Re: Infinitesimals



John Jones wrote:
herbzet wrote:
John Jones wrote:

So, an infinitesimal is not characterized by size, and if size is
necessarily represented by sequence,

Size is not necessarily represented by a sequence.

I was entertaining the thought that the activity of sizing is either
implicitly or explicitly dependent on, or equivalent to, the presence
of a sequence.

The real numbers in their natural order are ordered by size
(magnitude) but do not constitute a sequence. Though certainly
we can, in this case, pick out a sequence of reals ordered by size.
This will necessarily be only a proper subset of the reals.

I am using the terms "order" and "sequence" in their TECHNICAL
sense (see below).

> > then an infinitesimal is NOT in the sequence.

No, I would guess that infinitesimals are not sequence-able.

But is it not a common mathematical predeliction to impose an order
irrespective of whether order can be imposed?

Um, I don't think so. I would hazard that you misunderstand what
is going on.

I am quite sure that
even if inifinitesimals are not sized, somewhere a logician or
mathematician would put them in a sequence, assuming that it hasn't
been done already.

Assuming I is an infinitesimal, here's a sequence of infinitesimals:

{n + I} for n in N -- that is, I, 1 + I, 2 + I, 3 + I ...

It looks like this sequence is ordered by size.

Just BTW, it is usual to distinguish "order" from "sequence" since
an uncountable aggregate may have an order, but a sequence is always
a countable aggregate.

I don't understand how (the elements of) an uncountable aggregate can
have order.

"Order" has a technical definition, but certainly you can see that
for any two distinct real numbers, one is larger than the other:
the reals can be ordered by size, even though they are uncountable.

See http://en.wikipedia.org/wiki/Order_theory

which starts easy: stay calm and be _patient_.

Also take a look at http://en.wikipedia.org/wiki/Sequence

Quarks also, may not be physical objects, if all
physical objects can be sized. If they are not physical objects, then
I am thrown into a fog of my own making if I claim that 'the quark'
physically exists.

Well, true. If fire hydrants are not physical objects, then we'd be
wandering in the fog to claim that 'fire hydrants' physically exist.

But let that go.

Yes, if I claim that a quark is not sized and is physical, I will need
a foglamp.

I think we have a problem here with 'exists'. Do quarks exist?
Do fire hydrants, made of quarks, exist? Do infinitesimals exist?
Does 17 exist?

What is the criteria for object-hood?

Your question got me thinking

That's why they pay me the big bucks.

and I had to start a new post in reply.

Your usual dodge. No offense.

I have to write on this for a thesis proposal and it would be good to
air it generally for ....ah hem, 'feedback'.

Well, good luck with that.

Ta.

Ta.

--
hz
.

Relevant Pages

  • Re: Infinitesimals
    ... we can, in this case, pick out a sequence of reals ordered by size. ... This will necessarily be only a proper subset of the reals. ... Then let's say that 'order' is simply a memorable array. ... I would say that the numerical order you set up the infinitesimals ...
    (sci.logic)
  • Re: Infinitesimals
    ... we can, in this case, pick out a sequence of reals ordered by size. ... This will necessarily be only a proper subset of the reals. ... Then let's say that 'order' is simply a memorable array. ... was set up against their signs and not against infinitesimals ...
    (sci.logic)
  • Re: Infinitesimals
    ... Size is not necessarily represented by a sequence. ... If they are not physical objects, ... wandering in the fog to claim that 'fire hydrants' physically exist. ... Yes, if I claim that a quark is not sized and is physical, I will need ...
    (sci.logic)
  • Re: Multiple infinities - one more look
    ... continued for lager length of digit sequences without limit. ... infinite digit sequences... ... so the resulting reals have an order. ... (i.e. having a finite program to output their digits in sequence). ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... The way you get reals by a neverending process of generating digits ... and which thereby constructs a sequence of nested intervals whose ... It isn't *already* in the rationals you started with. ...
    (sci.math)