Re: The shocking truth about the naturals

J. Burse says...

aatu.koskensilta@xxxxxxxxx schrieb:

We can say anything we like. The upward L�wenheim-Skolem theorem,
however, shows that we could be mistaken about the naturals being
countable!

What does it mean "we can say"?

Lets say we want to have the naturals countable, we could
be mistaken about using FOL.

It always depends on how you diagnose a problem. I interpret
your statement that you fully trust FOL. And then if FOL
does yield non-standard models, you imply that naturals could
be non-countable.

No, he's <i>joking</i>. Or being sarcastic. Or being ironic. Whatever.

He doesn't <i>really</i> think that the
naturals could be non-countable. He's just saying that it makes
as much sense to say that as it does to say that the reals could
be countable. To use the downward LS theorem to justify saying
"The reals might be countable" makes as much (or as little) sense
as using the upwar LS theorem to justify saying "The naturals
might be uncountable".

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Is a line segment composed of points?
    ... In FOL PA, ... elements we consider as naturals, which would be /denied/ that status ... _missing_ standard reals and pi). ... I mention hyperreals often because many of TO and others' posts ...
    (sci.math)
  • Re: The shocking truth about the naturals
    ... Lets say we want to have the naturals countable, ... your statement that you fully trust FOL. ... i.e critizing the object and critizing ... A & B: Limitations First Order ...
    (sci.logic)
  • Re: The shocking truth about the naturals
    ... naturals could be non-countable. ... To use the downward LS theorem to justify saying ... "The reals might be countable" makes as much sense ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... All I know is that what we know about infinite ... > the sets I call finite have larges members. ... The set of all finite naturals is not infinite, ... Sets defined by mapping functions from the naturals to the reals which have ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> less than sqrthas no largest member. ... > The set of all finite naturals is not infinite, ... >> I WILL claim that your incomplete definition of cardinality ... > Sets defined by mapping functions from the naturals to the reals ...
    (sci.math)