Re: Calculus XOR Probability
- From: Virgil <vmhjr2@xxxxxxxxxxx>
- Date: Tue, 02 May 2006 17:11:34 -0600
In article <MPG.1ec1404cde84263398aca0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
Virgil said:
In article <MPG.1ebfda9786fc0aa298ac93@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
Of course IFR depends on the definition of quantity and arithmetic
formulas.
Consider those already defined as usual.
Defining them "as usual" requires the mechanism of the very set theory
that TO objects to, o that TO is already assuming all that he wants to
reject.
The usual definitions for arithmetic operations have nothing to do with
transfinite set theory, though that may be artificially woven into some
axioms.
The same axioms that give us transfinite set theory are needed to give
us the naturals, and from them the other number systems on which TO is
depending.
TO can't get to where he wants to go without the same foundation that
provides the structures he is trying vainly to avoid.
That's all quite well mapped out already in standard math. Just considerDefining them "as usual" requires the mechanism of the very set theory
f
and
g to define a bijection as mutually inverse functions.
that TO objects to, so that TO is already assuming all that he wants to
reject.
One can compare sets over a finite domain, or an infinite domain.
But not over an unknown domain.
Which is exactly my point about needing definite domains and ranges, and not
vague definitions like "all finites".
A domain is no more than a set, and the set of von Neumann naturals, or
the resulting sets of integers or of rationals, or of reals, are still
sets.
Without a specific set as domain and another as codomain for a giv en
function, there is no way to tell whether it is invertible or not.
If " f(g(x))=g(f(x))=x" for all relevant x, as TO claims, then they
have to be bijections from some set to itself.
No, they're mutual injections between the naturals and some other sequential
set, together defining a bijection.
The equation stated, "f(g(x))=g(f(x))=x" requires that
each element of the domain of f is also in its codomain,
each element in the domain if g is also in its codomain and
each element in the domain of either is in the domain of the other.
So it either domain is restricted to the naturals, both are.
They absolutely have the same cardinality, that's true, but when you
observe
them as sets of points on the real line, if one function always has more
elements than another within any given value range, then it seems
perfectly
intuitive and logical to say that it has more elements overall.
Then TO is saying that , for N being the standard set of naturals and S
being {1/(n+1): n in N}, there are more members in S than in N, as this
exactly meets his standards.
The most egeregious error in this regard is the equivalence
derived between the set of naturals and the set of rationals,
when there are clearly an infinite number of the latter per unit
on the number line, and only one of the former. So, while what
you say is true of cardinality, the conflation of that to "size
of the set" is absolutely unfounded and at the heart of your
"counterintuitive" results.
Does TO insist that the cardinality of the rationals is greater
than that of the naturals?
No, but the set of rationals is infinitely larger than the set of
naturals, your diagonalized bijection notwithstanding.
Larger in what sense?
One finite set is larger that another if and only if there is an
injection but no surjection from the "smaller" to the "larger".
TO is always going on about what is true for finite sets must
carry over to infinite ones, but balks at the most important carryover
of all.
That rule rule (that one set is larger than another if and only if
there is an injection but no surjection from the "smaller" to the
"larger") need not be constrained to finite sets.
..
No, that's what you get when you spew pure negativity, rather
than saying anything substantive. It's not a particular mapping
or function, but a rule regarding the relationship between set
size and mapping function.
Such "rules" are mappings.
Perhaps you could consider it a "mapping" from the mapping function
used to used to define the bijection and the value range to some
expression of the set size. I think of it as a formulaic
relationship.
In other words, a mapping.
.
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