Re: Model and straight line

"smn" <smnewberger@xxxxxxxxxxx> schrieb im Newsbeitrag
news:eb3f2c39-7e14-4450-a2f1-58386257ff25@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

"Dominijanni Simone" <simoned...@xxxxxxxxxxxxx> wrote in message
news:96b2235e-b97f-4557-bd07-232ce538d714@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Hi. I know that SL (straight line) is a elementary object and that
there is a bijection from R (real number) to SL.

This is only true if real numbers are overly acurate rational numbers as set
theory demands.

In school when the professor speak about of the SL says that
MODEL: SL is an infinite number of points in a row
This model makes the idea of a SL without holes.

The notion of infinity in set theory deviates from the original and clearly
understandable meaning.

If this is a right model of the SL, then for every point of the SL
exists "the successor point".

Every point of the SL? What arrogant ignorance! I do not deny that there are
reasonable statements using the expression every number, for instance: The
successor of an even number is definitely an odd one. However, in order to
leave the world of genuine (non-Hausdorff) continuum and enter the quite
different world of numbers, one has to either deliberately or tacitly
neglect what Cantor himself called an abyss. Uncountables are losing their
feature when they get approximated by countables.

To those teachers who are having problems to answer frequently asked
questions like yours, I recommend to be honest and understandable:
- Several mathematicians introduced approximative notions of continuity and
infinity.
- This arithmetisation is still considered inevitable in order to use
numbers in case of non-linear functions.
- Admittedly some tenets that are obviously wrong from the original, i.e.,
the pre-Dedekind point of view, are easily to swallow even for intelligent
people if one explains them in a winking manner. Infinity of modern
mathematics can be easily understood if one considers it like a huge
infinitely expandable bag.

[Just a comment to the teacher: It does not matter that Cantor denied this
bag-model. Well, he was able to think as if he was schizophantastic, eating
the cake and still have it. Tragically he himself believed in his
transfinite whole numbers.]

- Get calm. The best recommendation on can give to students is to learn
pertaining analysis like a text from bible. The axiomatic method detracts
from some otherwise obvious discrepancies between the immediately
understandable original notions and sophisticated, arbitrarily twisted
mathematical counterparts.
- As a rule, it is not wrong to treat variables that denote continuous and
infinite quantities as if they were really exact, i.e., (un)real and at a
time also rational.
- However do not believe that the belief of the mathematicians in what
Hilbert called "gewisse Zusammenhaenge" (= certain tenets) always fits to
more comprehensive and in particular physical relations.

Return to you: You are quite right: Without an end of the cue there is no
successor. Forget AC.
If one is able and ready to think consequently, then one gets aware that the
thinking by Dedekind as well as Cantor and their fellows is based on
admittedly missing or incorrect definitions and evidences.
Even Ebbinghaus did not only reiterate the naive theory, he also did not
comment on his quotation of Lessing which referred to an obvious but
valuable error.
I do not share the opinion that set theory is valuable. I feel it an
inconsequent or maybe dishonest cover of the impossibility to subordinate
the world of the ideal mathematical continuum to the likewise ideal world of
discrete numbers. What would not work without the alephs? Ask physicists
whether they are using reals or rationals. They will not agree on that. When
I asked mathematicians how to deal with zero when splitting IR into IR+ and
IR- I got as much answers as possiblities. None was free of arbitrariness.
I had to find a unique and compelling answer myself.


then, considering that exists "the successor point" of 0 (that I
denote with 0'),
0' = 0. infinite 0 and as "last digit" 1 (0.0..01 , .. = endless
zeros ).

How to distinguish it?

But I know that 0' isn't a real number because between two real
numbers there are infinite real numbers.
But, considering that 0' isn't a real number and the bijection from R
to SL,
doesn't exist the successor point in the SL of point identified with
0.
But through this reasoning I deduce that in the SL there are infinite
holes.

Stop mocking.

Both the sets of real numbers and natural numbers are totally ordered
for
the relation "less or equal than", meaning that for any pair of numbers
in
the set you can say if one is less or equal than the other.

You can say and believe it. For the "numbers" of a genuine continuum, this
is all you can do. Trichotomy ceases to be valid across the abyss.

This may lead to expecting that you can place all the numbers of the
set in
order (which you can) and go from one number to the next, which you can
for
natural numbers, but not for reals. Between any pair {a, a+e} there's
an
infinite number of other reals, like a+e/2. You'll always find numbers
between a number and a candidate successor. So no immediate successor
exists. But that doesn't mean there are "gaps".

Without gap no successor and vice versa, discrete or continuous, tertium non
datur.


Note that the set {1, 2, 3, 4, 10, 11, 12, 13} with the relation "less
than
or equal" doesn't have gaps either!

In the context of a line there are gaps between all elements, even 1 and
two.

The principle problem that I have, it consists in the affirmation:
there's a bijection from R to SL

If R is merely quasi-continuous, i.e., continuous in the sense of set
theory,
why not twisting the notion of a line accordingly?


Mathematics (derived from set theory and logic) can't prove that your
wire has no holes -its an atomic physics problem .R is a mathematical
object (a set with further structure which can be constructed to from
set theory in such a way as to have no holes (assuming the positive
integers exist (a set with a successor function satisfying th Peano
axioms) .To talk about the bijiction between R and and the wire you
really have to model the wire in set theory .That is done as part of
foundations of Euclidean Geometry (See Forder's book -Dover ) and
there it is specifically assumed in the theory (after 100 pages ) that
the model wire has no holes .If you were doing atomic Physics it is
probably wrong but for usual uses of wires the assumption is useful
and accepted (like in studying music on a voilin.
I am responding because I noticed that no one really saw what your
question was .i hope this helps. smn

Could you please also offer help to J. v. Neumann who in 1932 introduced
Hilbert space and already in 1935 admitted he did no longer believe in
Hilbert space?

Salviati:
... in ultima conclusione, gli attributi di eguale
maggiore e minore non aver luogo ne gl'infiniti,
ma solo nelle quantità terminate.
IR>|>IR+>|>IR


.

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