Re: infinitely many nn's = infinite nn's?

george wrote:

On Apr 3, 7:26 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote:

I think the reason that the infinity of nonstandard
analysis appeals to me so much more than Cantor's
is because it fits Zeno's paradoxes so much better.


Well, this explains a lot. The rest of us think that the
reason you think Zeno's paradox is a paradox is because
you are an idiot.

*** you, George.


In the Dichotomy, we cross the same set
of infinitely many halfway points that we cross in the Achilles, but in
reverse, starting from the door. If we let the complete set of halfway
points = the nonstandard positive unlimited integer x,
and define the point in the middle of the room as being
"1/2," with successive points
moving toward the door being 1/4, 1/8, etc., then after crossing
infinitely many (but not all) of the halfway points beginning at the
door, and reaching the point "1/8," we will have
crossed ALL BUT TWO of
the infinitely many points, or (x - 2) points.


If you are doing "how many", then EVERY infinite set has
the property that if you take 2 things away from it, IT STILL
HAS THE SAME infinite number of things in it. Deal with it.

Nope, not in nonstandard analysis. Deal with it.

Try to wrap your infantile brain around this simple -- I mean, REALLY simple -- series of FACTS. First, if I walk from the door to the 1/8 point, I am NOT at the 1/2 point, i.e., I am NOT in the middle of the room! That's a FACT, shithead, not some stupid oversight on my part. Second, it is a LEGITIMATE goal, and an understandable wish, to have mathematics that can not only agree with that FACT, but also explain that FACT. In nonstandard analysis, given an infinite number x, (x - 2) is NOT equal to x!!! That is consistent with the act of walking into a room. In contrast, the FACT that under Cantor's mathematics, you can, as you say, take 2 from an infinite set and still have the SAME NUMBER of elements, is NOT consistent with the act of walking into a room. That means, stupid dumbass, that when I say that the "concepts of infinity" found in nonstandard analysis fits Zeno's paradoxes better than the concepts of infinity found in Cantor's mathematics, I'M RIGHT!

WHAT is your problem with this??? Even my own thoughts are, SO WHAT??? Okay, Robinson's contributions to mathematics have a better fit to Zeno's paradoxes than Cantor's contributions. SO WHAT??? Are Zeno's paradoxes the beginning and end of mathematics??? I don't think so, but you ACT as if YOU do! Why do you ACTUALLY think that a system in which (x - 2) <> x, where x is an INFINITE number, is not at least APPARENTLY more consistent with the Dichotomy than a system in which (x - 2) = x? You know, it won't hurt your reputation any if you OCCASIONALY agree with some extremely minor observation on my part. What ON EARTH is your problem here? I'm serious, this is just WEIRD. I'm used to you being a shithead, I am NOT used to you being this WEIRD ... Your comments may usually be irrelevant, but they are almost never STUPID. You're throwing me off base here, WHAT is going on? Maybe you're a Cantor fanatic?


In nonstandard analysis we can say this,


That is far from clear. You Do Not Know enough about
nonstandard analysis (or, far more relevantly, nonstandard
arithmetic) to be mapping any nonstandard numbers to
any of the ENTIRELY STANDARDLY real points in the Zeno's
Paradox scenario.

Blah blah blah.


but with Cantor's transfinite cardinals (ordinals?),


No, cardinals, if what you are to say afterward is to make
any sense. Ordinally this will all just be wrong.

THANK YOU!!! This is honestly helpful, and I do appreciate it. You may not like to admit this, but I really don't think I know EVERYTHING, and I do appreciate getting corrections and help from others who know way more facts than I do. Of course, I may think that someone in possession of many FACTS cannot THINK, but that's a different matter ...


that just makes no sense, since even 2 * x = x.


That is NOT true. What IS true is if that x is an infinite
ordinal, 2*x will be an ordinal with the same infinite cardinality
as x. In fact, 2*w=w, if w is a limit ordinal. But this is not the
same as w*2, which is w+w, which is ordinally greater than w.
It will, however, have the same CARDinality as w, despite
being ordinally bigger.

Okay, thank you again. I freely admit that I am confused about cardinals versus ordinals in infinite sets, or numbers, or whatever the proper term(s) is. I understand that w+1 is "greater" than w in terms of the SEQUENCE of terms (w, w+1, w+2) that one sees, but the ordinal versus cardinal thing confuses me. If the cardinality of a set is simply defined as the transfinite cardinal that the set in question has a BIJECTION with, okay, but can the set's ordinality be diferent from that? I don't understand that at all ...


In fact, as long as the
infinite numbers in question belong to the
same -- cardinality? -- you know what I'm trying to say,


We know you have got to STOP saying "numbers"!
EVERYthing is a number, from one viewpoint or another!
There are BUNCHES OF DIFFERENT KINDS of numbers!
There are standard and non-standard integers, rationals,
reals, ordinals, cardinals, imaginary, and natural, and infinitesimal
NUMBERS. Hint : the next time you get ready to write "number",
just THROW AWAY the "number" part and write THE ADJECTIVE
TELLING WHAT *KIND* of number it was instead. THAT might
make you readable. It might also get you to THINK about what
kind of number it was. That would help; thinking is not something
you are currently doing enough of.

Well... that's not a bad idea! (the number part, NOT the thinking part, which I currently do MUCH better than you ever have.)


as long as two infinite values,


It would also help if you would stop saying "values";
that's even more vague and therefore more unconscionable
than "numbers". What you MEANT to say here was CARDINAL
numbers, or, obeying the new linguistic directive, cardinals.

Sorry, over my head.


x and y,

both have a cardinality of, say, aleph-0, then x*y = x,
and x^y = x, and even x! = x.


That was already refuted, but again, just google
Cardinal Arithmetic and you will become able to say
this coherently.

Now wait a minute, there are proofs, using a bijection between two sets, for everything I said here. And it is also true that NONE of these are equal in nonstandard analysis.

Phil
.

Quantcast