Galileo's Paradox
in [Math]
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From: MoeBlee on 12 Dec 2006 15:43 Tony Orlow wrote: > MoeBlee wrote: > > 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'. > > Sorry. Exists = Does not produce a contradiction. Then you have a PERSONAL definition of 'to exist'. By your definition, many things exist, simply by consistency, even though they are not proven to exist. For example, with your definition, in Z, a choice function would exist on every set, even withOUT the axiom of choice. You operate in a shell of your ignorance. You're a complete joke. > S is the set of all infinite values, xeS, yeS: > > ~(Ax Ey: ~(x=y) -> y<x) ^ (Ey Ax ~(x=y) -> y<x) > > You cannot have a smallest infinity, and also not have it. > > In the set Omega is not a member of the sets your'e talking about. But omega is an object proven to exist in the theory of which the sets you are talking about are only SOME of the sets. How many times does this have to be explained to you before you understand? Look, for example, omega is not in the set of complex numbers. But omega and the set of complex numbers, and each complex number are all objects in such theories as Z set theory. You have the most RIDICUOUSLY stubborn mental BLOCKS, because you are so DESPERATELY attached to things being the way you THINK they SHOULD be that you can't even recognize the most simple and basic things about the way things ARE in these mathematical theories. It is NOT true that Robinson eschews a least infinite ordinal. He explicity MENTIONS omega as he USES it to describe the order type of the non-standard *N. Just because you WANT non-standard analysis to be in conflict with ordinary set theory is not a reason to think that it IS. Robinson's non-standard analysis is ITSELF all derivable in ordinary set theory and mathematical logic, AS HE SAYS SO HIMSELF - IN THE VERY FIRST SENTENCE OF THE BOOK. For godsakes, Tony, you are utterly blocked.. > I certainly detect a discrepancy, yes. Would that you detect that you have a SERIOUS learning problem. MoeBlee
From: Virgil on 12 Dec 2006 15:45 In article <457EB19D.9040800(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 10:07 PM, Virgil wrote: > > In article <457D7067.7030006(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > That EB chooses not to consider measures of sizes of sets which can be > > applied to Dedekind infinite sets does not mean that no one else is > > allowed to do so. > > Not even Dedekind himself was legitimated to do so. There has been progress since. > > > > The Cantor definition of cardinality, at least in ZFC or NBG, > > Not a single axiom in ZFC provides Cantors definition of cardinality. > That is why one must define it separately. However the properties that it requires are all provided in ZFC and NBG.
From: Virgil on 12 Dec 2006 15:47 In article <457EB20C.5040708(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 10:09 PM, Virgil wrote: > > In article <457D70D8.9080902(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > >> Can you reveal just one illusion of mine? > > > > That you know more about mathematics than the thousands upon thousands > > of those who have studied it much harder and longer then you have done. > > Maybe, it needs some distance in order to overlook large things. EB overlooks too much. In the English sense of not seeing what is there to be seen.
From: Virgil on 12 Dec 2006 15:48 In article <457EB2E6.2020609(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 6:32 PM, Bob Kolker wrote: > > Eckard Blumschein wrote:> > >> No. Cantor again merely showed by contradiction that the power set is > >> not countable. The reason is: Already the entity of all natural numbers > >> is an uncountable fiction. > > > > By definition, the set of integers is countable. > > I know this definition, and it is even quite plausible if one only takes > the ordinary (Archimedean) point of view. > > > A countable infinite > > set is a set which can be put in one to one correspondence with the set > > of integers. > > I cannot imagine any reason why an intelligent person may reiterate this. In the hope that unintelligent readers may finally comprehend it.
From: Virgil on 12 Dec 2006 15:50
In article <457EB396.4020008(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 10:16 PM, Virgil wrote: > > >> No. Cantor again merely showed by contradiction that the power set is > >> not countable. The reason is: Already the entity of all natural numbers > >> is an uncountable fiction. > > > > Not in ZFC or NBG. > > Do they claim that? The inductive axioms of both declare the existence of sets among which is a countable one. |