Re: infinity

stephen@xxxxxxxxxx said:
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > stephen@xxxxxxxxxx said:
> >> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >> > *** T. Winter said:
> >> >> In article <MPG.1d8f52f4d88d036398a263@xxxxxxxxxxxxxxxxxxxxxxxxx> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> >> >> > David Kastrup said:
> >> >> > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> >> >> ...
> >> >> > > >> No, it does not follow. *If* there is a maximal length L, then
> >> >> > > >> there are finitely many strings with length less than or equal to
> >> >> > > >> L. There is however no L such that all strings are in length less
> >> >> > > >> than that L, so it does not follow.
> >> >> ...
> >> >> > > > To get an infinite set of all string less than or equal to L you
> >> >> > > > need infinite L.
> >> >> > >
> >> >> > > But the set of all strings is not a set of strings less than or equal
> >> >> > > to some L.
> >> >> >
> >> >> > For ANY finite length L, there are a finite set of strings of length L or
> >> >> > less. For which L in N is the set of all strings up to that length an
> >> >> > infinite set? For NONE of them. There is not ONE finite string which, in
> >> >> > the ordered set of strings, has an infinite set of predecessors. Therefore
> >> >> > the set is finite, since any infinite ordered set MUST have some members
> >> >> > (an infinite number) which have an infinite number of predecessors.
> >> >>
> >> >> The therefore does not follow, and the since part is what you are trying
> >> >> to prove.
> >> > The since part is NOT what I am trying to prove. Do you think I was trying to
> >> > prove that any infinite set must have elements with an infinite number of
> >> > predecessors? That should be obvious. Is it not?
> >>
> >> It is not obvious, because it is not true. The definition of
> >> 'infinite' does not mention predecessors. If you have some
> >> definition of 'infinite' in your head that is based on
> >> predecessors, please share it.
> > An infinite set is one with an infinite number of elements.
>
> That definition did not mention predecessors. But anyway,
> now you need to define 'infinite number of elements', and
> show how you determine what the number of elements in a set is.
Not finite and not zero. Barring restrictions of finiteness, the ability to
create a surjection with a proper subset suffices. The problem with the finite
naturals is the restriction of finiteness on the elements, which causes the set
to be finite.
>
> > If the set is
> > ordered, every element is a successor or predecessor (not immediate) of every
> > other element in the set. Therefore, each element either has an infinite number
> > of predecessors, or an infinite number of successors, or both, since all other
> > elements are either predecessor or successor to that element, and one cannot
> > divide an infinite set into two finite sets, by omitting a single element.
>
> Yes, each element either has an infinite number of predecessors,
> or an infinite number of successors. Each finite natural has
> an infinite number of successors. No finite natural has
> an infinite number of predecessors, but this is perfectly
> consistent with what you just said above.
Except that, for every element with an infinite number of successors, there are
an infinite number of elements with an infinite number of predecessors.
>
> So again, why do you think that an infinite set requires elements
> to have an infinite number of predecessors.
Because if x is an infinite successor to y (meaning that there are an infinite
number of elements larger than y and smaller than x), then y is an infinite
predecessor to x (meaning there are an infinite number of elements less than x
and larger than y). That infinite number of elements between x and y are
successors to y, and predecessors to x. This is why, if an element has an
infinite number of successors, then another has an infinite number of
predecessors.
>
>
> >>
> >> You seem to have this idea that a set is infinite if and only
> >> if there exists an element that has an infinite number of
> >> predecessors. If you could define this notion in a non-circular
> >> fashion, that is, define 'infinite number of predecessors'
> >> without any reference to 'infinite set', then perhaps it
> >> would be possible to translate your idea into standard
> >> terminology.
> > I defined a finite number (0<x<=1, etc.). An infinite number is one whose
> > absolute value is greater than any finite number. An infinite set is one with
> > an infinite number of elements. It doesn't get any more basic than that.
>
> But you never have defined how you determine how many elements
> a set has. You claim that the set of finite numbers has
> a finite number of elements. Therefore you think that there
> exists some x>0 such that 1/x equals the number of finite numbers.
No, I have stated repeatedly that, like the largest finite to which it is
equal, this finite number cannot be determined.
> What is that x? What does it mean for a set to have 1/x elements?
> How do you determine that a set has 1/x elements?
The definition of finite I put forth is for quantities in general. If you want
to talk about set sizes, you need to talk about whole numbers, really. Surely
you don't argue that for finite x, 1/x is finite? And surely you don't really
expect me to have some number which is the reciprocal of that number that we
all agree is unidentifiable? Just be satisfied that it is the smallest positive
number which is not zero. That's as good as it gets.
>
> In any case, it is trivial to prove that the number of finite
> numbers is larger than any finite number. Pick any old finite
> number x. The set {1, 2, 3, ... x, x+1} has x+1 elements,
> which is greater than x. Therefore the number of elements in
> the set of finite number is larger than any finite number.
That is not a proof of any such thing. This is a proof that there is no largest
finite. You assume some number which is the size of the set, which is also the
largest finite, and find a contradiction. That doesn't mean the set is
infinite. It means the set has no distinct size because it has no upper bound.
If, for every finite natural x, the set of finite naturals up to and including
x has x elements, then there is no finite x which denotes any subset which is
infinite, and if no subset is infinite, then the set is finite.
> Stephen
>

--
Smiles,

Tony
.

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