Re: Logarithm of transfinite numbers

Virgil said:
In article <MPG.1e9db50fda772c0b98abec@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1e9c6d5fba15cf4e98abe3@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

You have heard, perhaps, of Orlow's amazing H-Riffic Number System?

I have heard of unicorns and leprechauns, too.

Have you ever seen one?

I have never seen "Orlow's amazing H-Riffic Number System" either.


1. 0 is a real number
2. If x is a real number, then 2^x and 2^-x are real numbers.

You have seen it, and now you see it again, and you've seen other versions of
it as well.




There's
enough confusion, so the unit infinity is BigOne,

What is this "BigOne" supposed to measure?
It cannot be the "length" of the reals since there is no such thing.
Shouldn't it be spelled "begone"?


Actually, I decided last night the unit infinity is Big'un and the unit
infinitesimal is Lil'un. That gives it a nice twang. Big'un's the length of
the
entire real line measured in units

What sort of units does TO propose which are capable of measuring "the
length of the entire real line" in any system currently extant?


None is any widely accepted system, but in mine there exists the infinite unit
Big'un, which IS the length of the real line.


and so is also the number of unit intervals and natural numbers on
the real line,

First TO must show that all these things have "measurements", which
requires a definition of 'measurement', secondly that there is a unit of
measure suitable for all 3 measurements, thirdly that they all are equal.
TO so far has done none of these things.


Actually, I can axiomatically state these things and treat Big'un as a
primitive, and if I can derive no contradictions, and can derive useful
results, you have nothing to complain about.




Big'un measures the Universe of Quantity

What is the "Universe of Quantity"?

The Real Number Line, as my previous sentence went on to say. Why snip the
answer to your question and then ask it, instead of just reading the answer
like a normal person?




The size of a set is its Bigulosity

How does one compare the"Bigulosity" of two sets?

For infinite sets one compares the formulaic expressions defining the
size of each in terms of Big'un, and if one is greater than the other
for all n greater than some identifiable value, then one can say that
that order relation is preserved in the infinite case, and precisely
say that, for instance, Big'un> Big'un/2, and there are twice as many
naturals as even naturals.

Replacing one undefined expression by another does not advance things.

Oh. Define 0.


How does one determine these "formulaic expressions" for arbitrary sets?

Using the mapping function from the naturals, or N=S^L, depending on whether
your infinite set is quantitative or symbolic.


Cardinality has precise explanations of how to compare set sizes.
Bigulosity has gobbledegook.

"Precise"? Only in the sense that you always get SOME answer, but not in the
sense that it is able to distinguish sizes for many kinds of sets. Leave that
to Bigulosity. Your theory can't even talk about the KIND of a number of bits
required for your ill-defined set of naturals. That's not "precise" in my book.
It's schlock. :)





With cardinalities there is a nice clear method which gives a provably
consistent ordering of sizes for well ordered sets, and for arbitrary
sets given an axiom of choice.

But absent rules for comparison , and proof of a consistent ordering of
"Bigulosities", the mere word means nothing.


Please critique my wording above.

Done above!

"Replacing one undefined expression by another does not advance things."

That just sounds like it went over your head, as usual. Do you disagree that
one can consider a formulaic expression to be larger than another for infinite
variables if it is larger than the other for all finites greater than some
value? Yes or no?




, and we
have regular numbers, big numbers, and lil numbers.

I repeat, absent rules for comparison , and proof of a consistent
ordering of "Bigulosities", the mere word means nothing.

Sure, but I have put forth the statement above recently, and don't recall
you're ever having responded to it. DO you find anything unreasonable about
this use of formulaic comparison?

Until the rules for constructing these formulaic comparisons are clearly
set out, yes! Handwaving doesn't do it.

Well, I have been discussing the use of IFR lately, and went over a number of
examples using N=S^L some months ago. So, if I'm waving my hands, maybe it's
just to keep the flies out of my face that seem to congregate in this "garden".


Right, so you're asking again about the order relation, and I'll
reiterate that's a good point, which I think needs to be addressed by
the construction of the number line starting with 0 and 1, and which
I need to consider more in trying to form this axiomatic basis.

To anyone competent to build a consistent system, these elementary
issues would be obvious enough not to need pointing out by others.


I told you I was already thinking about that, and since I am trying, as
suggested, to build a system from the ground up to integrate all these ideas,
the axiomatic foundation has to be laid just right to avoid subsequent
issues.
Excuse me while I contemplate the nature of truth itself and the quantitative
foundation of logic, as well as the derivation of quantity from geometric
truth, so as to achieve this goal. Okely Dokes?

Only if you do it off line until you have it all done.

Come on, Virgil. If I disappeared, you know you'd miss me. I give your life
meaning, like a squirrel to a dog, like a mouse to a cat, like a Triple
McFatBurger to a disgusting piece of white trash. You love it!

Geometry->Quantity->Logic. Thoughts from the deep?



Does TO claim that the sum of an endless series of positive naturals has
any Dedekind finite natural as a sum, or even as an upper bound?

Unboundedly large but finite, as I've said dozens of times.

And TO has been wrong each time.

WRONG!! :D



Divergent series, such as these, never "achieve" any kind of value. That
is the point of "divergence".

In fact, it's quite intuitive to think about a completed infinity like
Big'un,

Intuition is bad mathematics unless it can be backed up by a logically
coherent system. So far, TO only has intuition with no logic and no
coherence and no system.

Would for your sake that that were so, and understandable that you see it as
such, for the King's most practiced archer dismisses the Zen archer's aim as
blind luck, and without consequence. The discipline that underlies the aim of
faith is not obvious to those wearing the uniform of the ranks. But, hey, that
goes without saying, of course! :D


The present systems, and there are of them, work fine. If TO thinks he
can do better, let him present his alternate system up front rather than
reiterating his unfounded and unsupportable claims.


I have presented plenty of aspects of it

TO wants to build only the top story to his structure without any
foundation. That leaves everything entirely conjectural.

Yes, well, building from the roof down has its drawbacks, but painting from the
floor up makes no more sense.


Go off and work on that foundation for a while, TO, and don't bother
trying to prop up the attic until you have a basement to stick under it.


Do you pour a foundation before you know what shape the house will take, or
what load the foundation must support? Do you assemble a car, starting with the
wheels? Is a baby born feet first? Well, you probably were.


For Virgil guards the Gate

The gate hardly need guarding from the efforts of fools like TO.
But it is rather fun to poke holes in the fabric of his illusions.

Well, that's true. There are dozens of ways to enter the garden, besides the
gate, like through the gaping holes in the wall. ;-)

The gaping holes are all in TO's conjectures. If the walls of standard
mathematics had such holes as TO alleges, on present evidence, TO would
be too dim to find them.


Yes, of course, you are right as always, O Wise and Terrible Virgil.

--
Smiles,

Tony
.

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