Re: infinitely many nn's = infinite nn's?

On Apr 2, 2:56 pm, Phil <toob-head...@xxxxxxxxxxxxx> wrote:
MoeBlee wrote:
On Mar 31, 3:11 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote to The
Ghost In The Machine:

I must emphasize that I'm not blaming YOU here, or accusing you of
anything, but you ARE a human being, so these facts and potential
oversights MUST apply to you as much as they do to the rest of us.

Meanwhile, you asked for a proof of a certain theorem, and I gave it
to you. Since then you've just gone on and on, most stubbornly, about
mathematics you've not even studied. Now I really have to doubt the
point of supplying proofs for your review when you won't even pick up
a basic book on the subject.

MoeBlee

I checked out four books today,

Great! I very much recommend picking one you like and working with it
proof by proof, exercise by exercise.

and I have a few already that review the
subject, albeit in less detail, which I have been examining. And as for
my being stubborn, I would not keep hammering at points IF I could see
that they were being fairly, and INTELLIGENTLY, addressed! Now I really
did appreciate your proof, but it did NOT work from the given premise of
infinitely many elements, and therefore was not a valid response.

What? A proof is not invalidated for NOT using a premise. I'm sorry,
but you aren't even close to having any grasp on what a mathematical
proof IS or how mathematical proofs work.

Why I
should have to point out such basic rules of logic and reasoning to
people who at least know far more facts than I do is something that
gives me genuine fear about our system of education.

You have no IDEA as to basic logic for mathematics.

You have not had
that many responses, and I consider you to be a straight-up guy, but if
you look, you will see, time after time, that my questions about whether
certain claims are all using compatible premises, are "answered" by
simply USING the premises to first prove one claim, and then to prove
another claim. They will NOT stop to SERIOUSLY examine the premises
themselves!

The ultimate assumptions are the axioms - the logical axioms and the
specfific set theoretic axioms, and one might say that each inference
rule is, broadly speaking, a kind of assumption. OF COURSE I have
thought a lot about those axioms and rules, as have most of the people
you are talking with in this thread.

IF there are infinitely natural numbers, then by the most basic
definitions of actual infinity,

We don't use YOUR "basic definition". The set theoretic definitions
though are ones you can find for yourself in those books.

there are INFINITELY many numbers which
are preceded by infinitely many other numbers

If by 'number' there you mean 'natural number', then what you just
said is YOUR assumption. It is NOT shown to be entailed by the axioms
of set theory, and it IS shown to contradict the axioms of set theory,
as, in the standard ordering of natural numbers, there is NO natural
number preceded by infinitely many natural numbers.

(if w is preceded by
infinitely many numbers, then so is w-1, w-2, etc.).

If 'w' stands for omega, the set of natural numbers, then you have not
DEFINED 'w-1', 'w-2', etc. In an axiomatic mathematics, we don't just
throw symbols together with an assumption that they properly refer.
Rather, first we prove a uniqueness and existence theorem upon which
we give a defintion that satisfies the criteria of eliminability and
non-creativity.

On the other hand, if 'w' is a variable ranging over integers, then
yes, every integer is preceded by infinitely many integers, but only
finitely many of the predecessors are natural numbers.

There HAS to be not
just ONE number, but INFINITELY many numbers, on the natural number
line, and in the set of natural numbers, that are preceded by infinitely
many numbers.

If by 'number' you mean 'natural number', then what you just said is
PURELY YOUR ASSUMPTION.

But this would mean, by the current beliefs, that there
are numbers which are both FINITE, and preceded by INFINITELY many other
numbers, and which therefore are NOT equal to the number of numbers
preceding them, which is impossible, as induction quickly proves. And
what is the typical answer? Ones like yours, where they "prove" that
there is no such thing as a number which is preceded by infinitely many
numbers, without ever bothering to ask how this can possibly BE
CONSISTENT with proofs which show that there ARE infinitely many
numbers.

I don't need to ask how it can be consistent for two reasons: (1) I
see that any Peano system is a model and (2) There is not even a hint
of a contradiction (a sentence and its negation both provable) shown.

You guys act like the existence of two proofs somehow "proves"
that the premises behind those proofs MUST be compatible. And that's
"good" reasoning???

WRONG. I "act" in no such manner. That two statements are both
provable is NOT taken as any kind of proof at all that the statements
are consistent with one another. You just have NO IDEA what is
involved in mathematical proof.

How about the one where I take an "infinite number" w (technically a
nonstandard infinite positive integer)

"TECHNICALLY" in WHAT theory?

which equals the number of
elements in the sequence S = {1/2, 1/4, 1/8, ...}, a sequence which sums
to 1,

The sequence doesn't sum to 1. The series based on the sequence sums
to 1. That is, the limit of the sequence of sums is 1.

use it to produce an infinitesimal g such that g = 1/w.

In what theory is this?

Therefore,
the infinite sum X = {g + g + g + ...}, where there are exactly w g's,
also sums to 1.

In what theory is this?

This gives us two infinite sums with equally many
elements (note that we CAN make a bijection between the two sets), both
of which sum to 1, in which EVERY element of S is INFINITELY greater
than EVERY element in X, all of which are the infinitesimal g, and yet
they sum to the same result. That's bad enough, but by removing the
first two elements from S and X, we now how two infinite sums, in which
EVERY element of S is INFINITELY greater than EVERY element in X, and
yet X sums to 1, and S sums to 1/4. Does that not sound like just a bit
a contradiction to you???

I don't know; I have no idea what theory you're talking about. If this
is supposed to be some kind of contradiction in set theory, then you'd
have to show me proper definitions and derivations from the axioms of
set theory.

Does that maybe make you wonder, just a bit,
about whether there MIGHT BE a problem with either the premises
currently being used in the natural numbers, or at the very least, with
the way in which mathematicians are using those premises?

Why should it? Your arguments are about your own informal way of using
mathematical terminology and are not defintions and proofs from the
set theory axioms. (Yes, there is non-standard analysis, which is not
shown to contradict set theory or mathematical logic, and is derived
from them, and which does not, as far as I know, lead to such
contradictions as you imagine.)

I doubt it,
and that's not a good sign, either for you, or for our system of
education. And note that the reasoning here is SO BASIC, that neither I,
nor anyone else, needs to have an advanced degree in mathematics in
order to THINK FOR OURSELVES about whether or not there MIGHT BE a problem.

Your reasoning is so basic as to not be reasoning at all. But please
do let me know when you have explicated a system of logic and
mathematical axioms such that, as with set theory, there is a
mechanical procedure to determine whether a formula is or is not an
axiom and whether a sequence of formulas is or is not a proof.

And notice one other thing: although anyone can come up with bull***,
purely dialectic objections to ANY proof, suppose, just for the sake of
mathematical investigation, that the contradictions just listed really
are contradictions, really do indicate a problem. If you take a set of
infinitely many numbers, infinitely many of which are infinite (leaving
potentially infinitely many finite numbers), then these contradictions
vanish;

Of course you don't have to worry about formal contradictions if you
don't (as you indeed you don't) even have a formal system to begin
with.

they have no power at all over such a set. Similarly, given a
set with potentially infinitely many numbers, all of which are finite,
you again find that these contradictions are suddenly helpless,
powerless. Why would that be? One explanation is that in the two sets
just listed, we are NOT using incompatible premises, nor are we using
them incorrectly! Is this a proof that something is wrong with the
current premises and/or the way in which the current premises are being
used? No, but it REALLY should give you pause, and make you want to
investigate further ...

*I* should investigate further? Let's see, I scrutinize every informal
mathematical proof I read to see that indeed it can be formalized by
the particluar system of logic and axioms (Z set theory or whatever
variants) that have a mechanical procedure for checking adherence to
well formedness and proof formation. And I study the proofs that show
that the logical system is sound and has certain other properties. And
I study the history, mathematics, and philosophy about paradoxes in
general and contradiction in certain of systems. And, as much as my
budget of time permits, I study alternative logical systems and
alternative mathematical axiomatizations, even to the point of wishing
to better understand such alternatives as relevence logics, free
logics, and even paraconsistent logic. And, as much as my budget of
time permits, I read philosphical commentary from a wide range of
viewpoints, including strong dissent from the classical mathematics
and set theory. And I am wholly frustrated and unhappy with the fact
that at such an amateur and beginning level of involvement as I have,
I am not even within several thousand hours of the study that it would
take for me to have the kind of scope of knowledge and understanding
that I could claim to be even moderately adequate for a mathematical
dilettante, let alone for both mathematics and the philosophy of
mathematics.

Meanwhile, you are about to read a textbook on the subject for the
first time, meanwhile as you ALREADY have been ignorantly spouting
paragraph upon paragraph upon paragraph of pure dogmatic confusion.

Yet, relatively, you choose to say that *I* am the one in dire need of
broadened perspective.

Annoyed Phil

Sheesh.

MoeBlee

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