Re: Calculus XOR Probability

In article <MPG.1ed53c8b33ee7fbe98acec@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

imaginatorium@xxxxxxxxxxxxx said:

That's how it comes about that for any particular pofnat P given at
the beginning of the argument, it is the case that the set of
pofnats up to P is a finite set. But there is no Q you can give at
the beginning of the argument such that Q is a pofnat, and the
entire set of pofnats does not include anything exceeding Q. There
is no "point" in the pofnats at which the set "becomes
[Tinfinite]", but neither is there any point at which the pofnats
stop. Thus those of us constrained not to use the i-word say that
the set of pofnats is unending.

Brian Chandler http://imaginatorium.org



I agree it's unending, and in that sense may be considered infinite,
as that is compatible with the definition regarding injection into a
proper subset. Within set theory, that works without contradiciton.

As there is very little, if any, of mathematics that cannot be
successfully embedded within set theory, TO has just given away his
entire argument.



However, I am speaking in more quantitative terms, and relying on a
set of rules which determine whether an arithmetic operation produces
a finite or infinite value depending on its arguments.

That is a part of set theory, at least if "quantitative" is in any way
related to the real numbers as quantities.


For instance,
if A and B are finite and nonzero, A+B, A-B, A*B, A/B, A^B and
logA(B) are all finite, and nonzero except for subtraction if A=B.
Given those rules, and N=S^L, we can say that if S and L are both
finite, that is, we have only finite length strings and a finite
alphabet, then N, the size of the language so defined, is also
finite.

But given that L is unbounded for natural and mathematical languages,
so is N. And it is that unboundedness which TO cannot deal with.



This relates directly to the naturals as well, through the
digital number systems, which are languages, since any finite natural
only requires a finite string, and if all strings are finite in the
set, the set is also finite, given any finite number base. There is
no bound on L, but if L cannot be infinite, then neither can N.

And if S > 1 and either is unbounded (which means infinite to everyone
except TO) then so is the other.

TO is vainly trying to separate unboundedness from infiniteness in
ordered sets.

And TO has no idea at all of how to deal with unordered sets, of which
there are too many in mathematics to ignore.
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >> Conway knows how to do mathematics. ... people have studied the maths of infinite ... anything else that is unending, but we do not assume that this unending ... >> The L set can include all pofnats up to n, some pofnat n, in which case ...
    (sci.math)
  • Re: Orlow cardinality question
    ... >> Tony Orlow (aeo6) wrote: ... and yet you claim it is infinite. ... Simply that the unbounded set of pofnats, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... Conway knows how to do mathematics. ... > It works if you allow the infinite naturals that are required for the set to be ... >>> Does a well ordering need to have a successor to every element? ... The L set can include all pofnats up to n, some pofnat n, in which case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... lartgest finite with an infinite successor. ... implies that there is some lartgest finite with an infinite successor." ... that not mean that there is no point at which the set of finite naturals is ... it is the case that the set of pofnats up to ...
    (sci.math)
  • Calculus XOR Probability
    ... lartgest finite with an infinite successor. ... implies that there is some lartgest finite with an infinite successor." ... I suppose what it boils down to here is infinite induction. ... it is the case that the set of pofnats up to ...
    (sci.math)