Re: Galileo's Paradox

Tony Orlow wrote:
Can you cite where he uses omega in the development of NSA please?

The existence of the set of natural numbers is used all over the place
in the development, as a well as countable infinity, and on page 52,
omega is used in a formula of ordinal arithmetic to describe the order
type of the non-standard *N.

You are just FLAT OUT INCORRECT that non-standard analysis is
incompatible with omega.

What are you TALKING ABOUT? Read Robinson (which means reading the
actual development, not just isolated passages), why don't you, instead
of ignorantly spouting about what YOU THINK he does and does not need.

There is no need for omega in nonstandard analysis.
Robinson works in classical mathematical logic and set theory, in which
omega exists. IST includes Z set theory, in which omega exists. Or, if
you want to point to so other treatment of non-standard analysis in
which treatment does not also entail the existence of omega, then
you're welcome to do it, but it ain't Robinson and it ain't IST.


He ignores it. It would contradict his internally consistent theory, and
that would bother set theorists.

No, he doesn't, you jerk. Read the damn book. And read his other works,
and read his summary of his mathematical philosphy in his essay
"Formalism 64".

There is no smallest
infinite allowed at all.

It's not a question of "allowed". You really understand NOTHING about
this. In particular sets and systems that are proven to exist, ordinals
are not members. So what? The ENTIRE theory in which this takes place
DOES prove the existence of ordinals. Look, no ordinal is a complex
number, but we construct the complex numbers in a theory in which
ordinals do exist, even if ordinals are not complex numbers. No ordinal
is a non-standard real. But the theory in which non-standard reals are
proven to exist does also prove the existence of ordinals.


Ahem. He "proves" it cannot exist, just like the monkeys prove there's
no largest banana.

What a complete loser you are. He's ALREADY taken set theory, and the
existence of omega, which is proven in set theory, as RUDIMENTARY.

There's no tiniest giant. He's talking about
something on the same continuum as the reals and naturals and rationals.
He's not talking about any extra dimensions of specification like with
complex numbers.

You competely missed the point of my analogy. No surprise there.

Ordinals and cardinals are naturals in their finite
state. Do the infinite forms of them jut off in other directions?
Apparently so. Robinson's do not.

Isn't it time Nonstandard Analysis became the Standard?

He makes reference to "countablility" but I
haven't seen any alephs about yet.

The ordinals themselves are not members of the non-standard number
system, but DERIVING the existence of a non-standard system takes place
in a theory in which ordinals do exist.

Yeah, over there, and they can't play this game. They don't belong to
this continuum.

You're hopeless. Apparently, you demand just to pontificate about an
author's work of which you even ADMIT you don't know the rudiments that
the author himself stipulates.

You can't just rip one part of
a theory, like a shard, out of a whole theory. Perhaps there is a
non-standard analysis that can be devised without classical
mathematical logic and ZFC, but Robinson's work does NOT do that. He
uses classcial mathematical logic and set theory all over the place in
connection with results in non-standard analysis. And IST includes
EVERY SINGLE theorem of Z set theory.

Ae you saying there is no contradiction between standard and nonstandard
analysis, no cnclusions that are different?

They are conclusions about DIFFERENT objects and different kinds of
objects that still exist in a single theory. Why do you REFUSE to
understand this?

Read the VERY FIRST SENTENCE in Robinson's book, why don't you.

Did you read "contemporary mathematical logic" to mean "transfinite set
theory"? You can do better than that.

Contemporary mathematical logic is inextricably woven with set theory.
The languages mathematical logic usually talk about have an infinite
set of symbols. The theorems of mathematical logic are flying all over
the place with resuts about countability and uncountability and
infinite sets of symbols, formulas, theorems, and axioms. Robinson uses
the set of natural numbers and countable infinity throughout his
developments. And he mentions understanding of "Abstract Set Theory" as
rudimentary for the book, and it is used, if you even bothered to READ
the damn book.

MoeBlee

.

Relevant Pages

  • Re: infinity
    ... > Here's a way to visualize countable ordinals that might ... The point 1 is not included unless you allow infinite n, ... > Infinite sets of naturals *don't* have a largest element. ... But that isn't mathematics. ...
    (sci.math)
  • knowledge is power
    ... Where was that extremely dense mass of the particle situated ... chain of infinite number of limited objects of timeless and spaceless ... that existence was created by a super deity of heavenly powers,we ... If our present material universe is 10 to 20 billion years of age,and ...
    (sci.geo.meteorology)
  • knowledge is power
    ... Where was that extremely dense mass of the particle situated ... chain of infinite number of limited objects of timeless and spaceless ... that existence was created by a super deity of heavenly powers,we ... If our present material universe is 10 to 20 billion years of age,and ...
    (sci.physics.particle)
  • knowledge is power
    ... Where was that extremely dense mass of the particle situated ... chain of infinite number of limited objects of timeless and spaceless ... that existence was created by a super deity of heavenly powers,we ... If our present material universe is 10 to 20 billion years of age,and ...
    (sci.astro.amateur)
  • knowledge is power
    ... Where was that extremely dense mass of the particle situated ... chain of infinite number of limited objects of timeless and spaceless ... that existence was created by a super deity of heavenly powers,we ... If our present material universe is 10 to 20 billion years of age,and ...
    (sci.physics.relativity)
Loading