Re: Calculus XOR Probability

Matt Gutting said:
Tony Orlow wrote:

<snip>

I am not complaining about the axiomatic definition of the naturals. I am
asking that you look parametrically at how the set grows in relation to other
sets, and at the reciprocal density of sets on the real line. This can only be
done with some notion of a value range. Hence Big'un.

Again, I'm not sure what you mean by the set "growing". Sets don't grow. They
exist.

If a quantitative set is mapped in ascending order from the naturals, with each
increment in the domain, the range increases by some amount. So, there is a
growth rate, as measured per element, relative to the standard growth of the
incremented naturals. The formula that maps from the naturals measures the
naturals relative to the mapped set, and the inverse measures the set relative
to the naturals.


<snip>

MoeBlee:
"length", "infinitesimal", "real", "interval", "unit", "divided",
"subinterval". Definitions, please. Actually, nevermind. Definitions
need to revert to axioms in a specified logistic system. And you STILL
have not specified a definitive set of primitives nor axioms nor even
what logic your system uses, though that doesn't keep you from waxing
poetic with your undefined vocabulary.

Are you sayign you can't understand what I'm saying? No, you're just being
deliberately obtuse.

No; considering that all these things can be defined in the standard model,
that your model is different from the standard model, and that you haven't
provided definitions for the terms as you use them, MoeBlee is perfectly
justified in asking for definitions.

Yes, the formal theory needs this, but pretending not to be able to make any
sense of what I'm saying can't be genuine.



Big'un is the standard length of the real line, but not the rgeatest number.
If Big'un is a whole number, Big'un^2 is too, and bigger than Big'un, being the
number of reals on the line.
Still in your theory now. As I recall, you had said that the existence
of Big'un is axiomatic. In set theory, the existence of the set of
natural numbers is axiomatic, but, as I understand, you object to that
as some kind of magic wand having. But then why wouldn't the axiomatic
existence of Big'un be magic wand waving?
For one thing, a set of values is different from a single value.
That doesn't make the existence of Big'un any more intuitive than the
existence of the set of natural numbers. In fact, you claim Big'un is
primitive. If that is the case, then you don't even need to claim its
existence, but rather you need to give axioms that give its properties.


Uh, yeah. Like it's the number of unit intervals, and the number of reals in
the unit interval. That's after declaring 0 and 1, and the interval in between,
and copies of it sequentially on either side of it.

The naturals
form a sequence, defined by a generating function.
The set of natural numbers also can be characterized in different ways.

Apparently, but wouldn't it be nice to formulate them in such a way that they
make the most sense?

The most sense to whom? They are already formulated in such a way that they
make sense to most of us.

Sure, except for all that "counterintutiveness".



Big'un is a primitive.
So, we could make omega a primitive. We don't, because we have an even
more basic primitive.

Well, you start with 0 or 1, and the successor operator, but then you dismiss
the successor operator when it comes to omega, and cause a discontinuity on the
line.

<snip>

No. Omega is not a natural number; thus, the successor operator doesn't apply
to it.

Yeah, I know that rule. It's a kludge. Sorry. I don't accept it as pertaining
to anything real.



I
haven't made any objections to the axiomatic definition of the naturals,
Then what's this business about "poof"?

It's about refusing to look at how the set is GENERATED, and this insistence
that the set "springs from the axioms, complete". Why would anyone refuse to
consider the functions that define sets and their relative desnities on the
real line?

At least partially, it's because density depends on order, and it doesn't make
a great deal of sense to say that changing the density of a function defining a
set changes the size of the set.

Density is about order only to the extent that one defines an interval with one
endpoint after the other. If we have 100 points on a 10-unit line segment, we
don't need to know what order they are in, or if there is any order, to know we
have an average density of 10 points per unit. If we have the average density
and the range, we can calculate the number of elements, and if we have the
average density and the number of elements, we can calculate the range.



only
to the subsequent logic that claims such a set cannot have infinite values.
The subsequent logic is pristine, impeccable first order predicate
logic. You have pointed to no flaw whatsover in the application of
first order predicate logic.

Uh, yes, I have pointed out a flaw in the argument. You are using a form of
infinite induction, making a claim for an infinite set based on all finite
initial segments of it. Because one increment cannot make a finite value
infinite, you claim that even an infinite number of such increments cannot
produce an infinite value from a finite one. That's flawed, for the very
reasons I will rereiterate. When you are proving "x is finite" you are proving
"x<oo", an inequality which becomes false when "x=oo". The difference upon
which your inequality is based disappears, invalidating the proof in the
infinite case. The equality between element count and value in the naturals
holds in the infinite case, showing clearly that the set is only actually
infinite when it contains infinite values.

No. When you are proving "x is finite", you are proving (with, for example,
the natural numbers) that "x is 0, or x is the successor of a natural number".
Inequality (or equality, for that matter) doesn't even enter into it.

It does if you consider inductive proof of an equality to be valid in the
infinite case, and note that if the limit of the difference establishing an
inequality has a limit of 0, then the inequality does NOT hold in the infinite
case.


<snip>

MoeBlee:
This is in your theory. "unit", "distance", "number line", "hyperreal
line". If you're not going to define them, then please at rhyme them,
would you?

origin:
E 0

finite unit:
E 1

infinite unit:
E Big'un

order:
0<1
1<Big'un

line:
E x,z: x<z -> E y: x<y & y<z
E x -> E y: x<y

distance:
x-0=0
x-x=0
x-y=x-z -> y=z
(x-y)+1=x-(y-1)
(x-y)-1=x-(y+1)

Rhyming "line" with "line" is lame. I suppose I could call something a
"flornge", for the sake of rhyming, but I haven't yet. I'm saving that as an
alternative name for the Orlow Water Bot, a bot fly I discovered that has an
aquatic larval stage. :)

These aren't really definitions. (I'm going to assume that, under the heading
"distance", the equation "x-0=0" should be "x-0=x".) It's possible, for example,
to define "Big'un" as 2 and have all your equations work out right.


Oops, yes, sorry, x-0=x. Those axioms should be the same as the standard
arithmetic axioms.



Right, that's the standard treatement. You look only at corresponding elements
without regard to how quickly each progresses along the line. What I am
suggesting is that if you look at the number of elements within a given value
range, that gives a measure of the density of the set on the line. Unless one
of the sets has greater range than the other, the set with the greater density
should be considered the larger set.
'Density' defined in what theory? Density is relative to an ordering.
You seem to overlook that a set can have more than one ordering, and
different orderings will make different subsets dense depending on the
ordering. Thus, cardinality based on density is not firm, since we
don't have a definition for an arbitrary set of its "standard ordering"
or "canonical ordering" or whatever you want to call it.
We don't have a notion of quantitative order, as demonstrated on the real line?
When it comes to quantities, we have a "standard ordering" defined by "<".
I said, for AN ARBITRARY SET. We do not have standard orderings for
sets other than a few such as the set of natural numbers, the set of
real numbers, et. al. So it doesn't work to base cardinality on
standard orderings, since sets in general don't come with little tags
on them specifying a standard ordering.

There are basically two kinds of infinite sets: quantities defined
formulaically and languages defined by concatenation of of elements of an
alphabet. When I suggest IFR, I am applying that to quantitative sets, that is,
sets of real numbers. They have an inherent order by quantity, and all lie on
the real number line, where density is a measure of the set over any interval.

What about, e.g., the set of all formal power series, or the set of all
functions from the reals to the reals? These are certainly not finite sets,
but they're neither quantities nor languages defined by concatenation of
elements. Where do these fit in?

The formal power series is essentially a digital number, which falls under the
category of a language, so it depends on your base for x. Similarly, the set of
all formulas that define functions from the reals to the reals may be viewed as
a language, depending on what operators you allow in your fromulas. However,
you are talking about all functions from the reals to the reals, and the set of
all functions from a set to itself is S!, where S is the size of the set. So,
the set of all functions in R^2 would be (Big'un^2)!. I suppose this rule
equating the number of functions from a set to itself as the factorial should
be explicitly derived as a theorem.



"In the infinite case" the "inductivley proven equality" between "the
number of terms" of sequence A and sequence B still holds!
Yes, in the infinite case of the set index, but not when set density is taken
into account.
In what theory? What sets? Sets don't just have density. ORDERINGS are
dense or not. And a set might have MANY different orderings.
When I speak of density on the real line, that is obviously in the natural
quantitative order, don't you think?
I'm talking about sets in general. The more often used sets of numbers
do have standard orderings, but sets in general do not.

Do you have an infinite set that doesn't have an order of generation, besides
the sets of all reals in an interval or altogether?

Sure. How about the set of all curves on the Euclidean plane passing through
a given point at least once? I don't see a particular way of generating this
set.

Nor do I at this point. Standard theory says what, that it's equal to R, or c,
yes? I'll give that some thought. I imagine it has to do with expanding the
continuum to two dimensions and mapping the points there in R^2. I'll think
about that. Good question.

Matt



--
Smiles,

Tony
.

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