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Topic: § 534 Finis
Replies: 1   Last Post: Aug 18, 2014 12:56 PM

 Ben Bacarisse Posts: 1,972 Registered: 7/4/07
Re: § 534 Finis
Posted: Aug 18, 2014 12:56 PM

mueckenh@rz.fh-augsburg.de writes:

> On Monday, 18 August 2014 05:33:04 UTC+2, Ben Bacarisse wrote:
>

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

>
> The set following beyond every finite natural or every FIS.

It's not very helpful to define one ill-defined word by using even more.
Is there any point in my asking you to define these other ill-defined
terms, or will you just use yet more? Mathematical definitions allow
the definition to be used. For example, the definition of the set limit
allows even you to determine it, despite the fact that you don't like
it. What you have said does not let me determine the "finite set" from
any set sequence. What is, for example, the "final set" of your
sequence s_n? Do all set sequences have a "final set"? Is it unique,
or do some set sequences have more than one?

>> >> 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.

>
> What do they apply to?

The limit operators are applied to different things, and the || is
applied to different things. Every set in this scenario is different
from every other one -- all the s_n are distinct, and the limit set ({})
is distinct from all the s_n. What on earth is it that you think is
being applied to "the same set"?

<snip>
>> > 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?

>
> If there is a final set "at omega", then this woud be natural.

Maybe. Do you define it that way? I only know how to define the limit
set, not the "final set", but if you give your definition of the WMglish
phrase "final set" in a useful form (i.e. not in terms of more WMglish
words) everyone could confirm that it does indeed behave as you want it
to.

> What does the limit of the cardinalities apply to?

It applies to the cardinalities, what else? It certainly does not apply
to the limit set, that's defined in an altogether different way. Only
someone very confused by the definition would think that it might. I
don't know of any useful kind of "final set" definition such that the
limit of the cardinalities applies to it. Maybe you can define one?

> Up to now you only
> said what is wrong. Now say please, what is the meaning of lim card? I
> don't accept the answer "nothing".

Of course it applies to something -- its the limit of a sequence of
cardinalities. But there is no set, derived form the set sequence,
which it measures. At least I don't know of any set sequence operator
which obeys the rule that lim |s_n| = |FS(s_n)|. One thing is for sure
(and you really should stop telling your students otherwise): FS is not
the limit operator as it is usually defined.

--
Ben.