Galileo's Paradox
in [Math]
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From: MoeBlee on 11 Dec 2006 18:18 Virgil wrote: > But, as I said, is about as non-existent as a string can get. Any less > existent and it wouldn't be a string at all. Okay, so there are degrees (degrees of more and of less) of existence? MoeBlee
From: Tony Orlow on 11 Dec 2006 18:27 Virgil wrote: > In article <457d965d(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <457cc0ce(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> I suppose now you're going to tell me I'm using nonstandard language.... >>> Almost always, as far as the meaning of standard mathematical terms goes. >> Did you at least learn what a formal language is now? I remember you >> taking issue with me about the "null string" as if no such thing >> existed. > > > Where pray tell did I ever do that?. > > Besides, in a sort of sense, isn't being a null string about as > non-existent as a string can get? The existence of the null string is uncontroversial, as opposed to the existence of my uncountable T-riffic strings. :)
From: Tony Orlow on 11 Dec 2006 18:28 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.
From: Tony Orlow on 11 Dec 2006 18:31 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.
From: MoeBlee on 11 Dec 2006 18:35
Tony Orlow wrote: > Is omega considered the smallest infinite number? Omega then does not > exist in nonstandard analysis. You'll have to define 'exist in non-standard analysis'. In the set theory that is presupposed for the work of non-standard analysis, omega exists. In classical mathematical logic that is the very hearth of non-standard analysis, we suppose the existence of the set of natural numbers. And in IST, omega exists. You seem to have the mistaken impression that non-standard analysis is some kind of mathematics that is separable from classical mathematics and set theory. MoeBlee |