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Topic: fom - 01 - preface
Replies: 18   Last Post: Dec 12, 2012 3:34 PM

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Posts: 1,968
Registered: 12/4/12
Re: fom - 01 - preface
Posted: Dec 11, 2012 9:21 PM
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On 12/11/2012 12:55 AM, WM wrote:
> On 10 Dez., 21:03, fom <> wrote:
>> On 12/10/2012 11:57 AM, WM wrote:


>> Yes. He did. But, Cantor's notion of a real
>> number was clearly found in the completion of a
>> Cauchy space.

> That is completely irrelevant for the result.

>> He found that more appealing
>> than Dedekind cuts. This is evident since
>> his topological result of nested non-empty
>> closed sets in a complete space is closely
>> related.
>> There are ordinal numbers in set theory given
>> the names of natural numbers.

> Only those which are finite.

>> Find a different criticism of Alan's remarks
>> if you must. This one is incorrect.

> So you disagree that 2 is a real number?

Since you like quoting the Grundlagen, try
transcribing long detailed passages from
section 9 where Cantor rejects definitions
that conflate logical priority as you have
been doing.

That is where he calls his construction

"... a fundamental sequence and correlate
it with a number b, TO BE DEFINED THROUGH

and goes on to say,

"Care must be taken on this cardinal
point, whose significance can easily
be overlooked: in the third definition
(his definition of fundamental
sequence, fom) of the number b, say,
is not defined as the 'limit' of the
terms a_v of a fundamental sequence
(a_v); for this would be a logical
error similar to the one discussed
for the first definition, i.e. we
would be presuming the existence of
the limit lim(v=oo) a_v. But the
situation is rather the reverse."

Cantor does not address the full
nature of how real numbers are
constructed. At the beginning
of section 9 he writes,

"I shall not discuss the introduction
of the rational numbers, because
rigorous arithmetic presentations
of this have often been given; among
those with which I am most familiar
I emphasize those of H. Grassmann and
J.H.T. Mueller."

In section 11 he explains his
principles of generation wherein
2*omega arises. It is based on
the "repeated positing and uniting
of unities" which is a far cry
from his "fundamental sequences".

In section 14 he explains multiplication
of his number classes but defers
explanation of a form such as
2*omega as previously explained.
This explanation is in section 3

"If one takes a succession, determined
by a number beta, of various sets which
are similar and which are similarly ordered
and such that each has a count of its
elements equal to alpha, then one obtains
A NEW WELL-ORDERED SET whose corresponding
count supplies the definition for the
product beta*alpha, where beta is the
multiplier and alpha the multiplicand."

So, this time let me suggest
that you open a new top-level
post openly apologizing to Alan.

That is what civil people do when
they are wrong.

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