Is zero even or odd?

[John W. Kennedy]

But the size of the set of real numbers is Aleph 1 (oo^2).

Aleph-1 is at least aleph-null^aleph-null.
--
John W. Kennedy
"The pathetic hope that the White House will turn a Caligula into a
Marcus Aurelius is as naïve as the fear that ultimate power inevitably
corrupts."
-- James D. Barber (1930-2004).

 

[Torkel Franzen]

Aleph-1 is at least aleph-null^aleph-null.

On what do you base this assertion?

 

[Dave Seaman]

Aleph-1 is at least aleph-null^aleph-null.

No, it's the other way around. Since aleph_1 is by definition the
smallest uncountable cardinal, and since the reals are uncountable, it
follows that c (= 2^aleph_0 = aleph_0^aleph_0), the cardinality of the
continuum, cannot be less than aleph_1. On the other hand, it could be
that c is quite huge among the alephs.

 

[Michael Mendelsohn]

Numbers are concepts. Graph paper drawings are only representaions of
numbers, not numbers themselves.

How would you go about representing a complex vector "space number"? Now
we need 9 dimensions for our graph paper.

Have you never heard of origami?

Cheers
Michael

 

[Michael Mendelsohn]

0/0 pops up now and then: The system does the right thing if
this is treated as 1.

If in measuring a resistor, we find 0.0A at 0.0V, is the resistance 1
Ohm, then?

Cheers
Michael

 

[Jamie]

I have found, however, that if one has a model that leads one to a
divide by zero problem one is suffering from one of two problems: one,
an insufficient model, or two, an inaccurate measurement (which one can
sophomoricaly lump into one, if one is so inclined).

It is always necessary to ask yourself "why am I allowing a division by
something that may go to zero?" "What does it mean if my 'x' is zero
here?", etc., and code accordingly.

if your dealing with microprocessor code a simple Bit test of bit 0
(first bit) will tell you if the value is Odd/even..
if its on its Odd value otherwise its Even. and that covers the
zero problem.

 

[Kevin Aylward]

And physicists think it an ugly bodge.

Actually, I think the physicists think its just a bit annoying, its the
mathematicians that think its the ugly bodge.

Clearly the infinities are
failures of the theory,

Or a failure of the mathematics.



Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Kevin Aylward]

Great, now we define what is a number. Bertrand Russel threw
up his hands and whats-his-name declared he can prove it can't
be done.

Numbers, as are what are not just in the head, are 1:1 mappings
to things.

1 apple, 2 apples ... but sqrt(-1)apples, why that's only
in your imagination.


"Modeled" is the operative word. I see only graph paper,
the rest is imagination = dreams = not really there.

sqrt(-1) is used in calculations to indicate orthogonality,

It can do, but that is not the only reason for sqrt(-1). Its certainly
not how it came about in the first place.

where never the twain shall meet and the values do not mix
indiscriminately. And never is heard a discouraging word ...
Oklahoma isn't on Usenet, I take it.

Can we settle on "sqrt (-1) is a different kind of number"?

No.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Nicholas O. Lindan]

Aleph-1 is at least aleph-null^aleph-null.

Quite right, slip of the fingers (probably the mind, but I
always blame it on the fingers).

Should read ... Aleph 1 (oo^oo) ...

 

[Nicholas O. Lindan]

No, it's the other way around. Since aleph_1 is by definition the
smallest uncountable cardinal, and since the reals are uncountable, it
follows that c (= 2^aleph_0 = aleph_0^aleph_0), the cardinality of the
continuum, cannot be less than aleph_1. On the other hand, it could be
that c is quite huge among the alephs.

I'm lost.

 

[Dirk Bruere at Neopax]

Actually, I think the physicists think its just a bit annoying, its the
mathematicians that think its the ugly bodge.




Or a failure of the mathematics.

Is there a difference?

--
Dirk

The Consensus:-
The political party for the new millenium
http://www.theconsensus.org

 

[Nicholas O. Lindan]

If in measuring a resistor, we find 0.0A at 0.0V, is the resistance 1
Ohm, then?

Touche.

But, yes, I'll say it is 1, just not in conventional ohms.
At 0.0A and 0.0V any scaling factor can apply to the volts
and amps without changing the measurement:

1 new volt / 1 new amp = new value of the resistor.

0 new volts / 0 new amps = 1 * (1 volt / 1 amp) = new value of the resistor.

Value of the resistor = 1 new ohm.

 

[Nicholas O. Lindan]

if your dealing with microprocessor code a simple Bit test of bit 0
(first bit) will tell you if the value is Odd/even..
if its on its Odd value otherwise its Even. and that covers the
zero problem.

Once more, a PIC saves the day!

Cheers from the crowds ...

 

[John Woodgate]

I read in sci.electronics.design that Nicholas O. Lindan <[email protected]>

I'm lost.

You need to study the math of infinities. Aleph-null is the smallest
infinity, and whatever you do to it with finite numbers doesn't change
it. Many operations with itself, even, don't change it. But raising it
to its power, {-}o^({-}o), creates a new infinity with different
properties. Although it's called aleph-one, no-one knows whether it is
the *next* infinity after aleph-null, or whether there are other
infinities in between.

No, I can't say I *understand* it either. The above was written in
parrot mode.

 

[k wallace]

One infinity, one zero. +oo == -oo; +0 == -0. Neither
actually exist and you can approach from the direction of
your choice.

lol- pre-coffee, I read this to say "neither of you actually exist..."
one of the better refutations I've seen on Usenet, I was thinking, and
then I reread.
Ok, continue on..
-kwallace

 

[Michael Mendelsohn]

Touche.

But, yes, I'll say it is 1, just not in conventional ohms.
At 0.0A and 0.0V any scaling factor can apply to the volts
and amps without changing the measurement:

1 new volt / 1 new amp = new value of the resistor.

0 new volts / 0 new amps = 1 * (1 volt / 1 amp) = new value of the resistor.

Value of the resistor = 1 new ohm.

When checking it turned out that some thief had actually stolen the
resistor where 0V,0A was measured. The circuit was broken, but noone
noticed because the voltage was zero.

Hence, vacuum/an insulator/air has a resistance of 1 new Ohm?

Cheers
Michael

 

[George Cox]

I read in sci.electronics.design that Nicholas O. Lindan <[email protected]>


You need to study the math of infinities. Aleph-null is the smallest
infinity, and whatever you do to it with finite numbers doesn't change
it. Many operations with itself, even, don't change it. But raising it
to its power, {-}o^({-}o), creates a new infinity with different
properties. Although it's called aleph-one, no-one knows whether it is
the *next* infinity after aleph-null, or whether there are other
infinities in between.

Whether (aleph_0)^{aleph_0} = aleph_1 or not isn't decided in the usual
set theories.

aleph_1 is the next infinity after aleph_0 (given the axiom of choice),
the question is, is 2^{aleph_0} the next infinity after aleph_0? (And
generally, is 2^{aleph_{alpha}} the next infinity after aleph_{alpha}?)

 

[John Fields]

When checking it turned out that some thief had actually stolen the
resistor where 0V,0A was measured. The circuit was broken, but noone
noticed because the voltage was zero.

---
AHA! Pilot error!

In truth, the E in

E
R = ---
I

refers to the voltage _across_ the resistor, (a shunt, was it?) which
you didn't measure. What you measured was the voltage from the low
side of where the resistor was supposed to be to ground, which gave
you zero volts which corresponded, also, to zero amps. Had you
measured the voltage _across_ where the resistor was supposed to be
you would have measured the entire supply voltage minus what was being
dropped across the load by the current flowing through the meter and
you would have concluded that by subtracting the meter current that
you would have had:

E E
R = --- = --- = oo
I 0

Which would have been right!

 

[Michael Mendelsohn]

In truth, the E in

E
R = ---
I

refers to the voltage _across_ the resistor, (a shunt, was it?) which
you didn't measure. What you measured was the voltage from the low
side of where the resistor was supposed to be to ground, which gave
you zero volts which corresponded, also, to zero amps. Had you
measured the voltage _across_ where the resistor was supposed to be

Well, I did - do you think I'm stupid?

The problem is, the measurement was automatic, and since there was
short-circuit somewhere (presumably in parallel to the supposed
resistor), the voltage was zero across the measurement points even
before the resistor was stolen. The current is of course measured the
proper way.

The software computed resistance by Nick's rules and hence never noticed
anything unusual.

you would have measured the entire supply voltage minus what was being
dropped across the load by the current flowing through the meter and
you would have concluded that by subtracting the meter current that
you would have had:

E E
R = --- = --- = oo
I 0

Which would have been right!

Unless E=0 too, in which case the result is 1 (says Nick).

On a short circuit you can detect no voltage, but you can measure a
current.

E 0
R = --- = --- = 0
I I

This leads to a contradiction when E=I=0.

Cheers
Michael

 

[Dave Seaman]

I read in sci.electronics.design that Nicholas O. Lindan <[email protected]>

You need to study the math of infinities. Aleph-null is the smallest
infinity, and whatever you do to it with finite numbers doesn't change
it. Many operations with itself, even, don't change it. But raising it
to its power, {-}o^({-}o), creates a new infinity with different
properties. Although it's called aleph-one, no-one knows whether it is
the *next* infinity after aleph-null, or whether there are other
infinities in between.

No, I can't say I *understand* it either. The above was written in
parrot mode.

It's a widespread belief (and one that is unfortunately perpetuated by some
popular expositions) that the cardinality of the reals is aleph_1. Not so.
The cardinality of the reals is 2^aleph_0, which is the same as
aleph_0^aleph_0. This cardinal is called c, for the cardinality of the
continuum. The proposition that c = aleph_1 is called the continuum
hypothesis, and it is known to be independent of the usual axioms of set
theory.

<http://mathworld.wolfram.com/ContinuumHypothesis.html>