not proven that the use of a set where n + 1 sometimes does NOT exist really is logically compatible with N being infinite, but the set of halfway points used in the Dichotomy is an infinite set, and clearly does have a last element, namely the point in the middle of the room, and 2 * 0.75 = 1.5, corresponding to a point that is NOT in the infinite set of points in the segment [0,1]. Also, the alternative is actually very similar to finite sets, giving N twice as many elements as E, so it is actually LESS "weird" than the current beliefs.

Third, it MAY actually be the case that the current beliefs, where if n is in N, then so is n + 1, are actually NOT compatible with true infinite sets! Now THAT would really be significant. Initially, of course, we should assume that the situation is similar to geometry, where we have two valid branches of "mathematics of infinite sets," but it MAY be that there is only one valid branch of infinite sets, and that in that branch, N = 2E. Hint, in the set of points used by the Dichotomy, as long as we cross a FINITE number of halfway points n, corresponding to an INFINITESIMAL distance into the room, then 2n exists, but as soon as we move any FINITE distance into the room, corresponding to an INFINITE number of halfway points n, with a FINITE number of halfway points remaining, then 2n does NOT exist. So, as long as we limit n to FINITE values, then the axiom, "if n, then 2n," is valid, but as soon as we deal with an INFINITE set of points corresponding to any finite distance into the room, well ... Think about it.

Phil
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