Re: Galileo's Paradox

stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
Why do you have a problem with the mere suggestion of an infinite value?
I don't have a problem with infinite values. However, you have to do more than merely say "positive infinite n". Assuming your positive infinite things aren't the same as something that we already know about (in which case, you should just say so), you either need to give a construction of these positive infinite things or you need to specify their properties. You haven't done either. For example, how many of these things are there? How do they relate to each other? How do they interact with the natural numbers? Are the operations of addition and multiplication defined for them?
Well, I have been through much of that regarding such specific language approaches as the T-riffic digital numbers, but that's not necessary for this purpose. It suffices to say that, if a statement is proved true for all n greater than some finite k, that that also includes any postulated infinite values of n, since they are greater than any finite k. I don't need to construct these numbers. Consider them axiomatically declared.
Then list the axioms for them.
(sigh)

infinite(x) <-> A yeR x>y
Is that your only axiom? If so, then state your first theorem about them and give the proof.


That's the only one necessary for what defining a positive infinite n. A whole array of theorems pop forth from infinite-case induction and IFR, such as that the size of the even naturals is half that of the naturals. That's a no-brainer. Go back to where I first answered your question at length, and read again, at length. It's not transfinitology, but it's also not nonsense.

Allow me to add another:

|{ x| yeR and 0<(x-y)<=1}|=Big'un.

That's the unit infinity.

What is the definition of | | ?

Stephen

"Size" of the set. Or, is "|...|" reserved only for "cardinality"? I think I can adopt the previous symbol for absolute value, as set theorists have, without stepping on too many toes.

Tony
.

Relevant Pages

  • Re: Galileos Paradox
    ... I don't have a problem with infinite values. ... construction of these positive infinite things or you need to specify ... Then list the axioms for them. ... such as that the size of the even naturals is half that of the naturals. ...
    (sci.math)
  • Re: Galileos Paradox
    ... Assuming your positive infinite things aren't the same as something that we already know about, you either need to give a construction of these positive infinite things or you need to specify their properties. ... A whole array of theorems pop forth from infinite-case induction and IFR, such as that the size of the even naturals is half that of the naturals. ... Using the inverse function, g=x*Big'un, over the range (0,Big'un], we get that there are Big'un^2 hyperreals, Big'un for each interval. ... This corresponds to the 2D array comprising the hyperrationals, if one replaces the redundant fractions with the equinumerous irrationals. ...
    (sci.math)
  • Re: Galileos Paradox
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    (sci.math)
  • Re: Godel, self-reference, and framework/object confusion
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  • Re: Galileos Paradox
    ... construction of these positive infinite things or you need to specify ... Then list the axioms for them. ... who cares if you can prove a whole bunch of theorems about ...
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