Re: Galileo's Paradox

Tony Orlow wrote:
MoeBlee wrote:
Tony Orlow wrote:
I have read Robinson. On what page of what book does he refer to omega -
1 in comparison to omega? I do not find any such reference.
He uses the assumption that any infinite number can have a finite number
subtracted, and become smaller, like any number except 0, so there is no
smallest infinite, just like you do with the endless finites.
Non-Standard Analysis, Section 3.1.1:

You are REPEATING the mistake you've made from the beginning regarding
Robinson, because you never bothered to read it and understand it but
instead just jumped around to find things you only THINK say what you
think they should say.

The fact that I'm repeating it might serve as a clue that it's not a
mistake.

You have an odd notion of what gives evidence. Your propounding an
error over and over and over is NOT evidence that you are correct.

In CONTEXT, when Robinson is talking about those infinite numbers, he
is NOT talking about ordinals such as omega. There are two DIFFERENT
senses of 'infinite' in play, one is infinite cardinality and the other
is that of a certain ordering of elements in certain sets, which is NOT
a cardinality ordering and does NOT apply to omega the way you have
answered.

No kidding. Omega is the smallest infinite. It doesn't exist in NSA. My
statement was that using omega as an infinite number in the context of
NSA is contradictory, and so NSA contradicts transfinitology. Get it?

It isn't an element of certain sets and number systems. But it does
exist in the same theory in which non-standard analysis is a part. For
example, as an analogy, omega isn't a member of the set of complex
numbers, but omega does exist in a theory in which we construct the
complex numbers. For that matter, omega doesn't exist as a member of
the set of natural numbers, but omega does exist in the theory in which
we "construct" (scare quotes important) the natural numbers.

And I can predict you're going to say something along the lines of that
you're not saying what I have said you said. But then your answer makes
NO SENSE as a response about OMEGA.


It's a comment on the uselessness of omega, and the fact that NSA is not
built solely upon transfinitology, or there would be no contradiction in
conclusions. It's an alternative view of infinite numbers, one which
actually makes sense.

Robinson's work is built SOLELY on classical mathematical logic and set
theory (and even though he himself doesn't use a first order theory for
the construction, it is still classical mathematical logic, which
includes not just first order logic).

READ THE VERY FIRST SENTENCE OF THE BOOK.

"There is no smallest infinite number. For if a is infinite then a<>0,
hence a=b+1 (the corresponding fact being true in N). But b cannot be
finite, for then a would be finite. Hence, there exists an infinite
numbers [sic] which is smaller than a."

Of course, he has no for omega. It's illegitimate schlock, like I said.

What does, "he has no for omega" mean?

"no need" Sorry.

Of course he does. Read his writings.

Please cite where he employs or even references omega in the book. I'd
be interested. I haven't finished it, so I can't swear he doesn't,
but....I'm pretty sure he has absolutely no use for a concept that is
impossible within his theory, at least while discussing his theory.

It's NOT impossible in "his theory", because "his theory" is all in
classical mathematical logic (even as he sometimes uses logic that is
not just first order) and set theory. READ THE VERY FIRST SENTENCE OF
HIS BOOK. As to specific passages, I would have to go to the library to
get the book. And, in addition, to the book, read from his collected
papers as they are published in bound volumes. In one essay he
explictitly endorses classical mathematics and his many papers are all
steeped in it. And as to other approaches inspired by his work, look at
the actual mechanics and axioms of IST, or at model theoretic
constructions such as given a nice synopsis in Enderton's text, or look
at the ultrafilter approaches - all in classical mathematics. Even
BEFORE Robinson, non-standard models were discovered through model
theoretic notions that observe Lowenheim-Skolem-Tarski and compactness
and all kinds of things about infinite sets.

ONE MORE TIME: Don't confuse two DIFFERENT senses of the word
'infinite'. One applies to cardinality, the other applies to certain
kinds of ordering. If you DON'T confuse these two different senses,
then you will see that there is NOT a conflict between non-standard
analysis and infinite ordinals.

MoeBlee

.

Relevant Pages

  • Re: Uncountability of the Rationals?
    ... My rules don't say such a set doesn't exist - all countably infinite ... The set omega still exists and is the ... TO intends his ICI ... TO states that "tav" could represent any set, ...
    (sci.math)
  • Re: Cantor Confusion
    ... And the number of digits is omega. ... why I insist that an actually infinite set of natural numbers must ... If your axiom contradicts this, ...
    (sci.math)
  • Re: Raatikainens critique of Chaitin
    ... in this case infinite complexity. ... Omega can be said to represent halting problem's structure ... Here is a definition of randomness in Kolmogorov complexity: ... A real r is Chaitin random if the information content of the initial ...
    (sci.math)
  • Re: infinity
    ... >> If by omega you mean the smallest infinite cardinal (usually referred ... that given a finite set, you can count it by singing a ditty, and using ... Suppose the attempt to sing the ditty never stops - in that case ... >> creates an infinite element, ...
    (sci.math)
  • Re: Raatikainens critique of Chaitin
    ... in this case infinite complexity. ... Omega can be said to represent halting problem's structure ... Here is a definition of randomness in Kolmogorov complexity: ... A real r is Chaitin random if the information content of the initial ...
    (comp.theory)