Re: computation and cardinality

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 01/18/05 

Date: Tue, 18 Jan 2005 08:07:31 -0600

William Elliot wrote:
> On Mon, 17 Jan 2005, Mike Oliver wrote:
>> Who mentioned certainty? I didn't ask for the answer with
>> certainty, just for the answer. It's alright for you to
>> be uncertain about your answer, as long as it happens to
>> be right.
>>
> You got an answer which you didn't like because it gave you a choice.
> So in your inablity to accept a lack of absolute certainity, you
> demanded to know which choice is right. My advise to you young man, if
> you can't make up your mind, is to go to Utah or to middle-East where
> it's right to marry both.

Doesn't have much to do with what I *like*. Your "choice" answer,
on the face of it, just doesn't make much *sense*. CH is an existential
statement about sets of reals -- "there is a wellordering of the
reals (coded by a set of reals) whose every proper initial segment is
countable". Well, is that set of reals there, or isn't it? *I* get
to choose? Flattering for me, too bad about the rest a y'all. Do
I also get to choose whether there's a 1908 Liberty Dollar somewhere
in the Grand Canyon, or whether there's snow in Toledo and freezing
rain in Niagara Falls?

Now, perhaps you don't think arbitrary sets of reals exist at all,
at least as real objects independent of our reasoning about them. Fine.
Then the most obvious position for you to take is that CH is *meaningless*,
since it uses terms that fail to denote.

There are respectable ways you can get around that, if you want, by
adopting some theory of mathematical meaning that does not require
the objects of mathematical discourse actually to exist. The costs
may be higher than you imagine, though, to the simplicity and
explanatory power of mathematics as a whole.

In any case your "certainty" remarks are way off base. *I* am
willing to countenance that CH has a truth value, even though
I can't currently be certain what it is, and may well never be.
*You* seem to be so unhappy with that situation that you make
up a new "choosable" truth value ad hoc, so you can be sure
CH has that one.

Relevant Pages

  • Re: Math as Religion
    ... Certainly what initially attracted me to mathematics was its shear beauty. ... I think also for a long time it conveyed a sense of inner truth to me, but since studying mathematical logic, and thinking a lot about its true foundations, I lost this sense. ... it is THE truth that magnitude is more fundamental than the reals. ...
    (sci.math)
  • Re: computation and cardinality
    ... > reals whose every proper initial segment is ... > explanatory power of mathematics as a whole. ... That you can imagine else, is wonderful self-deception, that what ... > willing to countenance that CH has a truth value, ...
    (sci.logic)
  • Re: Math as Religion
    ... I lost the sense that it is THE truth. ... Who has said that mathematics is THE truth? ... it is THE truth that magnitude is more fundamental than the reals. ... certainly any conflict is not allowed in this language. ...
    (sci.math)
  • Re: Math as Religion
    ... I lost the sense that it is THE truth. ... Who has said that mathematics is THE truth? ... it is THE truth that magnitude is more fundamental than the reals. ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... This is not a job the axioms were ever meant to do. ... It's actually a problem with language generally, not just mathematics. ... just procrastinated the problem of interpretation for one step. ... Consider for example the set of reals you considered in another posting, ...
    (sci.math)