Galileo's Paradox
in [Math]
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From: cbrown on 11 Dec 2006 21:10 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > > > >>> (T1) infinite(x) <-> A yeR x>y > > > >>> Tony Orlow wrote: > > > >>>>>> infinite(x) <-> A yeR x>y > >>>>> Is that your only axiom? If so, then state your first theorem about them > >>>>> and give the proof. > >>>>> > >>>> That's the only one necessary for what defining a positive infinite n. A > >>>> whole array of theorems pop forth... > >>> Before going there, you might want to start by adding the axiom: > >>> > >>> (T2) exists B such that infinite(B) > >>> > >>> Otherwise, who cares if you can prove a whole bunch of theorems about > >>> something that doesn't exist? > >>> > >>> Cheers - Chas > >>> > >> What do you mean by "exist"? > > > > That's what I get for letting sloppy notation confuse me :). > > > > I'll put it another way: When you assert "infinite(x) <-> Ay in R, x > > > y", what are we supposed to think you mean by "x > y"? > > > > For example, let T be an equilateral triangle with unit length sides. > > Is T > 1.72? > > > > Cheers - Chas > > > > Is T infinite? If you mean, does infinite(T) = 1, I don't know - that's why I'm asking. It's your definition. > Does "1.72" refer to the number of points? "1.72" refers to a real number; in other words, 1.72 in R. Cheers - Chas
From: Virgil on 11 Dec 2006 21:30 In article <457de9f7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457d95fb(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> Does TO claim that the infinite numbers of Robinson's non-standard > >>> analysis are in any way connected to the transfinite cardinals and > >>> ordinals of Cantor's analyses? Pray tell what ultrafilter generates > >>> standard cardinals and ordinals in the way that ultrafilters are needed > >>> to construct Robinson's non-standard reals from standard reals. > >> I claimed no such thing. I am saying his very reasonable approach > >> directly contradicts the very concept of the limit ordinals > > > > How does it do that? As there are n ordinals in his non-standard reals, > > and discussion of ordinality is irrelevant in his system. > > Is omega considered the smallest infinite number? Omega then does not > exist in nonstandard analysis. Omega is the union of all finite ordinals. It does not exist in standard analysis either, but that does not bar its existence, that only limits its locale. Just as the non-existence or polar bears on Antarctica does not bar their existence entirely.
From: Virgil on 11 Dec 2006 21:31 In article <457dea87(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457D8AC9.5060306(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/7/2006 1:27 AM, David Marcus wrote: > > > >>>> And which role has been envisioned for aleph_1? > >>> Kind of a silly question. aleph_1 is the first cardinal after aleph_0. > >>> That's its "role". > >> So far I was told aleph_1 means the continuum of the reals. > > > > Given the continuum hypothesis, it is. But that is not how it is > > defined, and absent the continuum hypothesis, one cannot say for > > certain. > > CH says that c, the cardinality of the continuum, is equal to aleph_1, > the first cardinal after aleph_0. That simple question appears to be > undecidable, leaving open the possibility that aleph_1 is less than c. Are you sure?
From: Virgil on 11 Dec 2006 21:35 In article <457dedaa(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> I claimed no such thing. I am saying his very reasonable approach > >> directly contradicts the very concept of the limit ordinals, which are > >> schlock, > > > > WHAT contradiction? Robinson uses classical mathematical and set theory > > all over the place. > > > > Wonderful. Then there must be a smallest infinite number, omega, in his > theory. Oh, but there's not. If there are ordinals in that theory they are not members of his set of non-standard numbers, though they might be needed in its construction. What do you know about ultrafilters, TO? > There is no need for omega in nonstandard analysis. There is no smallest > infinite allowed at all. He makes reference to "countablility" but I > haven't seen any alephs about yet. That is because TO doesn't know how to look.
From: Virgil on 11 Dec 2006 21:45
In article <457dedef(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > Just as Hilbert could improve on Euclid's geometry, those who follow can > > improve on what those who have gone before have wrought, but not the > > reverse. > > Hilbert's axioms need to be reviewed. I turned the first 8 into four > more powerful ones, and I'm not even a mathematician. The judgment that Hilbert had improved on Euclid's axioms that of a vast majority of all of the many contemporary and subsequent mathematicians who expressed any judgment on the issue. What jury does TO wish to subject his alleged improvements to for judgment? |