Re: infinity

In article <MPG.1db061f5f5f419d098a429@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> stephen@xxxxxxxxxx said:

> > Yes, any number that exists that is smaller than omega is a finite
> > number. You have not shown that <oo is a number, or that it
> > exists. Is <oo a number? If it is a finite number, then <oo+1 is
> > also a finite number, and the range of [0, 1, ... <oo+1] is larger
> > than <oo, so the range of a subset of the finite numbers is somehow
> > larger than the range of all the finite numbers.

> <oo is a specification of value range. It is interpreted as "the
> value range is less than oo", which means it is finite.

So it must also be both finite and larger than every finite at the
same time. Something only possible in the twilight zone of TOmatics.
> >
> >
> > > There does not need to be a largest one. If we say the range of
> > > [0,1) is <1, that means the differences have an LUB of 1, but no
> > > difference is equal to 1. If we say the range is <oo, it means
> > > the differences have an LUB of oo, but that no difference has
> > > that value. Less than infinity means finite, like less than zero
> > > means negative. Do we have to know the smallest negative value
> > > less than zero to know that all values less than zero are
> > > negative? Nope, we don't.
> >
> > All you are saying is that all finite numbers are finite. But that
> > is a different thing that saying the range is finite. There is no
> > finite number equal to the range. Just as there is no number less
> > than 1 that equals the range of [0,1). "<1" is not a number, or if
> > it is, it is some new type of number that you have made up.

> it is a specification of the range. For sets with no maximal or
> minimal member, we can say the range is less than the LUB of the
> differences.

LUBs, when they exist in the reals, are finite. If the range is to be
less than the LUB, assuming a LUB exists, the range must be fintie.
> >
> > Let's look at the properties of your new number "<1". Presumably
> > "<1"+"<1" equals "<2". I would guess that "<1"+1 also equals "<2".
> > What does that give us?
> >
> > "<1"+1 = "<2" = "<1"+"<1"
> >
> > Now what does "<1"-"<1" equal? For every real number, x-x=0. If
> > "<1" is a real number, then it is the case that "<1"+1 =
> > "<1"+"<1" "<1"-"<1"+1 = "<1"-"<1"+"<1" 1 = "<1" So if
> > "<1" is a real number, it is actually equal to 1. If it is not a
> > real number, then who knows what it is, or if it even exists.

> You do not add and subtract value ranges.

Whyever not?

If you have two sets and
> wish to know the value range of their union, take the union, find the
> extrema, and derive the range. In fact, one could have a range of <1
> and a range of <1 combine to make a range of 100. Consider the union
> of the intervals [0,1) and (99,100], which is [0,100] with a value
> range of x=100.
> >
> >
> > > The value range of the set of finite naturals is finite, as no
> > > difference in the set is infinite.

> >
> > According to your definition, but that is a pretty poor definition.
> > You claim the value range is finite, but the value range does not
> > equal a finite number. Seems pretty inconsistent to me.

> It is not necessary to know the largest finite in a set to know that
> if all elements are finite, then all differences are finite, and the
> value range is finite.

All the reals in [0,+oo) are finite so that its value range is finite,
according to TOmatics, even though it is an infinitely long interval.

Note that for every real in [0,+oo) there is a finite natural larger
than that real, so that TO is claiming indirectly that infinitely long
intervals have finite lengths.

Just another of the plethora of peculiarities of TOmatics.


> The value range is not infinite unless one
> element is infinitely greater than another.

A drawback not shared by set diameters.

"Value ranges" can apparently be smaller than diameters even where they
both exist, or "value ranges" can fail to exist at all for interesting
sets, whereas diameters exist nicely for all sets in metric spaces.

"Value ranges" seem to have no value at all.
.

Relevant Pages

  • Re: An uncountable countable set
    ... But until TO proves trichotomy for all his imaginings, they are not on any line. ... Note that one can prove the density and "continuity" of the reals (contunuity being the LUB and GLB properties. ... The set of standard reals is bounded above if there are any infinite reals, so must have a LUB, so what is it TO? ...
    (sci.math)
  • Re: infinity
    ... >> do not have a LUB. ... Omega is not a real number and thus not a LUB in the set of real numbers. ... >> is greater than every finite number and less than every infinite ... That conflicts with the definition of the reals as an Archimedean ...
    (sci.math)
  • Re: An uncountable countable set
    ... Accordingly TO must show that any non-empty set of his extended reals ... reals, so must have a LUB, so what is it TO? ... Then there is a gap between the finite reals and TO's supposed infinite ... That is only for the cardinality of set of finite naturals, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)