ALL numbers

[Imparcticle]

Yesterday I was enlightening a friend of mine concerning the many wonders of Fibonacci numbers and the golden ratio (let this friend be represented as X). As I was speaking with X, I learned that another friend of mine (let him be represented as V) was listening very attentively. Here is our conversation, as it will make things easier for explanation:


Me: As you can see, X, phi (in the form 1.6...) is the only number whose
square is (phi - 1). No other number, as far as I know, has this quality. Apparently, this is supposed to be true for all numbers.....

Friend V: No. You can't say "for all numbers".

Me: Ah, because by saying "all" I am quantifying an infinite set of numbers?

Friend V: Yes.



Unfortunately, we were unable to continue this conversation on that day. We are scheduled to continue a few days from now.
I actually disagree with V's assertion. This is because of a simple fact: there are certain properties of numbers that have been proved to be true for all numbers (right? :uhh: ). So I could say "all numbers".
But can I say "all of the subsets of an infinite set are such that x is always true for all of the subsets"?
is my usage of the word "all" quantifying my subject? What does the word "all" do in terms of how it quantifies?

[Alkatran]

x^2 = x + 1
x^2 - x - 1 = 0
(1 +- sqr(1 - 4*1*=1))/2*1 = 1/2 +- sqr(5)/2 = 1/2 +- sqr(5/4)
= 1.618 or -.618

(-.618)^2 =~ .381
-.618 + 1 =~ .382

Hmmm

[Imparcticle]

wow. thanks for the clarification.

[matt grime]

Well, what you said is badly phrased, but that's what happens with spoken English, and as alkatran shows incorrect, but it is perfectly possible to make a statement about something being true for an infinite set.

x^2>x for all x in (the infinite set) (1,infinity)

something is true for all elements in some set if the negation, that there is *an* element for which it is false, is false.

Something is true for all the quantified members to which it applies if it is, erm, true for them all.