Re: Review of Mueckenheims book.

MoeBlee wrote:
On Mar 7, 10:24 am, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
MoeBlee wrote:
On Mar 6, 5:25 pm, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
1. What does "arranged linearly" mean?
One after the other, sequential, like the naturals we were discussing
before someone threw kumquats into the equation.
See, it's good that we asked what your PERSONAL definition is,since
the usual definition of 'a linear ordering' does not require
sequencing, as 'sequence' itself has a mathematical definition.
Sure, that's fine. You don't consider a sequence to be possibly
uncountable,

I NEVER SAID THAT. Please do not put words in my mouth. In fact, I've
said many times, in posts to YOU, that there are uncountable
sequences, which are functions on uncountable ordinals. You don't even
read.


You may have said that, but then you're in the minority, from what I've seen around here. I meant "you" in a more generic sense.

and you consider a line to have an uncountable number of
points, so those concepts are incompatible for you.

Given appropriate definitions, a line has uncountably many points.
That is not incompatible with the existence of transfinite sequences.


If an uncountable sequence is acceptable, then why can one not have such a sequence of infinitesimals constitute a line? I think you knew what I meant when I said a linear arrangement anyway, since were were speaking of the naturals. What are we supposedly arguing about?

However, if you
consider Ross' (where is he, I wonder - on hiatus, too?) Finlayson
numbers, an uncountable sequence of infinitesimally spaced iotas, as he
calls them, the two concepts can be reconciled.

I don't need Ross's nonsense to reconcile what is not in need of
reconciliation anyway.


Okay. I don't need Diet Coke.

2. Why do you think "they" are arranged linearly?
Ask Peano.
Why not just look at any old proof that the standard ordering on
natural numbers is arranged linearly (even with your defintion of
'arranged linearaly')?
Why bother when it's obviously true? Shall I look up the definition of
"of" too?

YOU answered with a vague "ask Peano"; I just recommended that we can
do even better.


Do what better? Define the naturals? Sure, the definition goes beyond Peano, if that's what you mean, but the sequential nature of the "set" is established up front.

What's "infinite n"?
n| A meN n>m
What is that supposed to be?
n such that for all m in N, n>m

n is infinite <-> AmeN n>m.
Is what you're saying?
Yes, when you ask what "infinite n" is, I assume that you can accept a
statement referring to the definition which is not a whole statement
with the verb "implies". If you ask, "Where is the Babbo?" would you be
confused if I said, "Under the sink", rather than, "The Babbo is under
the sink"? I hope not.

What is YOUR definition of '>'?
That's a very good question, and one which has been among the
fundamentals stewing in the back of my mind. Essentially, '<' is defined by:

a<b -> ~b<a

Why do you KEEP doing that?! Someone asks you for a definition, and
you just answer with some other term that is just as much in need of
definition. Here you define '>' by way of '<'. Please just define one
of them!


Sorry, Moe, but I did not use '>' above. The behavior of '<' is defined above using a statement employing first order logic and '<'. Read over, and take a deep breath. I start with '<', and define '>' in terms of '<', but we can start with '>':
a>b -> ~b>a

And then say:
a>b <-> b<a


This does not imply that:

~b<a -> a<b

Well, duh. Given a usual definition of '<' and '>', we already know
that x not less than y does not entail that y is less than x, and
hardly need all this to explain it:


It's just an observation for subsequent application...relax

For it can be true that:

~b<a ^ ~a<b

or:

~(b<a v a<b)

in which case we deine '=':

~(a<b v b<a) <-> a=b

See? I used it to justify the definition of "=". :)


Since we can exchange a and b in this statement without changing it we
have a reflexive function between them:

a=b <-> b=a

We can also define '>' as:

a<b <-> b>a

And that's about all there is to say about '<', from a language standpoint.

PLEASE DEFINE '<'. Please don't play shuffle the pea pod. Nevermind
the obvious manipulations regarding 'less than', 'greater than', 'less
than or equal to' and 'greater than or equal to'. We ALREADY know all
about that. Just give a definition of '<' please.


From a language standpoint, that's the definition. Read on.... (sigh)

Of course, this really only applies where we have some space with a
dimension along which a<b, a=b, or a>b. This is a linear space, e.g. the
number line, and '<' can be interpreted geometrically as "below" or "to
the left" or "in the negative direction from".

"Can be interpreted as". I see you still are not able to produce
mathematical definitions.


The mathematical definition of the symbol consists of the rules of manipulation of it within the language. For '=', a=b <-> b=a. For '>' and '<', a<b <-> b>a and a<b -> ~b<a. As extensions to the basic logical symbols, that's how they work.

You can't explain what "infinite-case induction" is because you can't
explain what "infinite numbers" are.
I did, in the same basic way Robinson did, as a number greater than any
finite number.
'infinite number' in that sense is distinct from 'is infinite' in the
sense of cardinality.
Yes, that's true. Set theoretic infinity is defined using bijection with
a proper subset,

That is 'Dedekind infinite'. We also define 'infinite', which does not
require a bijection with a proper subset (and we use some form of
choice to show that such a bijection is entailed).

which includes countable infinities, which this
definition really does not.

WHAT definition? Robinson's. I keep telling you (and you could glean
for yourself in the source) that that is something very different from
cardinality and does not supplant cardinality. But you are IMPERVIOUS
not only to reason but to INFORMATION.


Relax.

But good to see you appreciate A. Robinson, whose work is built on
classical mathematics, including mathematical logic, ZFC, and is non-
constructive up the wazoo.
On the other hand, it's hard to make sense of your endorsement of
Robinson's non-standard analysis - built on classical and decidedly
non-constructive mathematics and ZFC, in the sense of using the
principles of ZFC including Zorn's lemma - when you claim ZFC is all
wrong.

I claim that pure sets cannot be infinite, and that set theoretic
infinity depends on something other than set membership, even if an
inductive set can be defined using only 'e'.

That's manifest nonsense. It is definable by 'e' alone, so it doesn't
"depend on" anything else aside from than the pure logical apparatus.


Not until you adhere to the rule that set size increases with additional elements.

Where it is, each element
is a proper superset of the last, and so in that sense, there is a
sequence, which is more than a set, and it depends in one way or another
of successive elements growing in "size" in one way or another. It's not
that set theory is "wrong" persay, just that it extends its claims of
truth beyond its actually capabilities,

"Wrong" is a fair paraphrase of whatever words you've used before. But
if you wish to back off from that strength of language now, fine.
However, "extends its claims of truth beyone its actual capabilities"
is just more Orlowian babble.


It's "Orlovian" babble, if you would.....

Anyway, again, if you endorse A. Robinson's non-standard analysis,
then you are endorsing work that does depend on classical mathematical
logic and set theory including the axiom of choice. However, Aatu
pointed to a constructive form of non-standard analysis, which you can
examine for whatever assumptions it requires, so as to see if those
assumptions are acceptable to you. A. Robinson's assumptions are not
acceptable to you, unless you do find the axioms of ZFC to be
acceptable to you.

where it ignores measure. Of
course,

Measure is developed in mathematics that can be expressed and
formalized in set theory.

MoeBlee





Sure, okay.
.

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