Re: Kuratowski Ordered Pair

On Dec 20, 5:16 pm, galathaea <galath...@xxxxxxxxx> wrote:
On Dec 20, 3:19 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

Please, ante up already. Please say EXACTLY what the Kuratowski
definition was INTENDED to do but does not do.

the intent is not necessary to an objection

The poster claimed that there is some intent of the definition that
the definition fails. I'd just like to know what intent the poster has
in mind. I mentioned the intentions of the definition (and a nice
discussion is given in Moschavakis's book), and indeed we prove that
those intentions are fulfilled.

but is it really that hard to see what might be their issues?

They mention various metaphors and ramble about various concerns. But
the only thing substantive has been the concern of defininig n-tuples
without going through numbers. I addressed that.

: I guess I cannot quite understand what people mean by an
: "unordered pair."  I can understand not knowing the order of a
: pair and I can understand superposition of all possible orders.
:  But, the connectivity of a pair without order is
: incomprehensible to me.

Yes!!!  Yes!  Yes and yes!!!

You start by challenging me to understand vague and metaphorically
motivated objections to some formally worked out mathematics and then
you turn around to endorse NOT understanding a perfectly simple and
intuitive formal definition. Rich.

I once got into a discussion with my topology teacher about which was
more
primary.  He argued that ordered pairs required more definition,
whereas my
point was that on all conceptual levels I could identify (visual,
auditory,
etc.), the ordering seemed to follow most naturally from the input,
and the
act of "unordering" seemed a latter abstraction.

a -> b is much more useful evolutionarily than, say, a = b.

when presented with a collection of "things"
  objects, impressions, ..
we appear to identify the collection through some ordering process

we select an object and pattern recognise it
  then another
    and so on
a temporal series of conceptual isolations

the same goes for counting

we _choose_ one
  then another

these orderings aren't a priori or associated innately with the
objects
  and they might change any time an identification is done
but the ordering seems integral to the conceptualisation

we read symbols in some order
  giving them some connective mereology as a data structure
  something we can iterate through

Meanwhile, I see no need for set theory to emulate your personal
opinions in metaphysics. I don't take set theory to even be about such
metaphysical concerns.

this is how turing machines work

The informal notion of a Turing machine is formalized in formal set
theory with tuples.

and then we look at set theory
  which takes collections as unordered
and a pair (a, b)
becomes {a, {a, b}}?

No, it does not "become". There's just a definition. We don't need to
impose metaphysics of 'becoming'.

and it still assumes the ordering for parsing
but claims an unordered semantics?

I don't know what you mean by 'unordered semantics'.

when a language mangles a natural concept so horribly
  there is plenty to object

You're welcome to offer an alternative formalization that satisfies
the intent (the theorems I mentioned). Would your alternative be as
simple as the Kuratowski definition?

  no matter the intent

No, we wish for the defintion to facilitate certain mathematics.

MoeBlee
.

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