```Date: Jan 27, 2013 4:42 PM
Author: Virgil
Subject: Re: ZFC and God

In article <364860fe-77b9-4673-a29b-71c565afdd4c@u20g2000yqo.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:> On 27 Jan., 18:44, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:> > >> > > 0.7 is terminating.> > > if 0.777...777 with n digits is terminating, then also 0.777...7777> > > with n+1 digits is terminating. Therefore there is no upper limit for> > > the number of digits in a terminating decimal. This fact is usually> > > denoted by "infinite" and abbreviated by "...".> >> > Are you suggesting that 0.777... is *both* an infinite and terminating> > expansion?> > In my opinion infinite has only one meaning, namely potenial infinity.You opinion is of no value outside Wolkenmuekenheim .> The set N is potentially infinite. There is no upper threshold> although every FISON is finite. - And there is nothing else.> > By the AxInf we get a second infinity, namely a set N that has A> elements, where for every n in N: A > n.ONE measure of the finiteness/infiniteness of a set S is determined by whether it can be injected to a proper subset of itself.Sets like {1}, and {1,2}, and {1,2,...,n} cannot be injected to any proper subsets of themselves, and sets which cannot be we call "finite".However sets like |N CAN be injected into some of their proper subsets, for example n -> n+1 injects |N to {2,3,4,...} which is a proper subset of |N. Such sets are called "not finite", or more briefly, "infinite". In standard mathematics "injectability to some proper subset" and "non-injectability to any proper subset" are too awkward, s we replace them with "infinite" and "finite" respectively.We also note that one can prove that from the above definitions of finiteness and infiniteness of sets one can show that Set S is infinite if and only if her is an injection from |N to S.But note that the existence of a set |N is not needed to definie finiteness versus infiniteness of sets. > > If someone believes that this axiom is better than the axiom that> there are 10 different naturals with sum 10, then he should try to> find evidence. Hitherto I have not seen eveidence provided by you that> AxInf would be useful or required.> > > Anyway, you haven't proved that there is a function> >> >   f:{1,...,k} -> {0,...,9}> >> > as required by *your* definition of terminating decimal, so you have> > not shown that 0.777... is a terminating decimal.> > You are wrong. Perhaps in Wolkenmuekenheim, but nowhere else.> Can't you understand? All natural numbers are finite.> Why the heck should I define a single k?Unless you can sow that a last k exists, you lose!> >> > > Note, there is another meaning of infinite, namely "actually> > > infinite". Those who adhere to that notion *in mathematics* should> > > show that it differs from "potentially infinite" *in mathematics*,> > > i.e., expressible by digits.> >> > Well, I don't understand why anyone would wish to show that.> > Perhaps in order to show that matheology is not complete nonsense?Comparing what WM calls our matheology and I call his WMytheology, I find far less senses and more foolishness in WM's WMYTHEOLOGY.  > > > But,> > regardless, this is beside the point.  I'm asking for a proof that> > 0.777... is terminating according to the definition of terminating> > that you agreed to.> > I did this in my last posting. Please look it up there. Well as I have> it just at hand, here it is again:> 0.7 is terminating.> if 0.777...777 with n digits is terminating, then also 0.777...7777> with n+1 digits is terminating. Therefore there is no upper limit for> the number of digits in a terminating decimal. This is written as> 0.777...> > This is the definition that I agreed to.Then, according to WM, a terminating decimal need not have a last digit, and can thus be represented s an infinite series..ONe wonders how WM would define a non-terminating decimal;> > Regards, WM--
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