Re: Orlow cardinality question

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> stephen@xxxxxxxxxx said:
>> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>> > Randy Poe said:
>> >>
>> >>
>> >> Tony Orlow (aeo6) wrote:
>> >> > Randy Poe said:
>> >> > > Again, you are insisting on an axiom that doesn't exist,
>> >> > > in this case that if A < B, then |A| < |B|. That is, if
>> >> > > A is a proper subset of B, then the cardinality of A is
>> >> > > smaller than the cardinality of B. As you have been told
>> >> > > many times, there is no such requirement on infinite sets,
>> >> > > and indeed a property of infinite sets is that there
>> >> > > exists A < B with |A| = |B|.
>> >> > Cardinality purports to describe the sizes of infinite sets.
>> >>
>> >> Bzzzzt. Wrong from the starting gate.
>> >>
>> >> Cardinality purports to be a method to assign an ordering
>> >> to infinite sets. This it does remarkably well.
>> > An ordering in terms of what? Element values? Standard deviation? No. It
>> > purports to be a way of comparing the sizes of sets, that is, it purports to
>> > distinguish between a few different infinite numbers of elements, and to prove
>> > that no other distinction can be made, which is wrong.
>>
>> > Try defining the word cardinality in five words or less.
>>
>> That would be silly. It has a precise mathematical definition.
>> Using anything other than the precise mathematical definition
>> is pointless, unless you just want to make vague hand wavy
>> arguments.
>>
>> Stephen
>>
> So, you can't summarize what it is that cardinality is supposed to accomplish,
> but claim it is other than determining the size of a set? What is the basic
> purpose of this theory, if not to measure sets?
> --
> Smiles,

> Tony

Cardinality is not a theory. It is a definition. Two
sets have the same cardinality if there exists a bijection between them.
I am not sure what you mean by "purpose", or what you would
accept as a valid answer. What is the purpose of the "size of a set"?

Stephen
.

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