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Topic: Infinity: The Story So Far
Replies: 298   Last Post: Apr 7, 2014 7:30 AM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Infinity: The Story So Far
Posted: Feb 27, 2014 11:14 PM

On 2/27/2014 3:03 PM, Virgil wrote:
> mueckenh@rz.fh-augsburg.de wrote:
>

>> My book gives the correct axioms.
>>

>>> For over a century now, I think Peano's axioms have been the standard,
>>
>> Like Cantor's nonsense. Where do they define that the difference must always
>> be 1?

>
> They don't. They only requires that each member of a Peano set have a
> unique successor member, but do not require any specific 'difference' up
> front.
> It is only later, after all the basic axioms are satisfied, when
> defining addition that any "1" appears.
>

That is actually problematic.

Willard's work in arithmetic looks to systems that
focus on division and difference rather than

But, Mr. Gabriel has it wrong. In the axioms that
he did not understand in multiple languages, there
is a subtle definition of even numbers. Even numbers
are those measured by 2 equal parts. Odd numbers
are those that are not.

Then, Euclid observes that an odd number differs
from an even number by a unit.

Equals are compared with respect to even numbers
in the sense that equals measure even numbers
by reflection as a part.

So, there is a natural grouping of pairs of odd
numbers with their enclosed even number congruent to
2 mod 4 into triples. The "difference" between
consecutive odd numbers is 2 equal parts which
are units.

Now, if the triples are interleaved with the
numbers congruent to 0 mod 4, then the "difference"
between extrema for consecutive triples is
2 equal parts which are not units.

Notice how I am using "equals" "units" and
"not units". The number labels make it
easier to understand the model, But, once
you understand the model, the labels can
be obfuscated.

Consider the graphs,

v_01 = { 1, 2, 3 }
e_01 = { { { 1 }, { 1, 3 } } }

v_02 = { 4 }
e_02 = { { } }

v_03 = { 5, 6, 7 }
e_03 = { { { 5 }, { 5, 7 } } }

v_04 = { 8 }
e_04 = { { } }

v_05 = { 9, 10, 11 }
e_05 = { { { 9 }, { 9, 11 } } }

v_06 = { 12 }
e_06 = { { } }

v_07 = { 13, 14, 15 }
e_07 = { { { 9 }, { 9, 11 } } }

v_08 = { 16 }
e_08 = { { } }

etc.

The even and odd numbers are organized as I have
described from Euclid's axioms. Now, let the
vertex set,

v_all = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ... }

Have the following ordered pairs constituting its
edge set,

e_all:

{ { { 1 }, { 1, 3 } } }
{ { { 2 }, { 2, 4 } } }
{ { { 3 }, { 3, 4 } } }
{ { { 4 }, { 4, 5 } } }
{ { { 4 }, { 4, 6 } } }
{ { { 5 }, { 5, 7 } } }
{ { { 6 }, { 6, 8 } } }
{ { { 7 }, { 7, 8 } } }
{ { { 8 }, { 8, 9 } } }
{ { { 8 }, { 8, 10 } } }
{ { { 9 }, { 9, 11 } } }
{ { { 10 }, { 10, 12 } } }
{ { { 11 }, { 11, 12 } } }
{ { { 12 }, { 12, 13 } } }
{ { { 12 }, { 12, 14 } } }
{ { { 13 }, { 13, 15 } } }
{ { { 14 }, { 14, 16 } } }
{ { { 15 }, { 15, 16 } } }

etc.

You will find that this structure is a semiorder.

http://en.wikipedia.org/wiki/Interval_order

you will find the statement,

"An interval order defined by unit intervals
is a semiorder."

Now, Mr. Gabriel laughed at my very complex axioms, and,
no doubt he shall again. However, the 20th axiom uses
a formula derived from Goedel's pairing function to

x + w

in the context of both

x + x and w + w

Note that these would be instances of "equals"
in the sense of Euclid

And, in the 21st axiom, the relation which did that
is associated with every additive relation and then
bounded above by differences given in terms of
units.

In axiom 20, 'v' can be 1 or any prime. The same is
true for 'w' in axiom 21. But, in axiom 21, the expression,

forces p = 1 and the relations,

-(p,v,u,t) /\ -(p,u,v,t)

are interpreted as

t - u = v and t - v = u

respectively with respect to differences by
units.

Axiom 20: (binary partition asymmetry relations, w + x => w^2 + x^2 +
2wx + v(w + x) + (x + x), x + w => x^2 + w^2 + 2xw + v(x + w) + (w + w) )

AvAwAxAyAz( P(v,w,x,y,z) <-> ( Ar( At( r adiv t ) -> As( ( ( AS(r,r,s)
/\ ~Et( AS(r,r,t) /\ AS(r,t,s) ) ) /\ ~( s midv v ) ) -> ( v adiv v ->
( EoEpEq( ( ( +(r,w,x,p) /\ +(r,v,p,q) ) /\ *(q,p,o) ) /\ EjEkEmEn( (
+(r,w,w,j) /\ +(r,x,x,k) ) /\ ( ( +(r,o,j,m) /\ +(r,o,k,n) ) /\ (
*(s,y,m) /\ *(s,z,n) ) ) ) ) ) ) ) )

Axiom 21: (addition relations recast with unit atomic differences
bounding partition asymmetry relations)

AwAxAyAz( +(w,x,y,z) <-> EtEuEvAp( Aq( p adiv q) -> ( P(w,x,y,u,v) /\ (
-(p,v,u,t) /\ -(p,u,v,t) ) ) )

The asserted existence is that of both 1 and 2 simultaneously,

Axiom 7: (pointed set domain assertion, there exists a unit atom and a
unit atomic-successor to the unit atom)

ExEy( Az( x adiv z ) /\ AS(x,x,y) /\ ~Ez( AS(x,x,z) /\ AS(x,z,y) ) )

Axiom 8: (asserted identity, x is identical to y if x and y have the
same unit atomic-successors)

AxAy( x = y <-> EvAu( Az( u adiv z ) -> ( ( AS(u,x,v) /\ AS(u,y,v) ) /\
( ~Ez( AS(u,x,z) /\ AS(u,z,v) ) /\ ~Ez( AS(u,y,z) /\ AS(u,z,v) ) ) ) ) ) )

1 and 2 must be asserted together because 1 and 1 + 1 are necessary
to assert the identity of 1.

As I said, the Mr. Gabriel failed to understand Euclid's axioms
in several languages.

Date Subject Author
2/25/14 Dan Christensen
2/25/14 Brian Q. Hutchings
2/25/14 ross.finlayson@gmail.com
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2/26/14 Leo Sgouros
2/26/14 ross.finlayson@gmail.com
2/26/14 William Elliot
3/1/14 Dan Christensen
3/2/14 Brian Q. Hutchings
2/26/14 thenewcalculus@gmail.com
2/26/14 Dan Christensen
2/26/14 thenewcalculus@gmail.com
2/26/14 Dan Christensen
2/26/14 Virgil
2/26/14 Dan Christensen
2/26/14 Martin Shobe
2/26/14 ross.finlayson@gmail.com
2/27/14 Dan Christensen
2/27/14 Martin Shobe
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/26/14 thenewcalculus@gmail.com
2/26/14 Dan Christensen
2/26/14 thenewcalculus@gmail.com
2/26/14 Dan Christensen
2/26/14 thenewcalculus@gmail.com
2/26/14 Dan Christensen
2/26/14 Brian Q. Hutchings
2/26/14 mueckenh@rz.fh-augsburg.de
2/26/14 Dan Christensen
2/26/14 mueckenh@rz.fh-augsburg.de
2/26/14 Dan Christensen
2/26/14 Brian Q. Hutchings
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Dan Christensen
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Dan Christensen
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 fom
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Tanu R.
2/27/14 Tanu R.
2/28/14 Virgil
2/27/14 Virgil
2/27/14 Tanu R.
2/27/14 fom
2/27/14 thenewcalculus@gmail.com
2/28/14 Virgil
2/28/14 mueckenh@rz.fh-augsburg.de
2/28/14 Virgil
2/28/14 mueckenh@rz.fh-augsburg.de
2/28/14 Virgil
2/28/14 fom
2/27/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Tanu R.
2/27/14 thenewcalculus@gmail.com
2/27/14 Virgil
2/26/14 Virgil
2/26/14 mueckenh@rz.fh-augsburg.de
2/26/14 Virgil
2/26/14 Dan Christensen
2/26/14 fom
2/26/14 mueckenh@rz.fh-augsburg.de
2/26/14 fom
2/26/14 Dan Christensen
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/27/14 Dan Christensen
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 fom
2/27/14 Peter Percival
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2/27/14 Peter Percival
2/27/14 fom
2/27/14 thenewcalculus@gmail.com
4/7/14 Aatu Koskensilta
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
2/27/14 Virgil
2/26/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/26/14 thenewcalculus@gmail.com
2/26/14 Brian Q. Hutchings
2/26/14 Virgil
2/26/14 mueckenh@rz.fh-augsburg.de
2/26/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
2/27/14 Virgil
2/27/14 mueckenh@rz.fh-augsburg.de
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2/26/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
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2/27/14 thenewcalculus@gmail.com
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2/27/14 Martin Shobe
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2/27/14 thenewcalculus@gmail.com
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2/27/14 Martin Shobe
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2/27/14 Martin Shobe
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2/28/14 Martin Shobe
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2/28/14 Sal Honda
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2/27/14 fom
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2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
2/27/14 Dan Christensen
2/27/14 thenewcalculus@gmail.com
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2/28/14 Dan Christensen
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2/28/14 Harman Kardan
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3/1/14 thenewcalculus@gmail.com
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3/1/14 Dan Christensen
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3/5/14 Dan Christensen
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3/5/14 Dan Christensen
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3/11/14 Martin Shobe
3/11/14 thenewcalculus@gmail.com
3/11/14 thenewcalculus@gmail.com
3/11/14 Martin Shobe
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