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Topic: § 534 Finis
Replies: 3   Last Post: Aug 18, 2014 1:41 PM

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Ben Bacarisse

Posts: 1,388
Registered: 7/4/07
Re: § 534 Finis
Posted: Aug 17, 2014 11:33 PM
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mueckenh@rz.fh-augsburg.de writes:

> On Saturday, 16 August 2014 23:55:02 UTC+2, Ben Bacarisse wrote:
>

>> Yes, that's the one. Do you think the definition of a set sequence
>> limit is wrong?

>
> No, but this definition does not apply to a final set.


What is a final set?

> If you believe
> in Cantor's set of all algebraic numbers, then you believe in a final
> set.


What is a final set?

>> > Set theory names the final set of all naturals omega.
>
>> Don't try to change the subject by saying something else wrong in the
>> hope that I'll chase down another rabbit hole. Set theory says that the
>> limit of the set sequence s_n is {} and that the limit of the |s_n| is oo.

>
> Obviously this would be a contradiction, if both apply to the same
> set.


Yes. Fortunately, they don't.

>> You find this puzzling for some reason.
>
> Do you think that the cardinality of an empty set can be infinite?


No, of course it don't, but the question does illustrate your
misunderstanding very clearly. Do you think the cardinality of the
limit set should be the limit of the cardinalities? Why are you so
puzzled by the consequences of a simple definition? Is it the word
"limit" that is confusing you? If it where called the set of "almost
all members" and denoted by AAM s_n would you be arguing that |AAM s_n|
= 0 and lim |s_n| = oo is contradictory?

>> Your claim: "the sequence of
>> cardinalities should have limit 0 if set theory was right" shows a
>> misunderstanding of what set theory says.

>
> If lim |s_n| = oo is the cardinality of lim s_n = { }, this would be a
> contradiction (in my opinion). Therefore this cannot be true (in my
> opinion).


It would indeed be very odd, and you should stop saying that set theory
says any such a thing -- particularly to students -- because it does
not.

>> > It is also Cantor's misunderstanding.
>>
>> You are the historian, so I won't argue. If you know that Cantor
>> thought that the limit of the cardinalities of a set sequence should
>> measure, in some sense, the cardinality of the limit set, then he did
>> indeed share your misunderstanding.

>
> Cantor used the limit set of all FIS called it |N and said that its
> cardinality is larger than every finite number. Did he conclude this
> from the limit of the sequence of cardinalities? From what else?


As I said, you're the historian, you tell me. He *could* conclude it
from the usual definition of a set sequence limit, but I don't know if
there was one at the time. It is, after all, perfectly true that

lim[n->oo] { i | i <= n } = N

Anyway, it seems from your reply that Cantor did *not* subscribe to the
same confused expectation about set sequences and their limits that you
do.

--
Ben.



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