Re: Calculus XOR Probability

MoeBlee said:
Tony Orlow wrote:
And what allows you to conclude that this ordinal is the size of the set,
especially when it is not any definite sort of a number?

I didn't use "size of set".

Look above. See my question? You answered it with the following nonsense, as if
that referred to the size of the set. So, my question is relevant, and your
inability to see the relevance isn't my issue.

Your question was what allows one to define the size of a set in such a
way.
Tony:
"Consider N is the length of the real line in unit intervals, and the number of
naturals."

Matt:
"What allows you to conclude that the real line has a definable length?"

Tony:
"It is an axiomatic statement in my system that there is a unit infinity N,
which is the length of the real line, and therefore the number of unit
intervals, or naturals, on the real line, as well as being the number of
nipotent infinitesimal reals in the unit interval. What drives me to this
declaration, which is perfectly allowed in an axiomatic system, is that it
works with the Inverse Function Rule, which works perfectly well for finite and
infinite mappings to the point of eradicating the Continuum Hypothesis and a
number of other unanswerable questions in your system, in the context of this
system. If we can agree that there is a real number line, then I can simply
call its length N.

What allows you to conclude that the set of naturals has a definable size?"

Matt:
"Well, basically, I could define a property of sets; I'll call it
"size". I would show that is well-defined for an arbitrary set S. Then,
knowing that the set N of natural numbers is a set, I conclude that the
set N of natural numbers has a definable size."

MoeBlee:
"Why do you keep asking questions whose answers have been typed out for
you over and over and over again? The answer to that question is:
Because you don't actually read and think about those answers.

The answer to your question is again: It is a theorem that every well
ordered set S is 1-1 with an ordinal. Thus, there is a least ordinal
that S is 1-1 with. And 'the least ordinal 1-1 with S' is the
definition of 'the cardinality of S'. The set of naturals is a well
ordered set, so it is 1-1 with an ordinal. The least ordinal that the
set of naturals is 1-1 with is itself. Therefore, the cardinality of
the set of naturals is the set of naturals.

Please read a book!"

Tony:
"And what allows you to conclude that this ordinal is the size of the set,
especially when it is not any definite sort of a number?"


MoeBlee, you can go back over the history of the thread yourself and confirm
that this is the chain of events that leads to this point. If you don't think
that you have defined a real *size* for the set, then why are you jumping at a
question regarding the definition of set size, and then talking about ordinals
and cardinality, and then saying this?


My answer was that I didn't use the words "size of a set". What do you
want from me? You might as well ask the proverbial, "When did you stop
beating your wife?" My views do not require me telling you how I
conclude that the size of set is anything at all, since I NEVER SAID
ANYTHING about the "size of a set". And NO I did NOT answer "as if
that referred to the size of the set". Please do not put words in my
mouth.

hahaha ho ho hum. Put words in your mouth.....that's a good one. I asked Matt a
specific question about the set having or not having a definable size, and you
answered what you though was a different question, and by pointing that out,
I'm putting words in your mouth? You don't have to answer EVERY question, you
know, especially with the answer to a different question. Sorry to be so blunt.


Then, the part you call irrelevent meandering was just my explaining to
you, yet again, even as you still do not understand, that by using the
word 'cardinality', makes NO SUBSTANTIVE difference in the theory.

I am not arguing with the mechanics of your cardinalities. They're carved in
stone. I simply mean to point out that they fall short in some respects
regarding what we might wish to expect from something we regard as the "size"
of a set. In my mind, it is most fundamental that this measure we call "size"
always change with the addition or removal of elements, if it is to be
considered at all precise. Otherwise, it's just a broad classification types of
infinities. Hey, that's better than nothing, a step in the right direction, but
not something I could feel confident about describing as the "size" of an
infinite set. Cardinalities are cardinalities, and happen to be equal to what
we consider "size" for bounded finite sets.


As to "definite sort of number", you'll have to define that.

Being as it's not part of my theory, no, I don't have to define why aleph_0 is
not a definite number.

You're claiming that set theory fails to provide a definite number? Is
that right? Then whether 'definite number' is part of your system or
not,f or us to evaluate whether set theory fails to provide what you
seem to be asking about (definite number), we have to know what you
mean by 'definite number'.

I mean a number that denotes some number of some sort of units. Something you
can perfom arithmetic on. Something that when you add to it or subtract from
it, it changes. Something that has SOME relation to quantity as we know it. You
know, that sort of a thing. A NUMBER.

I hope that was clear. :D


No. First you have to define 'real line', and 'length of' and then
prove that there exists a uniuqe object that satisties the description
'the length of the real line'.

Yes, and thathas to be built up from primitives, and I'm working on that, in
the rare moments when I have time to think. Tonight I get a bit of a break.
we'll see what I can accomplish with that time.

Then until you come up with something, you're unreasonable to think
that people should see your notions as anything at all, let alone
embodying basic truths upon which mathematics is to be axiomatized.

Perhaps. In the meantime, it's sort of fun to discuss and defend, and if it
succeeds, in the end it will make a good story, or maybe even a movie. Yeah,
Well Ordering the Reals II: Night of the H-riffics. Whom would you like to play
your part? I was thinking maybe James Earl Jones..... ;-)


MoeBlee



--
Smiles,

Tony
.

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