Re: Orlow cardinality question

Virgil said:
> In article <MPG.1d23768e72f73b01989e9c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
>
> > We have already discussed the problems with declaring the whole
> > numbers to necessarily be finite, and I have rejected that notion.
>
> I do not know how whole numbers differ from natural numbers in TO's
> mind, but the set of "finite" natural numbers, where a natural is taken
> to be finite if the set of naturals up to and including it is a finite
> set by the Cantor definition, is , by the Cantor definition, an infinite
> set. The "size", meaning cardinality, of that set of naturals is not a
> member of the set.
>
>
> > Without that restriction, the size of the set is N.
>
> Then the size of the set is not a member of the set because TO agrees
> that the set cannot have any maximum member, but that size would be a
> maximum member if a member at all.
You are confused. Your pofnats have no maximal member. My N does, and it's N,
and that's the size of the set, but that number has a successor, in one of two
ways.
>
> > What? You comment in the middle, but when I derive the expected result in
> > alternative ways you don't respond? I guess no one has any problem with this
> > part. It's interesting people complain, but never agree or concede.
>
> One never should agree to nonsense or concede a falsehood to be true.
Maybe if one thinks the robustness of a system is wrong, perhaps they should
comment, instead of snipping.
>
>
>
> > Every specific number is either finite or infinite, depending on the
> > finiteness of the position of its most significant digit. Any number
> > you can specify is one or the other, but no finite distance separates
> > one group from the other. it's the ever-unfolding lotus. Heard of it?
>
> Not in connection with the standard natural numbers.
Ah, well, now you have! Many happy blessings for you.
>
> And TO's notion of a numeral with infinitely many digit in some
> allegedly linear order between its "first" digit and "last" digit is a
> creation that fits better into fairy tales than mathematics.
"Once upon a time, there was a bad little weasel named Virgil. He was mean to
all the other forest creatures, especially the tigers, whose toes he would bite
from his hole as they walked by to sunbathe on a rock by the river..."
>
> TO posits a discrete linear order with first and last elements and
> infinitely many in between.
Even worse, I posit that the first may come after the last, and the last before
the first!
>
> And then TO says WE are dreaming to posit an infinite set of finite
> objects, when we have actually produced examples of infinite sets of
> finite objects.
>
Not separated by finite distances in a finite overall space, you haven't!

--
Smiles,

Tony
.

Relevant Pages

  • Re: Orlow cardinality question
    ... to be finite if the set of naturals up to and including it is a finite ... set by the Cantor definition, is, by the Cantor definition, an infinite ... Then the size of the set is not a member of the set because TO agrees ... finite objects. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... >>> proof you gave for any finite set having a maximal element. ... If the set is infinite, and has a maximal element E, then ... If an ordered set has a maximal member then appending any ... >> naturals is itself not finite, and thus need not have any maximal ...
    (sci.math)
  • Re: infinity
    ... >>> infinite set with both a largest and smallest element, ... But there is an infinite sequence of finite naturals, ... >> What is the finite number that equals the range of the finite ... Why MUST it be a member of the set of differences? ...
    (sci.math)
  • Re: Orlow cardinality question
    ... >>> be taken to check the results with infinite series. ... Any set satisfying the Peano axioms is well-ordered. ... has a last or largest member. ... There is no set of naturals with a maximum member that is not finite. ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... the positive integers which I defined the other day. ... A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. ... We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. ...
    (sci.math)
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