Re: infinity

stephen@xxxxxxxxxx said:
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > stephen@xxxxxxxxxx said:
> >> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >> > stephen@xxxxxxxxxx said:
> >> >> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >> >> > stephen@xxxxxxxxxx said:
> >> >> >> Here is a question for you Tony: is the following
> >> >> >> statement true?
> >> >> >>
> >> >> >> if x is a finite natural, then there exists a finite natural
> >> >> >> y such that y>x
> >> >> > For any GIVEN finite natural x, yes. There is no SPECIFIC finite natural for
> >> >> > which this is not true. To assume that there is produces a contradiction.
> >> >>
> >> >> But there are presumably UNSPECIFIC finite natural numbers
> >> >> for which this not true? What exactly is the difference
> >> >> between a SPECIFIC finite natural number and an UNSPECIFIC
> >> >> finite natural number?
> >> > Uh, whether or not you have specified it.
> >>
> >> That is not an answer, and besides you keep claiming
> >> there are numbers that I cannot specify, but somehow
> >> still exist.
> > What do you think "specific" means? Go look it up, and find me the first order
> > expression for it. The existential quantifier implies it.
>
> I do not know what you think "specific" means. From
> your usage and your last sentence you seem to think that "specific"
> means "exists" and "unspecific" means "does not exist".
>
> So when you say that there is no specific largest finite,
> you seem to be saying that the largest finite does not exist.
> If it does not exist, it is not a finite number, as all
> finite numbers exist.
I thought the problem you fellows had with me was that I do NOT equate "can't
be identified" with "doesn't exist".
>
> >>
> >> >>
> >> >> Anyway, you must also agree that the contrapositive is true:
> >> >> if there does not exist a finite natural y such that
> >> >> y>x, then x is not a finite natural
> >> >>
> >> >> So does there exist a finite natural y such that
> >> >> y is larger than the largest difference between any two
> >> >> finite naturals?
> >> > if x is a SPECIFIC finite natural, then there exists a SPECIFIC finite natural
> >> > y such that y>x
> >>
> >> > is equivalent to
> >>
> >> > if there does not exist a SPECIFIC finite y such that y>x, then x is not a
> >> > SPECIFIC finite natural number.
> >>
> >> > These statements I can live with.
> >>
> >> Can you translate those statements into first order logic?
> > A x in N E y in N y>x
>
> So for every x, there exists a y such that y>x.
>
> This is true. But this does not make use of your
> specific/unspecific definition.
>
> of course
>
> E y in N A x in N y>x
>
> is false. But you keep claiming that this is true
> for some UNSPECIFIC y.
No, what I am saying, really, is that it doesn't matter which is the largest,
if we know they are all finite. It doesn't matter which difference between them
is the largest, if we know all differences are finite. There cannot be a truly
infinite number of elements (as in reciprocal of some zero) unless there is
some difference between elements that is infinite, allowing for an infinite
number of elements to lie between those elements. When I speak of an arbitrary
largest element, it means we are assuming some x is the largest in some set. If
x is the largest element in a set of positive naturals, there cannot be more
than x elements in that set. So, if the largest element in any set of finites
is finite, whatever that element may be, the set cannot have more elements than
the value of that element. So, how can a set of all finite positive whole
numbers have a size that is greater than every single one of those elements?
Your answer is really that there aren't an infinite number of elements in the
infinite set, which makes me wonder what you think an infinite set really is.
It seems you think it means boundless.
>
> >>
> >> You have to define what a SPECIFIC finite number is.
> >> Apparently in your world there a SPECIFIC finite numbers,
> >> and UNSPECIFIC finite numbers, but you have not described
> >> what they actually are.
> >>
> >> Consider this specification for a number:
> >> z = the largest finite number
> >>
> >> I 'specified' a number, but you of course claim
> >> that this is an UNSPECIFIC number. Why does this
> >> not specify a number?
> > That does specify a number, and in so doing, causes a contradiction, as you
> > well know.
>
> So what is an UNSPECIFIC number?
Some number, any number, pick a number. No, wait, DON'T pick a number.
>
> Stephen
>

--
Smiles,

Tony
.

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