Re: Relative Cardinality


> > Without injection/bijection (pairing off members from two sets to see
> > which, if either, runs out first) there is no counting at all.

> I do not wish to count. I prove that there are not more elements of set
> A than of set B.

Virgil
Can't do it without some mechanism for pairing off members. Which
requires something that establishes either an injection or bijection.

WM
Pray, tell me to the best of your knowledge and belief where the
infinitely many more irrationals shall be accommodated. Not even
Hilberts hotel could house them. But you insist they must be squeezed
between nowhere and nothing.

Virgil
WM's "relative cardinality" at least as defined, has not been shown
able
to do this.

WM
It makes clear, to any thinking individual, that Cantor's bijection is
without value, old-fashioned and simply out.

Virgil
And WM has declared that if there are not more in A than B and not more
in B than A they need not be the same 'size'. Which means that there is
no point to it.

WM
That is not my fault, but a property of infinite (i.e. sizeless) sets
which, however, makes this method consistent.


Virgil
Then nothing of mathematics is real as none of it "exists" in space or
time.
WM
Without space and time nothing at all can exist. Without at least one
human being the notion human is void and the set of humans is an empty
set. Without at least one living being the notion of life is absent.
Without at least one sample of the number 3, be it a fundamental set or
that number in any n-adic system - written, read, felt, heard, shouted,
thought - there is no abstract notion "number 3".

Virgil
If they are not real in WM's sense, there can be no objection to their
being infinite.

WM
No objection from the standpoint of reality, that is undisputed. But an
objection by the simple and efficient tool of relative cardinality.

Tell the truth: Do you really believe that in spite of this fact the
set of irrationals comprises more elements than the set of reals?



Virgil
Counting up in cents to the U.S. national debt is impossible, so WM
declares that it does not exist?

WM
You can identify every number between 0 and this debt. Hence you can
count to it, not only in cent but even in lira.

A number with 10^10^10^10 digits 1 does exist too, but you cannot count
to it. And if it happened to be a prime number, then you would not know
how many preceded it.

All these things are very easy to grasp and their truth lays open. Why
do you pretend to not understand them?

> > Then WMCard does not measure set sizes.

> > Set sizes must satisfy: If a <= b and b <= a then a = b

> That is a fairly primitive rule extrapolated from finite sets.

Virgil
What other "primitive" rules of ordering is WM's silly relative
cardinality to be exempt from? Until WM can provide a list of those
order, and other, properties which hold and those which do not, his
definition is without provenance.

WM
Can you imagine in spite of my proof that there are more irrationals
than rationals?

> Potentially infinite sets

Virgil
Sets never exist in WM's real world so it does not matter in the least
what WM claims about their behavior in that world. Sets only exist in
the world of ideals in which WM's strictures are of no effect.

In the ideal world, where all sets live, there is no such distinction
between potential and actual as WM posits for his matereial world.

WM
Even all my students, who are not mathematicians, know the distinction
between potential and actual infinity. It was the first question of the
examination, and each one of 22 gave the correct answer to this
question.


G. Cantor
Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und
aktualen Unendlichen, indem ersteres eine veränderliche endliche,
über alle Grenzen hinaus wachsende Größe, letztere ein in sich
festes, konstantes, jedoch jenseits aller endlichen Größen liegendes
Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das
eine mit dem andern verwechselt wird.


> What about ordering the rationals by magnitude?

Virgil
What about it? If, as claimed, Cantor cardinality is independent of
ordering, then ordering the rationals by "magnitude" does not change
the
Cantor cardinality of the set. Does WM suggest that it does change the
Cantor cardinality?

WM
The question is void, because it is impossible to order the rationals
by magnitude. It would even remain impossible, if an axiom allowed it.
You see, wrong axioms are conceivable (like AC).
> I use the axioms of the field of real numbers, complete induction,
> limits and logic. That is enough. Relying on bijections only, is to
> deliberately limit ones means of recognition.

Virgil
Who said anything about *only* relying on bijections?

WM
So, why do you reject relative cardinality? Would you do it also if it
supported Cantor's results?

> > Now that is really "Emperor's new clothes". A theory of cardinalities
> > that does not produce any cardinalities. A theory which is not known to
> > be self consistent and does not produce any useful results, and whose
> > only attribute is that it makes its author feel dressed like an emperor.

> I see that the ostensible dressing of alephs and omegas is absent.

Virgil
So are any useful consequences of WM's non-cardinality.

WM
You underestimate the value of truth and of doing justice to Kronecker
and Poincaré and a few others who kept their heads clear while most
mathematicians were lulled into security, bewitched by the great
potentialities of different infinities.

> Are you playing with toys? Or are you in search for truth? If my proof
> is true, then Cantor's is not.

Virgil
What "proof"? None has been presented. At least none that is
mathematically valid.

WM
Why shouldn't it be valid? If you deny it then you should point to an
error in it. But it would be hard to smuggle an error in such an easy
and lucid proof.

> And my proof is unavoidably and inescapably true as long as the axioms
> of the field of reals are valid!

Virgil
Actually, the axioms of the field of reals, notably the greatest lower
bound and least upper bound properties, invalidate WM entirely.

WM
What you mentioned are not axioms. But would you dispute that there is
always a terminating rational between two irrationals?
Look here: If we have two irrationals like 3.1415xxx and 3.1515xxx
where xxx is an infinite sequence, then the terminating rational 3.15
keeps them separate. It is even by definition: two real numbers, not
separated by any terminating rational, are not different but are one
and the same number. Would you deny that? Perhaps you imagine that
things are different in the infinite?
That would be silly, but just the quality of arguing which is now
requested to maintain set theory. You may agree or not: Be sure, clear
heads will more and more recognize and accept. This process has already
started. The only question is whether you will belong to the iron-heads
which don't understand till the end so that you will be made a laughing
stock or if you are among the first to understand.


> But my relative
> cardinality works independently of any assumptions about the character
> of infinity.

Virgil
So does Cantor's injective/bijective definition, as it precedes any
consideration of finiteness or otherwise. The finite/infinite
definition
comes afterwards.

WM
Cantor's definition requires finiteness or actual infinity in
distinction from potential infinity. My definition of relative
cardinality works in every conceivable case: finite, actually infinite,
potentially infinite. I admit quite frankly that my definition yields
less fantastic results but they are sound and true.

Regards, WM

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