Cantor's theorem:

Scott, if you really are sincere about this subject, then please
consider this very detailed post:

You wrote:

Okay, I'm an idiot in writing proofs.

Writing proofs is the LEAST of your problems. And that is NOT a
gratuitous insult, and I'll say more about that later in this post.

I'm going to indulge you yet again by commenting on your latest
"proof".

First, just for keeping on file, proof of Cantor's theorem:

Cantor's theorem: For all S, S is stricly dominated by PS.
(Where 'P' stands for 'the power set of'.)

Proof:

g(s) = {s} is an injection from S into PS. So S is dominated by PS.

Now suppose f: S -> PS. Let D = {x | x in S & ~ x in f(x)}.

If there is an s in S such that f(s) = D, then, by plugging-in s for x
in the definition of D, we get (s in D <-> ~ s in D).

So there is no s such that f(s) = D. So D is not in range(f). So f is
not onto PS. So S is strictly dominated by PS.

So every set is strictly dominated by its power set.

Now the first lines of your argument:

Proposition: For every non-empty set S, for every function f:S->P(S),
there exists D_f subset of S such that there exists s in S, f(s)=D_f.

As Patricia mentioned, given your previous formulations, it is not
clear whether you intend for D to be fixed as the diagonal subset per
f, or whether it is, per f (whatever that means this time) just any
subset of S. It wouldn't make much sense for you to be struggling to
convince us of the latter, since it is trivially known to us, so I take
it that you mean the former. So here's a correct way to say it:

Claim: For any non-empty S and any f:S -> PS, there exists an s in S
such that f(s) = D.
(Where D = {x in S | ~ x in f(x)}.)

Proof: Let S and f be given as in the statement. The set

D_f = { x in S : x notin f(x) }

exists by Separation and by the Axiom of the Power Sets is an element
of P(S).

Just call it 'D' like I did. And, yes, it does exist by the axiom
schema of separation and it is a member of PS. But don't mention the
power set axiom. That is not relevent since we've already granted the
existence of PS by the power set axiom. You don't need the power set
axiom to assert that one set is a subset of another set. Proving that
one set is a subset of another hardly ever has anything to do with the
power set axiom, which is an axiom to prove the EXISTENCE of the SET of
all subsets of a set, not to prove that a certain set is a subset of
another set. So skip everything else you said in this part and just
say:

D subset_of PS.

To prove that "exists s in S, f(s)=D_f", let s in S. Then (s in D_f) or
(s notin D_f) by the excluded middle.

Okay. But, we all know excluded middle, so all you need to say is:

Suppose s in S.

Assume (s in D_f). Then (s in D_f), so by definition of D_f, (f(s)=D_f)

WRONG. VERY VERY VERY WRONG. At this point, your "proof" is over. It's
wrong. End of "proof".

If s in D, then it DOES NOT follow that f(s) = D. The claim that it
does follow is pulled by you from right out of NOWHERE. It's just
another non-sequitur you're throwing up.

In fact, at this point, it's time to move over, Rover, and CANTOR take
over:

What YOU need to SHOW is that, for some s in S, D = f(s). You didn't do
that. You just CLAIMED it. What CANTOR doesn't just claim, but SHOWS is
that, for NO s in S does D = f(s). Suppose, as you mistakenly do, that
for some s in S, f(s) = D. Then, like you said, law of excluded middle:
Either s in D or s not in D. But just since you happened to mention the
definition of D, let's use that definition by plugging-in s for x.
Oops, since D = f(s), if s in D, then s not in D; and if s not in D,
then s in D. So, oops, the assumption that there is an s in S such that
D = f(s) gave a contradiction. Thus, there is NO s in S such that D =
f(s).

For a while I really did think you were putting on a hoax. But now that
you've taken the trouble to write so much notation, I'm wondering
whether you might actually be trying to come up with a proof. I'm going
to take it on faith that you are. I spent a good amount of time last
night and today thinking about this thread. I very much hope that you
will give my posts here some real thought, and not just blow by them in
your rush to vindicate your conviction about Cantor's theorem.

My point is not to make a fool of you. I really do want to help you
understand what you're doing, though in trying to do that I am almost
bound to lose patience with your stubborness and foolishness. And that
is not a comment on your intelligence (for all I know, you're smarter
than I am) but rather on the fact that you have terribly misguided
yourself and, whatever your intelligence, you are acting very unwisely.

What you are doing is, intellectually, very wrong. I am not saying that
it is wrong to criticize set theory or to wonder about finding an
ingenious proof about it, as long as you at least know what set theory
IS. But what is wrong is to have a PRE-conception that you can prove
the negation of Cantor's theorem when you don't even know what set
theory IS. What is worse, you know virtually nothing about the formal
logic that is the setting of set theory and you know virtually nothing
about mathematical proof; indeed, you don't even know what mathematical
proof itself IS.

It would help to know what it is your really want to accomplish. Some
of the questions I have may seem insulting, but (at least for now, at
least for the duration of this post and before I lose patience again) I
don't intend insult but instead to be frank as is much needed here, all
in a hope that I can understand the context of what you're doing, and
possibly, just possibly, to help you understand something about this
subject (even though I am at a beginning level myself, I have made it a
point to make sure I have a very very solid grasp of the basics of
working in first order predicate calculus and doing proofs in set
theory at a level that at least includes the level of this thread). So:

1. WHY do you want to prove what you think you can prove? What
motivates you?

2. Where did you hear about Cantor's theorem and what made you feel
that you have a basis to contest it?.

3. You mentioned a text you have on set theory. What text is it?

4. THIS IS THE MOST IMPORTANT QUESTION YOU SHOULD BE ASKING YOURSELF:
What are you most interested in? Is your real motivation to UNDERSTAND
something or is it to make some point that you think the mathematical
world needs to hear?

From your point of view, the best thing would be to both understand the
subject and accomplish your goal of proving the negation of Cantor's
theorem. But you won't prove the negation of Cantor's theorem. You
might as well hope to walk out of your house tommorrow morning and, by
sheer power of your will, make the sun revolve around the earth. This
is not one of those "you can achieve anything if you set your mind to
it" kind of things. And you blew right past that point with your
insultingly glib reply to my analogy of someone randomly tearing apart
the engine of a race car and leaving the parts in a heap thinking that
that would make the car run thousands of times faster. And even IF you
could prove the negation of Cantor's theorem (which you can't, but IF),
the way to even THINK about doing that is not just to keep throwing up
a bunch of half-thought notation. If you're serious, the way to do
approach your goal is to learn how set theory works. Back to analogies,
the Wright brothers didn't make the plane fly without having studied
the mechanics of the problem and the properties of flight. But you're
like one of those guys who thinks he can fly just by running off a
cliff flapping his arms.

And this leads to why some people who have tried to help you have
gotten sore at you. You're wasting your time and the time of anyone who
wants to help you because - instead of trying to UNDERSTAND this
subject even if only through the responses that are written for your
benefit - you just keep throwing up more junk notation. That's not the
way to learn a subject that is built systematically like mathematical
logic and set theory. It's just inefficient and UGLY to try to prove
things by just throwing up one ill-conceived idea after another. Your
approach is the million monkeys typing approach: Sure they MIGHT type
the collected works of Shakespeare in a year, but, probably not, eh?
Sure, you MIGHT defy odds of a trillion to one (or whatever) by
throwing up a bunch of ill-conceived formulas, but it's insulting to
people who actually THINK in this subject of set theory.- not like you,
not like a monkey tearing apart the wiring of a race car.

And, as I said, the problem is not the notation, but rather that you
don't understand the MEANINGS here and how set theory WORKS and even
what a mathematical proof IS.

Also, when Arturo Magidin asked you to respond more to his points, you
answered that you don't need to, because you're moving on while having
agreed in your own mind with those points. But very often it helps to
post that you recognize that the other person is correct, point by
point, as that lets the person know that communication is being
achieved and that we can trust that these are then points that are
indeed agreed upon.

/

In the meantime, I IMPLORE you to get a book on basic predicate
calculus and to learn it by working the exercises. My own
recommendation is 'Logic: Techniques Of Formal Reasoning' by Kalish,
Montague, and Mar, which requires no previous background in logic nor
mathematics.

And in the meantime, in my remarks, consider that I alway include a
tacit assumption that set theory is consistent. The consistency of set
theory means that there is no formula such that both it and its
negation are theorems of set theory; i.e., there does not exist a proof
of a contradiction from the axioms of set theory. I do not prove that
set theory is consistent; but rather, as I said, I assume it. For me to
even broach a discussion of what would be involved in a proof of the
consitency of set theory would require that you know much more about
the set theory than you know, especially since you know virtually
nothing; and proving the consistency of set theory is even beside the
point, especially since I don't claim to prove it. However, just for
context, you should understand that Zermelo set theory is just about a
hundred years old now and has been examined inside out, upside down,
and everywhich way to Sunday, in excuciating, even truly incredible
detail and elaboration, by thousands and thousands and thousands of
mathematicians (and even with computers), from beginners to the very
most advanced, and by by both supporters and detractors of set theory -
and no contradiction has been found. So while you think you're going to
achieve the impossible dream or something like that, you should at
least not take glibly what is involved in it.

And unless you can give a different formulation of your goal,
mathematically speaking, whether you even understand this or not, we
must take you as claiming that you are attempting to derive the
negation of Cantor's theorem from the axioms of ZFC (or ZFCR, if you
want to add the axiom of regularity). And if that is to be
mathematically meaningful, it requires using only first order predicate
logic and the axioms of ZFCR in a rendering of a proof that ends with a
formula and its negation. That is what you must do to prove your claim.

And to be precise about that, below I'm posting a formulation of first
order logic and of ZFCR. Of course, proofs don't have to be formalized,
as usually just lucid mathematical reasoning is good enough. But all
proofs must, in principle, be formalizable. Below is is just one of
many equivalent formulations (it is probably not even the easiest to
work with, but it is concise and my point here is just to enter it for
the record so that we are at least clear that we can appeal to at least
one certain formulation):

The primitive predicates:

=
Read as: is equal to.

e
Read as: is in.

The logical axioms are all and only those sentences that are
generalizations of:

Any tautology.

AxP -> P[t|x]
(Where P is a formula and t is a term free for x in P.)
Read as: If for all x, P is true of x, then P is true of t in
particular.

Ax(P -> Q) -> (AxP -> AxQ)
(Where P and Q are formulas.)
Read as: If for all x, P is true of x implies that Q is true of x,
then, if for all x, P is true of x, then for all x, Q is true of x.

P -> AxP
(Where P is a formula in which x does not occur free.)
Read as: If P (which doesn't mention x) is true, then for all x, P is
true.

x=x

(B & x = y) -> B'
(Where B is an atomic formula, and B' is just like B except zero or
more occurrences of x have been replaced by y.)
Read as: If B is true of x and x = y, then B is true of y.

The rule of inference:

From P and P -> Q, infer Q.

The axioms of ZFCR (ZF plus choice plus regularity) are all and only
those sentences that are generalizations of:

Extensionality
Az(z in x <-> z in y) -> x=y.
Informally: If x and y have the same members, then x and y are equal.

Replacement (schema)
Aj(j in z -> AvAw((P[v|y] & P[w|y]) -> v=w)) -> EbAy(y in b <-> Ej(j in
z & P))
(Where P is a formula in which b does not occur free.)
Informally: Given any set z and a "class operation" on z, there exists
the range of the operation (i.e., there exists the image of z under the
operation). The formulation looks complicated (it may even look
incorrect at first glance) but it boils down to nothing more
complicated than as just described informally, except that the formula
that "defines" the operation may not have the free variable b or
whatever variable is used to designate the "range", lest there be
"circularity" that causes inconsistency.

Power Set
ExAy(y in x <-> y subset z)
Informally: For any set z, there exists the set whose members are all
and only the subsets of z.
('subset' is previously defined: y subset z <-> Ax(xey -> xez))

Union
ExAy(y in x<-> Ez(z in c & y in z))
Informally, for any set c, there exists the set whose members are all
and only those sets that are members of members of c.

Infinity
Ew(0 in w & An(n in w -> nu{n} in w))
Informally, there is a set that has 0 as a member and is closed under
the successor operation.
(0 is previously defined as the unique c such that Ay ~yec, which
unique set is proven to exist. 'u' is previously defined as the binary
union operation, which is proven to apply to any two sets. {} is
previously defined as the singleton operation, which is proven to apply
to any set.)

Choice
Ab(b in T -> ~ b = 0) -> Ef(f is a function & dom(f)=T & Ac(c in T ->
f(c) in c))
Informally: If no member of T is the empty set, then there exists a
function whose domain is T and such that for all c in T, f(c) is in c.
('function' is previously defined in the usual way. 'domain' is
previously defined in the usual way and proven to exist for any
function. parentheses as functional notation is previously defined in
the usual way.)

Regularity
~ x = 0 -> Em(m in x & x/\m = 0)
Informally: Any non-empty set x has a member m such that no member of x

is a member of m.
(/\ is previously defined as the binary intersection operation, which
is proven to apply to any two sets.)

Note:We use the term 'set' as if it is presupposed that all objects of
the theory are sets. But if 'set' is defined: x is a set <-> x = 0 v Ey
x in y, then it is provable that all objects in the theory are sets.

MoeBlee

.

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