Re: Existence of reals and observation of them

On Dec 11, 1:01 pm, LauLuna <laureanol...@xxxxxxxx> wrote:
On Dec 11, 8:06 am, cbr...@xxxxxxxxxxxxxxxxx wrote:



But I meant Cantor could simply not realize that a finite or countable
set of symbols can only yield a countable set of finite chains.

One the contrary; he /explicitly states/ in the passage that if every
real were finitely definable using a language with a countable number
of symbols, this would /contradict/ the theorem that the reals are
uncountable: it would imply that |R| = aleph_0. This is why he says
Konigs argument /must/ be wrong for /some/ reason; although his guess
as to /why/ it was wrong and what its relevance was to Konigs
argument was mistaken.

Cheers - Chas

Let me quote Cantor again:

[QUOTE] If Königs statement was "correct",
according to which all "finitely definable" real numbers form a
collection of cardinal number aleph_0, this would imply the
countability of the whole continuum; but this is obviously wrong.[END
OF QUOTE]

So, Cantor says that the set of all finitely definable reals is
uncountable.


Right; and that is supposed to follow from a claim of Konig's (see
translation below for numbered brackets).

Cantor is arguing that /Konig/ claims [2] he can partition R into two
disjoint sets F and I, F being the "finitely defined" reals, and I
being the "infinitely defined" reals. Konig's argument relies on his
claim that "the least element of I" is a finite definition;
contradiction.

But Cantor says that if we accept [2], then I must be empty; because
"infinitely defined" is a contradiction of terms like "a square
circle": so if [2] is to be taken as true, then there is /no/ real
number whose definition is an "infinite definition", and thus no
"least element of I" to worry about.

Thus, if Konig's claim [2] is true, then F = R; and every real has a
finite definition; therefore there are an uncountable number of finite
definitions since R is uncountable. But this conradicts the conclusion
we draw from [8] below about "finite definition" which says that F is
countable, so some assumption ([2], [8], or uncountability of R) must
be false: they can't /all/ be true.

There is a possible error here, but it's not clear whose it is. If the
justification of [2] were the LEM, the negation of "there is a finite
definition of x" is "there does not exist a finite definition of x",
and not "there exists an infinite definition of x". So some /other/
principle is being called into play here: "there is a definition of x,
which is either finite or infinite". But was it Konig who claimed
this, or was it Cantor, or was it both? I sure don't know!

I personally would reject [2] over [8]; but it is taken as true in
this argument.

This means he believed there was an uncountable set of finite
definitions of reals.

This is only possible if there is an uncountable set of different
symbols.

This can happen in no human language.

So far so good.


Agreed.

But if Cantor is not referring to a human language, what sort of
language is he referring to?


A language defined by an abstract /set/ of /terms/, not by words that
literally come out of a person's mouth or that they literally write
down on a piece of paper. The collection of all such latter entities
is at most finite, and bounded by the physical constraints of our
universe.

This is certainly /not/ a problem for a platonist: sets of terms exist
independently of minds "observing" them, or any other physical
constraint.

It's important that he never refers to an uncountable set of symbols
but to an uncountable set of concepts.


The word "Begriff(e)" seems to have multiple meanings; along with
"concept" and "notion", "term" is also a meaning (and all the online
translation engines I consulted translated it this way in this
context). This translation seems to be more in line with describing
sets of terms B1, B2, etc.; since concepts and notions are not
generally precise things, whereas terms are.

Could it not happen that he overlooked the fact that those concepts
have to be defined by means of a finite set of symbols so that he
committed a trivial error?


He didn't "overlook the fact" of it; he showed that if we remove the /
assumption/ that it /is/ a fact, we no longer have a /contradiction/.

As distasteful as it might be, /something's/ got to go. No matter how
much we might wish otherwise, [2] and uncountability and the
assumption [8] produce the contradiction [5].

I suggest he could have overlooked the finite nature of any human
language.


I think it's pretty clear that he isn't talking about any "human
language". He's talking about something more abstract.

As far as I can tell, the translation of the german text you posted
is:

-------------------

[1] I gladly would have spoken with you, among other things, about
Koenig's and Poincaré's attempts in set theory, which are based on
wrong principles in my opinion and are only suitable to arrange
confusion.

[2] Koenig wants to differentiate between two different kinds of real
numbers: those with "finite definitions" and those with "infinite
definitions".

[3] However, any definition must after all be finite; i.e., it
expresses a term using a finite number of already known terms B1, B2,
B3..., Bn. [4] "Infinite definitions" (not accomplished in finite
time) are absurdities.

[5] If Koenigs statement, according to which all "finitely definable"
real numbers form a collection of cardinal number \aleph_0, was
correct this would imply the countability of the whole continuum; but
this is obviously wrong.

[6] The question is now what fallacy the alleged proof of his wrong
theorem is based upon.

[7] The fallacy (which also appears in the note of a Mr. Richard in
the last issue of the Acta mathematica, which Mr. Poincaré emphasizes
in the last issue of the Revue de Métaphysique et de Morale) is, it
seems to me, the following:

[8] It is assumed that the system {B} of terms B, which have to be
used for the definition of individual real numbers, is at most
countably infinite.

[9] This assumption must be the fallacy, because otherwise we would
have the wrong theorem: "the continuum of numbers has cardinality
\aleph_0".

[10] Do I err, or am I right?

----------------------

In the above, Cantor didn't "overlook" that natural languages have at
most a finite number of terms; nor did he "forget" that a countable
language can produce at most a countable number of definitions; nor
did he propose that there is a "humanly manageable" way to construct
all sets, nor did he make the "trivial error" that natural languages
are not uncountable.

He /questioned/ whether the assumption [8] that "defined" = "defined
over a set of terms which is at most countable" was a /logically
consistent/ assumption, in the context of the proof of the
uncountability of the reals and Konig's claim of [2].

Regardless of how surprising and non-intuitive it might be, Cantor
proposes that the resolution to the paradox is that we /must/ accept
that definitions /can/ be composed of terms from an uncountable set of
such terms; and then [2] does not result in a contradiction. But then,
paradoxes /often/ have surprising and non-intuitive resolutions; that
is the nature of mathematics.

Did he err, or was he right? He erred, but not because he "forgot"
that the sentences of a countable language are at most countable. And
to give him credit, he did identify that the issue has to do with
examining the meaning of "finite definition".

Cheers - Chas

.

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