Re: infinity

Daryl McCullough said:
> Tony Orlow says...
>
> >No, you are equating "x is larger than any finite I specify" with
> >"x is larger than ALL finite values."
>
> That's because they mean the same thing. If x is not larger
> than *ALL* finite values, that means that there is some
> finite value that is larger than x. Call that value K.
> x is *not* larger than K, so x is not larger than any
> value I can specify.
>
> >When you state the first, I say that doesn't mean x is
> >infinite
>
> Yes, it does. If x is finite, then x+1 specifies a number
> that is larger than x. So if x is finite, then it is *false*
> that x is larger than any number you can specify.
>
> Think about it: I write some number down on a piece of
> paper. I don't tell you what it is, but I tell you that
> it is larger than any finite number you can name. If
> what I wrote down was a finite number, then what I said
> was *false*; if what I wrote down was a finite number,
> then you can name a number that is 1 bigger than the
> one I wrote down.
Right, for any number I choose, you can always find a larger one, which is also
finite. That is the crux of your argument, to which I am responding.
>
> The only way for me to make sure that my number is bigger
> than any finite number you can name is if my number is
> infinite.
Right, but why do you want a number BIGGER than all the differences in the set,
or BIGGER than all the values in the set? Are you trying to find a number that
is BIGGER than the size of the set?
>
> >It doesn't prove that value is infinite, just that it's bigger
> >than any finite you specify.
>
> If x is a finite number, then "x+1" specifies a finite number bigger
> than x. So if x is a finite number, it is *false* that it is
> impossible to specify a finite number bigger than x.
Right, for any GIVEN finite x, there is a y that is greater and finite. That
doesn't make y infinite, and it doesn't make the range infinite if for any
given finite range you can name a larger one either. If you want to say that
issue makes the largest element nonexistent, then in the name of consistency,
say the size of the set is also.
>
> >> If X is greater than any finite number, than X cannot be
> >> finite. You claim this is false, but your argument only
> >> makes sense if you change the orders of the quantifiers.
> >No, if you say "x is greater than EVERY finite number", THEN it's infinite.
>
> That's what people are saying. The set of all finite naturals
> has a size that is greater than *every* finite number.
No, it has a size equal to its largest element, which is finite. If you start
at 0, it has a size one greater, but still finite. For any finite natural you
choose, there is only a finite set of predecessors. There is no infinite set.
We have established that. This is bellyaching.
>
> >This is equivalent to saying, for any specific finite x, the set
> >size y is larger than x. If the statement were that the set size
> >is larger than EVERY finite value
>
> Look, if y is a finite number, then y+1 specifies a finite number
> that is greater than y. So if y is a finite number, then it is
> *false* that you can't specify a finite number greater than y.
Yes, and?
>
> What you seem to be saying is that y isn't actually a number,
> but is some kind of placeholder for a number. It's a variable,
> together with a set of constraints on it. So if I guess
>
> "Is y equal to 15?"
>
> then you say "No, it's larger than 15" and you write down
> the constraint "y > 15". If I guess "Is y equal to 1 billion?"
> then you say "No, it's larger than 1 billion". and you write
> down the constraint "y > 1 billion".
No, y is the largest non-infinite. y<oo.
>
> >> It might help you if you "Skolemize" the formulas.
> >> 'for any finite Y there is a finite X greater than Y'
> >> would become something like
> >> 'there exists a function f such that for any finite Y
> >> f(Y) is finite and f(Y)>Y'
> >> This might help you from continually making the mistake
> >> of thinkg that there is a single entity X that is greater
> >> than any Y.
> >Ummmm, isn't that what YOU are saying, when you say the set size
> >is larger than any/every finite natural? Isn't X your set size?
>
> No, he's saying that the set size is aleph_0, which is larger
> than *every* finite number.
According to the pattern, if you start with 0, the set size if the largest
element, plus 1. This system of limit ordinals is a distraction from the facts.
>
> --
> Daryl McCullough
> Ithaca, NY
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... >>> FINITE number greater than any finite you specify. ... > infinite, because there is always a larger finite value than any finite value ... they are equivalent in everyday English. ... then it would be equivalent to saying it's infinite. ...
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  • Re: infinity
    ... > Tony Orlow wrote: ... I have this number that I claim is bigger ... > This is very different from saying "if you pick a number, ... So, you found a number larger than all differences, and it's infinite. ...
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  • Re: infinity
    ... >>> finite you specify, that does NOT mean it's infinite, since there ... >>> is always a FINITE number greater than any finite you specify. ... just that it's bigger than any finite you specify. ... And every bound on the set of finite naturals must be greater ...
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  • Re: abundance of irrationals!)
    ... >>> infinite, but not close to any infinite value? ... A verrry important ... >> Saying that something gets bigger than any finite number at some point ... >> does not say that it gets bigger than all finite numbers at any point. ...
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  • Re: infinity
    ... but why do you want a number BIGGER than all the differences ... >That doesn't make y infinite ... We're not saying that it makes y infinite. ... Aimplies ~Bis equivalent to ...
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