On Wed, 7 Jan 2004 14:07:31 +1000, "Tony Thomas"
verdigris@iprimus.com.au> wrote:

The principle of one-one correspondence is indispensable to the theory of
infinite sets.
That it is contentious goes without saying.

Only among crackpots on the internet.

If two infinite sets can be
shown to logically map one-one without appeal to the inductive principle
of
one-one mapping then this is the preferable method.

Even if one-one mapping, as usually applied to infinite sets, cannot be
refuted as a procedure neither can it be asserted as necessarily valid.
If,
however, it can be shown that the consequences of inductive one-one
mapping
contradict a more defensible method, then it should be abandoned.

The following examples illustrate the weakness of Cantorian Set Theory
and
its variants on this account.

D1: Z; The set of all integers
D2: Z+ The set of positive integers
D3: Z- The set of negative integers
D4: Ze The set of even integers
D5: Zo The set of odd integers
D6 {0} The unit set containing zero
D7: #S The cardinal of a general finite or infinite set S
D8: v Set union
D9: ^ Set intersection
D10: + Arithmetic addition
D11: - Arithmetic subtraction

Theorem: A + B = #(AvB) - #(A^B) where A and B are finite or infinite
sets*

This is false. In fact it's "not even false" - the _difference_ of two
infinite cardinals is undefined. Below you do not demonstrate
any problems with actual set theory - the problems you demonstrate
are just problems with your misunderstood version of set theory
(you give an illustration of _why_ the difference of two cardinals
is undefined.)

Proposition (i): The cardinal of Z is odd

There is also no notion of "odd" or "even" for infinite cardinals
in _actual_ set theory.

Z- v {0} v Z+ = Z

LHS accounts for all the integers.

#Z+ = #Z-

The cardinal of Z+ is equal to the cardinal of Z- without recourse to the
inductive principle of one-one mapping. Using the principle of symbolic
substitution "+" -> "-" demonstrated this assertion.

If #Z+ has even parity then #Z- has even parity, since #Z+ = #Z-.
If #Z+ has odd parity then #Z- has odd parity too.
In either case, #Z will have odd parity.

Proposition (i) has been demonstrated.



Proposition (ii): Either the set of even numbers or the set of odd
integers
can be mapped to the set of all integers but not both.

Ze v Zo = Z

By theorem (i), The cardinal of Z is odd

The cardinals of both Ze and Zo must have different parities

If Ze can be mapped one-one to Z then Zo cannot and vice versa.

Proposition (ii) has been demonstrated.

* Objectors must show why this theorem cannot be applied to infinite
sets.

No, first you need to say what these things mean. Give a _definition_
of the difference of two infinite cardinals, and then give a _proof_
of that "theorem".

************************

David C. Ullrich

What I still don't understand is, how to make sense of the claim that a
function
is a bijection between infinite sets, without recourse to the inductive
principle.

can understand clearly that f(n) = -n is a bijection for some finite sets,

and I
readily believe that it is a bijection for some infinite sets, but I do not
understand
how that second belief makes sense without invoking recursion.

Two sets are defined equinumerous when there's a bijection between them.

In Information Technology a number has even parity exactly when its
binary representation has an even number of 1's. This is clearly not the
same concept at all, since in the sense I am used to both 5 and 6 have
even parity

whereas 4 and 7 both have odd parity.

But could you explain to me how one evaluates bijection over infinite sets
without using recursion? Or perhaps you can point me to the right area of
study to work it out for myself?

Here's another exercise.

Thanks Autu, you get the point.

Aleph1, Aleph2 and successive cardinals must be of even parity because they
are powersets.
The high priests dare not admit that infinite cardinals have parity because
it introduces antinomies.

On Wed, 7 Jan 2004 07:11:39 -0000, "ChrisX" <spambin@asarian-host.org
wrote:


"Barb Knox" <see@sig.below> wrote in message
news:btg5o9$uoh$1@lust.ihug.co.nz...
In article <3ffb86be_1@news.iprimus.com.au>,
"Tony Thomas" <verdigris@iprimus.com.au> wrote:

#A + #B = #(AvB) + #(A^B)

Can you give an example where this fails?

(Hint: no you can't. If A and B are both finite it's true.
If A is infinite and B is finite then both sides are
equal to #A. If A and B are both infinite then both
sides equal max(#A, #B).)

unless #(A^B) = 0




************************

David C. Ullrich

This seems to show that the difference between two infinite 'numbers' can be
finite.

: In article <3ffb86be_1@news.iprimus.com.au>,
: "Tony Thomas" <verdigris@iprimus.com.au> wrote:
: >* Objectors must show why this theorem cannot be applied to infinite
sets.

Objectors need show only that Tony wouldn't know what a Theorem was if
it walked up to him and slapped him.

you have to have axioms. For this treatment, Tony has to DEFINE
"integer", "cardinal number" (not just the operator, but its result-type),
and "the principle of symbolic substitution". That principle cannot just
be alluded to in natural language; it has to be EXPRESSED in formal
language.
That is going to take you very far afield from traditional logic, which
does NOT care about the names you use for things. If you decide to define
numbers as strings then you are going to have to import a whole string
theory into this in order to express the principle. YOU WILL Be every
bit as ugly as ZFC before this is over. It is NOT worth it.

If I quote pythagorus theorum I and make use of it, would you expect me to
prove it or even be able to prove it?

Yes. But actually that's IRRELEVANT here, asshole.

Similarly #A + #B = #(A u B) + #(A i B) is a well known theorem of Boolian
algebra and requires proof no more than a(b + c) = ab + bc does.

Holly shit! PLEAZE shut up, idiot!!! (You might better use your time to

"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:eesnvvsd079qdmhn7li3h1kp5d8hmtaqp7@4ax.com...
On Wed, 7 Jan 2004 07:11:39 -0000, "ChrisX" <spambin@asarian-host.org
wrote:


"Barb Knox" <see@sig.below> wrote in message
news:btg5o9$uoh$1@lust.ihug.co.nz...
In article <3ffb86be_1@news.iprimus.com.au>,
"Tony Thomas" <verdigris@iprimus.com.au> wrote:

[...]

Theorem: A + B = #(AvB) - #(A^B) where A and B are finite or infinite
sets*

I expect you mean something like #A + #B = #(AvB) + #(A^B),
since "^" and "v" only apply to sets and "+" only applies to numbers.

It seems false to say either

#A + #B = #(AvB) - #(A^B)

That's "not even false", because the difference of two infinite
cardinals is undefined, so the assertion is simply meaningless.
(Elsewhere in the thread we see _why_ the difference is
undefined...)

Thank you David, I should not have said it seems false, I should have
said it does not seem true.


or

#A + #B = #(AvB) + #(A^B)

Can you give an example where this fails?

(Hint: no you can't. If A and B are both finite it's true.
If A is infinite and B is finite then both sides are
equal to #A. If A and B are both infinite then both
sides equal max(#A, #B).)

I take your point regarding infinite sets David. With finite sets I already
gave an example where it fails:

unless #(A^B) = 0

because for finite sets, if A^B is not empty, #(A^B) < #A + #B

Wishing you love and respect
Chris

Thanks for pointing out my gross errors.

Can I take it from your remarks below that you accept that #A + #B = #(AvB)
+ #(A^B) applies to infinite sets and their combination with finite ones?

You said; "If A is infinite and B is finite then both sides are
equal to #A."

This may be so, assuming Ao +1 = Ao, but this is what I am arguing against,
so its use in this context amounts to a petitio principii.

Consider the following example:

Let A = N (the set of natural numbers including zero)
Let B = {-1} (the unit set containing -1)

#N + #{-1} = #(N v {-1}) + #(N ^ {-1})

#N + 1 = #(N v {-1}) + #{}

#{} = 0

#N + 1 = #(N v {-1})

#(N v {-1}) - #N = 1

This seems to show that the difference between two infinite 'numbers' can be
finite.
Of course this can be arranged in the form Ao - Ao = 1

according to your
theory but surely this is too objectionable a proposition.

Ao - Ao = 1
Ao + 1 = Ao
Ao - Ao = -1.
1 = -1

(did I divide by nought somewhere?)

I await your instruction.


Tony Thomas

"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:eesnvvsd079qdmhn7li3h1kp5d8hmtaqp7@4ax.com...
On Wed, 7 Jan 2004 07:11:39 -0000, "ChrisX" <spambin@asarian-host.org
wrote:


"Barb Knox" <see@sig.below> wrote in message
news:btg5o9$uoh$1@lust.ihug.co.nz...
In article <3ffb86be_1@news.iprimus.com.au>,
"Tony Thomas" <verdigris@iprimus.com.au> wrote:

#A + #B = #(AvB) + #(A^B)

Can you give an example where this fails?

(Hint: no you can't. If A and B are both finite it's true.
If A is infinite and B is finite then both sides are
equal to #A. If A and B are both infinite then both
sides equal max(#A, #B).)

unless #(A^B) = 0




************************

David C. Ullrich

David C. Ullrich wrote:

There is also no notion of "odd" or "even" for infinite cardinals
in _actual_ set theory.

But if we extend the usual definitions to give

1. A cardinal k is odd if and only if there is a cardinal number
l, s.t. k = 2*l+1

2. A cardinal k is even if and only if there is a cardinal number
l, s.t. k = 2*l

we can easily prove not only that every large cardinal is odd, but also
that all of them are even! In face of this horrid contradiction, the
whole edifice of contemporary large cardinal research crumbles.

The idea that a proper subset can be 'equal' to its containing set

was
contentious prior to the rigor mortis of othodoxy represented by ZFC. A
quick glance at the history of mathematics will convince you of this.

Your ad hominem is an indication of the panic that sets in when orthodoxy is
challenged even when the challenger is clearly too puny to cause any harm.

As to your final point, I think that it is for the 'defenders' to explain
why common sense is inferior to its opposite. A possible defence for
introducing a counter intuitive method or principle is that it leads to
something unexpected or worthwhile. In the case of infinite 'arithmetic' the
idea of one-one correspondence as usually applied seems to lead to a barren
wasteland full of improbable objects.

The 'reasonable' assumption I made was that infinite sets should be treated
like finite ones unless proved otherwise. The idea that an infinite set can
have a total is highly suspect. The use of the word 'all' in English allows
statements that fall outside its evolutionary use on finite collections. The
idea that it must be associated with an arithmetical total of some kind is
difficult to shake off. Synthetic constructions like aleph-null are neither
flesh nor foul. They raise their ugly heads from the swamp but dare not show
themselves for fear of revealing that they have no body ie that they cannot
be treated like ordinary numbers but must remain esoteric chimera.

In addressing these questions kit is worth asking what kind of infinite sets
correspond to the five possible Eulerian relations between two sets.

R1: Identical sets
R2, R3: proper subsets
R3: partial intersection
R4: Disjoint sets.

If such relations appear to exist then I see no reason why infinite sets
should not conform to Boolian logic.

Tony Thomas

"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:d5snvvkif2imf5pores7ladkqc7ajc6d2r@4ax.com...
On Wed, 7 Jan 2004 14:07:31 +1000, "Tony Thomas"
verdigris@iprimus.com.au> wrote:

The principle of one-one correspondence is indispensable to the theory of
infinite sets.
That it is contentious goes without saying.

Only among crackpots on the internet.

If two infinite sets can be
shown to logically map one-one without appeal to the inductive principle
of
one-one mapping then this is the preferable method.

Even if one-one mapping, as usually applied to infinite sets, cannot be
refuted as a procedure neither can it be asserted as necessarily valid.
If,
however, it can be shown that the consequences of inductive one-one
mapping
contradict a more defensible method, then it should be abandoned.

The following examples illustrate the weakness of Cantorian Set Theory
and
its variants on this account.

D1: Z; The set of all integers
D2: Z+ The set of positive integers
D3: Z- The set of negative integers
D4: Ze The set of even integers
D5: Zo The set of odd integers
D6 {0} The unit set containing zero
D7: #S The cardinal of a general finite or infinite set S
D8: v Set union
D9: ^ Set intersection
D10: + Arithmetic addition
D11: - Arithmetic subtraction

Theorem: A + B = #(AvB) - #(A^B) where A and B are finite or infinite
sets*

This is false. In fact it's "not even false" - the _difference_ of two
infinite cardinals is undefined. Below you do not demonstrate
any problems with actual set theory - the problems you demonstrate
are just problems with your misunderstood version of set theory
(you give an illustration of _why_ the difference of two cardinals
is undefined.)

Proposition (i): The cardinal of Z is odd

There is also no notion of "odd" or "even" for infinite cardinals
in _actual_ set theory.

Z- v {0} v Z+ = Z

LHS accounts for all the integers.

#Z+ = #Z-

The cardinal of Z+ is equal to the cardinal of Z- without recourse to the
inductive principle of one-one mapping. Using the principle of symbolic
substitution "+" -> "-" demonstrated this assertion.

If #Z+ has even parity then #Z- has even parity, since #Z+ = #Z-.
If #Z+ has odd parity then #Z- has odd parity too.
In either case, #Z will have odd parity.

Proposition (i) has been demonstrated.



Proposition (ii): Either the set of even numbers or the set of odd
integers
can be mapped to the set of all integers but not both.

Ze v Zo = Z

By theorem (i), The cardinal of Z is odd

The cardinals of both Ze and Zo must have different parities

If Ze can be mapped one-one to Z then Zo cannot and vice versa.

Proposition (ii) has been demonstrated.

* Objectors must show why this theorem cannot be applied to infinite
sets.

No, first you need to say what these things mean. Give a _definition_
of the difference of two infinite cardinals, and then give a _proof_
of that "theorem".

************************

David C. Ullrich

The high priests dare not admit that infinite cardinals have parity because
it introduces antinomies.

If I quote pythagorus theorum I and make use of it, would you expect me to
prove it or even be able to prove it?

Similarly #A + #B = #(AvB) + #(A^B) is a well known theorem of Boolian
algebra and requires proof no more than a(b + c) = ab + bc does.

The requirement that everything must be defined in an argument is tantamount
to saying one must append a dictionary explanation of every word the first
tim it is used in a sentence. Provided sufficient information is provided to
make sense of an argument then that should suffice, pending any request for
clarification.

It is not so much that ZFC is ugly, rather it is barren. Surely the
mathematics of the infinite should be 'infinitely' richer than the
mathematics of the finite. The objects of ZFC (the Alephs) seem like so many
mountains in a featureless plain.