Re: Logarithm of transfinite numbers

Tony Orlow wrote:
Ross A. Finlayson said:
Virgil wrote:
In article <MPG.1e9df60b9a559eac98abf0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

MoeBlee said:
Tony Orlow wrote:
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.

That would be good. But since you criticize other mathematics on
the basis that its axioms aren't true as statements about a
fundamental reality, then your own axioms are subject to such
scrutiny too. I don't know why you would think it is so manifestly
true that there is an object that exists as a fundamental reality
that is the length of the real line but (if you do hold this:) that
there isn't an object that is the set of counting numbers.

Yes, that's not a vacuous point, and one I can appreciate. Assuming a
length to the real number line when it has no discernible ends does
seem somewhat arbitrary, and that's why it needs to be assumed a
priori, since it's not really a derivable value.

Thus TO is claiming that it is legitimate for HIM to assume things
arbitrarily but not for anyone else?


But, if we say there
exists this infinite line, then there is SOME length to it

Why does something that does not have ends have to have something that
requires that it have ends?



It appears that the assumption that this
value is not only the length of the line, but also the number of
points within any unit segment of it

It does not appear so to us, so that we will require formal proof of
that claim.





Oh. Define 0.

The unique x(Ay ~yex).

That sounds like a definition of the null set ala von Neumann, but
not necessarily going to the heart of what 0 is. Really, I was
referring to its use in the Peano axioms, where is is taken as a
primitive. I see no reason why it can't be taken as an assumed
primitive and a starting place, and actually see this as a natural
starting place for mathemtics as a whole. Start with nothing.

That is precisely what von Neumann did, and is precisely what TO has
violently objected to in the past only because it showed how stupid one
of TO's arguments was.

What's your point?

We don't need protection from Tony here, Virgil, the above being one of
your less abusive posts. I think Tony is sincere and quite pleasant,
knowledgeable and sincere.

That's because I don't call you "Kerberos". Heh heh. Virgil's okay. He needs to
bark. That's his job. I only get annoyed when my arguments are twisted into
silliness dishonestly, which he doesn't seem to be doing TOO much these days. I
think he has a right to defend what he believes. This is debate. Rock on, Ross!


I suppose you won't reply. Do you have a point? It's probable that
you should assume, say, a basically high level of mathematical
sophistication of your readers, were you rational, perhaps in
combination with social.

Don't you have anything positive to say? That does not reflect well,
because of the negative things, where saying nothing is not necessarily
the same thing as saying nothing nice.

You know Ross, Virgil serves his purpose. His challenges are welcome, when he
gets specific about what questions he expects answered and what definitions
need to be established in order to make the theory even remotely acceptable.
If, heaven forfend, I can one day satisfy Virgil's requirements, it seems
almost impossible that anyone else could have the tiniest objection. So, if he
sets the bar high, good, that gives me something to shoot for.

Your denegerate intervals and Han's infinitesimal probabilities are vital to
IFR and the solution to the Continuum Hypothesis and more. It's a good thing,
this. :)


There's a good point in saying that there are as many reals, on the
line, between zero and one as there are naturals. Then when you sum
their values over the naturals as is the generally understood method in
the integral calculus or "Cholera bacilli analysis", that gives perfect
and generally perfectly expected results. That's where the integral
bar is an S for summation.

Absolutely right, and what Han's been arguing, alas, to no avail. It seems such
a straightforward concept, and yet, mathematics has made a concerted effort to
eradicate these bacilli, to the extent that it has destroyed its own beneficial
E. Coli and can no longer disgest new ideas without difficulty. I think we
offer a form of mathematical yogurt, a central staple in the diet of any vital
168 year old Georgian horseman.


Basically post-Cauchy-Weierstrass in the quest for foundations of the
eighteenth and nineteenth centuries certain results and counterexamples
led to basically denial of the quite real utility, and thus in a way
quite obvious formal soundness, of the infinitesimal analysis. That is
where at the same time its justification is disguised as
delta-epsilonics, with basically the exact same notion and quite
widespread usage of the Leibniz notation, how in the limit via infinite
induction it works, including ratios of infinitesimals, eg dx/dy.
Consider L'Hospital(e), and the quadrature (squaring, in 2-D), for
different reasons, and the same.

Obviously in my theory the points are polydimensional, with being one-
or two-sided on the line, in continuity one-sided and with weight over
the summation index of the index' reciprocal, scalar, in the
polydimensional: inverse, with geometry in the large and small, the
very.

There's something new, for you.

Yes, I haven't quite integrated your one- and two-sided points yet, though
multidimensional points are not unreasonable in the limit.


The universe is infinite, infinite sets are equivalent. Familiar? How
about the notion that there is a universe, would you agree that there
is a universe. The more comprehensively it's examined the larger it
gets. The more closely subatomic particles are examined the smaller
they get. (Time stops for particle/waves with the HUP. Please explain
gauge invariance.) Right: start with nothing, don't care. That's so
it goes.

Obviously in my theory of axiomless natural deduction, currently A
theory, there's only one theory with no axioms. Conveniently, only and
all true statements are theorems.

I would like to see derivations of some theorems from first priciples and pure
logic. That's what I'm considering now in the my attempt at foundations. Have
you managed to derive theorems formally this way? We need to do this, as they
have, if we wish to get the Garden to flower again. Also worms. Lots of worms.
We need worms. ;)


Ciao,

Ross F.



--
Smiles,

Tony



Hi Tony, hey how's it going,

In terms of Virgil, your repartee, which was not necessarily
unentertaining, to at least one external observer was not. When you
two hit the mud slide there it detracts somewhat from the mathematical
discussion, and particularly from yours. This is sci.math, humor is
irrelevant but appreciated, it's also irrelevant, and the less sociable
basically derision, particularly where unwarranted as are some of
Virgil's comments to you, and in the past me, about such notions as,
say, rationality and mathematical creativity, are unacceptable.
Virgil, don't get me wrong, I would stand for your right to speak, my
fist ends where your nose begins.

Nah, that's wishy-washy, Tony, I guess it's okay, the point is that
you're okay with that, and that you as well respect even Virgil here.
My problem is with Virgil's disrespect. I think he's chosen his own
role much as you have, a different one for different reasons, because
he purposefully never ventures.

Tony, with your N = S^L, or as I say b^p, that basically happens in the
finite. In the infinite, where it's sufficient for b or p to equal
one, S or L, i.e. unary or base infinity, then the real numbers of the
unit interval exactly correspond in a 1-1 bijective mapping to the
natural integers, and there is an obvious well-ordering of them, their
natural ordering. Otherwise there is none, yet there is a
well-ordering of the reals.

With regards to the powerset result and the antidiagonal argument to
coded powerset in those terms, the powerset is successor is order type,
in ubiquitous naturals, or in a pure naive set theory, of the elements
that quantify the universe. With the mapping f(x) = x + 1, number to
successor, that is the powerset mapping and illustrates how as objects
are discovered in the universe, the functions between them emerge. In
ubiquitous ordinals, the powerset is order type is successor, and the
powerset is simply a "new" kind of ordinal, an ornate ordinal.

That allows there to be a universe, and quantifiers over said universe,
without Cantor's paradox. (That's not the same as Russell and the Liar
and their conflation.) The universe is infinite and infinite sets are
equivalent. There is a universe, there is no universe in ZF, the
universe is the constructible universe, or actually the constructed
universe. There is a universe and over sufficiently primitive objects,
based on the empty set, quantification over all objects in the universe
results in the universe, thus ZF is inconsistent, as a simple
expression of mathematical logic about mathematics, because no
predicate always resolves to true, yet everything in ZF is a set.
Every thing in ZF is a set, thus everything in ZF must be a set.

That leads to notions of infinities as irregular, not well-founded.

The notion of the unit infinity and infinitesimal, or scalar unit
infinity and infinitesimal, I am happy to say that others who were in
previous, say, years adamant against their consideration are not so
disinclined today. It's an ancient notion that returns again and again
after its denial, because it applies.

So, where reapplication of those intuitive notions, buttressed with
mathematical logic justifying their existence, leads to reevaluation of
primary results in measure theory, and the possibility of an entire new
field of analytical results in the neo-classical, that's good.

Regards,

Ross F.

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