Re: Orlow cardinality question

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> stephen@xxxxxxxxxx said:
>> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>> > stephen@xxxxxxxxxx said:
>> >> 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
>> >>
>> > Size is one measure of a set. The purpose of a set is to act as a collection.
>> > That collection may have many attributes, but as a set, one attribute it always
>> > has is its size, or the number of elements it includes.
>>
>> That really does not answer what the purpose of the "size of a set" is.
>> It applies equally well to any measure of a set, such as cardinality.
> The size of a set is the one measure which applies to sets in general. The
> purpose of cardinality is supposed to be to provide that measure in as general
> a way as possible, otherwise, I don't know what the purpose of it is. When I
> speak of the evens as being half as big a set as all integers, Cantorians tell
> me that I am wrong because of cardinality, so obviously cardinality is thought
> of as being a means of measuring set sizes that applies to infinite sets as
> well as finite sets. Certainly, for finite sets, it is precisely the size of
> the set. Is it something different for infinite sets? If so, has it missed its
> mark?
>>
>> > The problem that is
>> > supposedly addressed by cardinality is the measure of set size, for both finite
>> > and infinite sets, hopefully addressed in some consistent manner.
>>
>> What if they cannot be both addressed in a "consistent manner"?
>> I put "consistent" in quotes, because I am not really sure what
>> you mean. There is nothing inconsistent about cardinality.
>> It upsets your sensibilities and intuitions, but your sensibilites
>> and intuitions are not the standard of "consistent".

> There has certainly been admission here that a proper subset can always be
> thought of as somehow smaller than its superset, but cardinality ignores this
> in most cases.

That is not an inconsistency. As has been repeatedly pointed out
to you and others, "size" is not a well defined term, nor
is "smaller". Two things can have the same volume but different
weights. In one sense they are the same size, in another they
are different sizes. Yes a subset is "somehow smaller" than
its superset, but it can also be "somehow the same size".

> In finite sets of naturals, you freely admit that the set size
> can't be larger than the maximum element value, but somehow this flies out the
> window at infinity.

In an infinite set of naturals there is no maximum element.
Obviously the "set size" cannot be defined in terms of something
that does not exist. But of course you seem to think there
is a largest finite integer, or some strange gobbledygook like that.

> In Cantor and the Binary Tree, there seems to be all sorts
> of confusion regarding what happens to the binary tree when it is infinitely
> deep, because of all the other exceptions and changes in approach that suddenly
> are applied to infinite sets, so that suddenly there are uncountably many paths
> and countably many branches, when in fact there are half as many paths as
> branches. Inconsistency in this area is rife.

> I offered a system for sets defined by mapping functions that works precisely,
> both for finite and infinite sets. It wasn't welcome.

For one reason, it only applied to some sets. You could
not explain what the "size" of a set of Turing Machines was, or
how it related to the "size" of a set of functions. Are there
"more" functions than Turing Machines according to your system?

>You'd rather have your
> kludge of inconsistencies that violate information theory and infinite series,
> and claim that you have an internally consistent system that works, when the
> conclusions it draws are obviously wrong. That's your privilege. Enjoy.

You have yet to show any inconsistency. Nothing about cardinality
violates information theory or infinite series. It sounds
like you are just stringing buzzwords together in a nonsensical
fashion.

>>
>> > Unfortunately, it is not consistent enough for my tastes, with all the
>> > exceptions claimed for infinite sets, as if they aren't even sets anymore, and
>> > as if cardinality is suddenly NOT supposed to be a measure of set size. If
>> > cardinality of infinite sets is not a measure of their size, if adding or
>> > subtracting elements doesn't change the cardinality, then what the heck is it
>> > supposed to signify? That's what I'm asking.
>>
>> It signifies an ordering of sets based on whether or not
>> a surjection exists between one set to the other. It signifies
>> whether each element of one set can be uniquely mapped to
>> each element in another set.

> That's the methodology. The root purpose is to order sets by size, hopefully
> with the most precision possible. If we just divided numbers into finites vs
> infinities, without distinguishing between countable finites, what kind of math
> would we have? "0, 1, uhhhh, oo!" That's about how useful cardinality is for
> infinite sets.

It is far more useful than that. Again, note your phrase "with the most
precision possible". I have yet to see a way of ordering all sets
(not just sets of numbers) more precisely.

>>
>> What does "size of a set" signify? "Number of elements" is
>> not a satisfactory answer because the two are apparently
>> synonymous. You seem to be assuming that "size of a set"
>> is some fundamental concept that needs no explanation.
>> Given that it is not a well defined term in general, that
>> seems strange.

> That's exactly what it is. Synonymous words or phrases are what definitions
> ARE. The size of the set is the number of elements in the set. It doesn't get
> much simpler than that. If that doesn't make sense to you, then I don't know
> what you think we're talking about.

So you have a circular justification for "size of a set".
You cannot say what it signifies other than it signifies itself.
Why then does cardinality have to signify something other than itself?
Your question "what does cardinality signify" seems to be meaningless.

> A set is a unit that consists of a number
> of units. The size of that set is the number of units of which it consists. If
> nothing is know about the units themselves, then this is all we know about the
> set, and is really the only pure measure of a set.

But "number of elements" is meaningless for an infinite set until
you define some "infinite" numbers. There are no infinite natural
numbers, despite your confused ramblings on the subject, and
there is no reason that infinite cardinals need to behave exactly
like finite cardinals.

Stephen
.

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