Re: Cantor's definition of set

On Oct 27, 11:07 am, LauLuna <laureanol...@xxxxxxxx> wrote:
On Oct 25, 10:25 pm, John Jones <jonescard...@xxxxxxx> wrote:
Where on earth do you get the idea that the members of a set need have
nothing in common?? Take some set of people: the members of the set
have something in common (being human beings, for a start).

OK; then I can have a set of humans. Does this meet the criterion that
a set has unique members? If the answer is yes, then
1) we must presume that all humans are uniquely different. But
wouldn't this make a set dependent not on logical principles but on
material contingencies?
2) a set of humans describes what they have in common more than what
is different about them. should we redescribe the set so that it
mirrors the letter of the law as it were, and describes only what is
different about humans?
3) Can we have a set of differences?

When I try to make sense of what you are saying, it sounds to me as
though you are complaining about the atomistic, 'structureless',
conception of set in Cantor, or better, in current set theory. The
members of a set are considered apart from any possible relationships
obtaining between them.

Now relations are set theoretically understood as sets of pairs,
triples, etc. which in turn are interpreted as unordered sets, e.g.
<a,b> = {a, {a,b}}, all this boiling down to extensionality in a
certain haze of oddity.

If this is what you mean, perhaps you are ponting to an actual
problem: how can mathematics deal with what is ultimately non
extensive in nature?

Regards- Hide quoted text -

- Show quoted text -

(You are not referring to extensionality as the set of objects of a
concept )
ZF Extensionality proposes that if x and y have the same members, then
they are the same set. I will tell you what I think is wrong with this
idea and then say whether it represents my complaints elsewhere.

1) I have bought two sets of the same sheffield cutlery. By ZF theory
they are the same set (so I might think that, according to ZF, I had
miscounted them and that I actually have only one set).

2) Let us say that I have made a simple mistake. I have misconstrued
the sense of the phrase 'the same members', and that what is meant by
'same' can, in fact, only be decided by a consideration of spatio-
temporal superposition. With this new criterion, each set of cutlery
is the same set IF their elements (the knives, and the forks) are
spatio-temporally limited to one place/event and not two.

My objection follows:

3) In other words, in order to identify whether sets are 'the same' I
need to stipulate the way in which I positively re-identify the
elements of a set. These stipulations are not always the same. For
spatio-temporal sets these re-identification criteria are spatio-
temporal. For non-physical sets (such as a persons 'character traits')
these criteria are not spatio-temporal.

Therefore, the axiom of extensionality proposing that "if x and y have
the same members, then they are the same set" is an axiom that is
dependent on the sense given to 'same'. It is not, therefore, an axiom
- at least it is only partially formulated, and appears to rely on
charitable interpretation based on knowledge of the world's objects.
So, the ZF definition of extensionality - as it stands - is not
rigorous on that account.

More observations and objections follow:
a) The above objection would apply as much to lists or groups, etc.,
as to sets. So the above problem of 'sameness' is not limited to sets.
b) More importantly, ZF extensionality has merely displaced the
problem of 'sameness of sets' to 'sameness's in the world'. In which
case ZF 'sameness' does not relate to sets but directly to objects in
the world.

I cannot finish this post here and now as I have to go, but it seems
that my issue is primarily to do with the way members are identified
in a set, and that the problems I have are partially represented by
the problem of 'sameness' and hence 'extensionality', outlined above.
I will return to this.

.

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