Re: Where's respect? was Re: Corrective interpretation of real numbers

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 02/06/05 

Date: 6 Feb 2005 11:59:39 -0800

One thing you noted was a consideration that a proper subset, of a
finite or infinite set, might be considered lesser. For an infinite
set, cardinality does not make that distinction. Some argued in the
past that it was a meanginless measure. I don't because if A < B then
B\A =/= {} and A\B = {}, and under all circumstances non-null X > null.
 That was vociferously denied by several enthusiastic participants of
this discussion. Then, Fred Katz wrote that instead it was true, as
outlined for his reasons in his dissertation, and thus the vociferous
deniers about the charge that a proper superset is definably larger
than a set, for example Dr. Ullrich here, the giggling and babbling
layabout hedgehog, changed direction in that way. For even vis-a-vis
all integers you can look to asymptotic density from number theory.

I read your posts casually, but not all of them, I must keep in mind
that conversely I have described a variety of logical structures that
you have not encountered, and I can not necessarily expect you to know
that.

About the infinitesimals in R, the hyperreals are often proffered as an
example of a nonstandard representation of the real numbers that
contain infinitesimals. There are others besides Robinson's popular
construction. As well, you might note the transfer principle between
the standard and nonstandard definitions, and utilities, that allow
results in one to be verified in the other.

There are different perspectives or views of the real numbers that
allow various methods upon them. Timothy Little last week or so dubbed
the reals with iota-values the pseudo-reals, but I just call them the
real numbers, as I do the hyperreal or surreal numbers. That's a
statement that the hyperreals are only the real numbers, in part
because of the completeness of the reals, a specific property related
to the representation by each of the real numbers of continuity.

So, you might consider why infinitesimals necessarily _are_ a part of
the real numbers.

About the naturals containing infinite integers as do the
hyperintegers, well, er... I'll just repeat "infinite sets are
equivalent".

The idea with the iota-values is to discuss the contiguity of points on
a line.

Eray, there are a variety of nonstandard constructions of the reals,
and hyperreals, while useful and convenient, suffer most of the same
problems as reals with regards to discretizing them. A plane can be
specified with a pair of intersecting lines. Did you mean airplane? I
think you want to learn about interval arithmetic for your numeric
approximations, please symbolically reduce before crunching.

Regards,

Ross F. 

--
"Zero, zero, zero, zero, zero..."
"You don't have to be cruel to be kind." - Spacehog

Relevant Pages

  • Re: Wheres respect? was Re: Corrective interpretation of real numbers
    ... finite or infinite set, might be considered lesser. ... deniers about the charge that a proper superset is definably larger ... the reals with iota-values the pseudo-reals, but I just call them the ...
    (sci.math)
  • Re: Cantors Diagonal Argument
    ... This is about denying the existence of infinite ... MUST be a proper class, in the opinion of that model. ... > cardinals both exist at all then there are certainly more ... > cardinals than reals. ...
    (sci.logic)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)
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