Re: An uncountable countable set

Tony Orlow wrote:
MoeBlee wrote:
Tony Orlow wrote:
Ahem. I said that Robinson's analysis seems to have nothing to do with
transfinitology.

You agreed it is not an ordering by cardinality.

Yes, and you asked me what I would think if it was derived from
transfinite set theory. You're tripping over your tail.

So what? You agreed that it is not a cardinality ordering. And so the
fact that there is no least infinite (in a different sense of
'infinite') in one ordering, which is not a cardinality ordering, is
not a contradiction or "incompatiblity" with there being a least
infinite in a cardinality ordering.

Read Robinson for real and then comment. See how quickly you can absorb
his actual section on logical tools, and whether you might skip ahead
and then back.

That's the point. You can't easily (or even at all) absorb that
material in the chapter on logic without having studied the basic
material that is in undergraduate logic texts.

You don't learn the
mathematical logic and set theory that are the basis for the material
and even pretty much ignore the mathemtatical logic and set theory that
the author himself summarizes in the book. (You need to start by
learning how to work in the predicate calculus), and (2) you don't
listen when someone tries to warn you about the confusions you are
making due to your not understanding the basis and context.


Well, it doesn't help when they freak out because they can't handle
questioning certain iffy logical constructs. You're getting a tad edgy
lately, which I understand. I'm not floating on a cloud myself. Life is.

It's got nothing to do with me "questioning logical constructs". The
problem is that you spout about non-standard analysis while you haven't
even BEGUN to address the rudiments of the subject matter, while you
instead SKIP the rudiments without which an understanding of the more
advanced material cannot be had.

Because, AGAIN, for the TENTH TIME, 'smallest infinite' regarding
ordinals and cardinals is a DIFFERENT SUBJECT from 'smallest infinite'
regarding elements in certain non-standard models.


Why does the very mention of no smallest infinite elicit such bile,
then?

Whose bile? I'm not rankled by there being no least infinite member of
certain domains (again, that is not a cardinality 'infinite') and
wouldn't even be put off by an alternative theory that somehow had no
least infinite cardinality. But what does irk me is that after I
cleared the way for you NOT to conflate two different contexts of
'least infinite', and after YOU even granted the difference, you
conflated them anyway, thus to spout nonsense on the Internet yet
again.

So, either smallest has two meanings, or infinite has tow meanings, or
both.

Yes, they have at LEAST two meanings, as they are English used in
different contexts. A student of the subject will understand the
differences in context, but a phony baloney like you will be oblivious
to those differences because a poseur like you has no interest in
actually studying and understanding the material but rather is just
looking for quotes and and bits and pieces of ideas for fodder to put
into Internet posts defending his half-baked commentary on a subject of
which he knows nothing.

"to fuse into one entity; merge:

Right, you merged one sense of 'least infinite' with another.

1 a : to bring together : FUSE b : CONFUSE
2 : to combine (as two readings of a text) into a composite whole [all caps original]

You CONFUSED one sense with another as you failed to distinguish them,
as you combined them into one while they are separate. You conflated
them.

I would agree that the terminology is unfortunate, since there are two
different senses of 'infinite' in play.

Two? Which? Countable and uncountable?

How IDIOTIC of you to ask that. I spent about three posts already,
using all-caps even, to say that you are conflating the cardinality
sense with the sense of points in a certain ordering in the universe of
a non-standard model. Not countable and uncountable - those are
cardinality. AGAIN, for about the TENTH time: You are confusing
'infinite' in the sense of cardinality with 'infinite', in the very
special sense of the context of non-standard analysis, of certain
orderings that are not necessarily cardinality orderings.

Maybe the word 'position' will help. You are confusing 'infinite' in
the sense of cardinality with 'infinite' in the sense of position in
certain orderings where we CALL certain members of a universe 'finite
members' (though, again, this is a DIFFERENT sense of finite, not
necessarily a cardinality sense) and other members of the univers
'infinite' since the "infinite" members are all related from the left
by ALL of the finite members.

It has to do with a relation on the universe of the non-standard model.
If R is the relation, and S is the set of standard (finite) members of
the universe, then the infinite members are the x's such that for any y
is S, we have <y x> is in R. But R is NOT the relation of 'y is of less
cardinality than x'.

Also, as to 'countable' and 'uncountable', those are not different
senses of 'infinite'. First, 'countable' includes finite. And the
difference between 'denumerable' and 'uncountable' (which is, between
'countably infinite' and 'uncountable') is not a matter of different
senses of 'infinite' in the way I'm mentioning regarding the difference
between cardinality and position in certain orderings of non-standard
universes. Rather, the difference betweeen 'denumerable' and
'uncountable' is a difference between infinite sets that are still
inifnite in the SAME sense of 'infinite' as a kind of cardinality.

Your raising 'countable' and 'uncountable' here does not come from you
sincerely wanting to understand non-standard analysis or set theory or
mathematics, but rather is an irrelevent diversion so that you can
continue to argue about this from your position of ignorance.

Oh, so it's okay for me to say the set of naturals in not infinite in MY
sense of the word?

Use words however you like. But if you want people to understand you,
then you need to set up your context and definitions. And if you have a
book, organized and systematic, in which you set up your context and
definitions, but people ignore that chapter (such as you for all
practical purposes ingore the chapter in the book we're talking about,
but more importantly, you ignore the basic body of mathematical logic,
set theory, and mathematics that that book uses as that book is written
for an audience that has already worked through the basics that the
book only summarizes in the logic chapter), then you'll have every
right to complain about such sloppy readers of your book as I am
complaining about your sloppy reading.

Anyway, I said myself that I think the double sense of 'infinite' is
unfortunate. Personally, I wish a different word had been chosen for
non-standard analysis, and I wish there were more uniformity among
authors and mathematicians in the natural language terminology so that
readers would not have to be so very mindful of context and overall
vagaries of terminology that occur in mathematical writing. However,
that being my personal feeling, it is still not not an excuse for your
not studying the basics of the subject so that at least the obvious
differences in context would be clear to you.

Yes, no kidding. There were references to terms and definitions which I
could at first glean fairly well, but as they built upon each other, the
references got buried upon each other, and I lost track. I have never
been a graduate mathematics student. Pardon me. I am not sure that works
to my disadvantage in this particular instance. My shackles are
non-mathematical. :)

Forget about graduate level for now, is my very point. Please just
learn the basics of the mathematical logic and set theory, which are
the references that built up too fast for you in that chapter, so that
you will understand that chapter and other graduate level mathematical
logic (including non-standard analysis) as you move forward to study
it.

This is why, when I choose a thread and participate, it takes forever to
end. I am looking in every direction, not to the front of the lecture
hall, to find the exotic critters under each rock and in each log. There
are underlying logical questions which you and others consider settled,

I told you a long time ago that there is a tremendous amount of debate
in thousands of books and journal articles about the fundamental
logical and mathematical-philosophical foundations. But you need to
understand the rudiments of the subject so that you can understand what
the different debators are saying.

Then perhaps your arrogance manifests itself in other ways. You might
want to think about that.

I do. But if you ever catch me being an arrogant smart ass with people
who know a lot more about this subject than I do and who have taken his
time and effort to help me with explanations and formulas and proofs,
then please let me know.

Welcome to the club, and while you're getting defensive about not being
the repository of all human or other knowledge, you might want to
consider, as I do while seeing othersget bent out of shape, that it
might be due to real world concerns, distractions, and needs.

I am not judging you on how much time you don't put in. I'm judging you
on how much you DO put in to post ignorant trash. Not even on the time
you put in to post ignorant trash, but rather the ignorant trash
itself.

shouldn't even be wasting my time on this, by any normal sensibilities,
given what's been and remains on my plate. Maybe that's true of you. Or
maybe you have no excuse at all. But I suspect you do. So, take you
time. But do check out Non-Standard Analysis, when you get the time. I'm
doing Boole and Robinson. We all choose our paths.

This is not a matter of "many paths". You can't properly do this
without learing the rudiments. If you "take a different path" that
skips the rudiments, then eventually you will have to come back to
learn the rudiments anyway, and in the meanwhile you'll be burdened
with a MISunderstanding of the advanced material, which is probably
worse than no familiarity at all with it. Okay, probably there are a
very few gifted people who can jump into the middle of advanced texts
without having studied the basics, but I am not one of them, and I can
tell from your POSTS that you are not one of them.

MoeBlee

.

Relevant Pages

  • Re: Distinct linear orderings on Z
    ... >>very accurate description of the size of an infinite set, ... >>integers, and no more integers than odds, based on cardinality measure? ... > used by mathematics in universal terms.They're talking about tangents. ...
    (sci.math)
  • Re: Galileos Paradox
    ... Note, I said 'infinite', not 'infinity'. ... about the term "cardinality", but I will respond to that later. ... The words 'prime' and 'even' have meanings outside of mathematics. ... correspondence) and less squares than naturals. ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... >>And then at some point this led us to deciding that cardinality and ... >>meant for an infinite set to have a certain number of elements. ... always goes hand in hand with subtraction. ... Does the interval on the reals have to include 1? ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... >> So you claim that all infinite sets have the same cardinality? ... >What I don't understand is what the notion of cardinality really is. ... on about, it's not mathematics, it's a confession of faith (and ... or on whether "axioms" in one system or another ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... all Cantorian approaches like this, that they are in the same general ... >>Not contradiction according to Cantorian cardinality. ... Why should it not be true of infinite ... And what finer distinction does it make than between sets that are ...
    (sci.cognitive)
Loading