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From: Archimedes Plutonium on 16 Jan 2010 02:43
I suppose that is a proof, of sorts, although I am not enamored by it.

I want to prove that the only way to well-define or precisely define
Finite is to
pick a large number and say that is the end of Finite. But once that
is accomplished
the Infinity no longer has a mathematical Algebra because
multiplication over the
infinite set containing infinite-numbers is impossible to have a
precise multiplication
or addition operators.

Proof: Infinite set with infinite-numbers always has a largest
infinite number, since
every number and every set has both a FrontView with a BackView.
Infinity in
mathematics is always able to be formed into the middle of the set or
the middle
of the number. Just as a Finite set or finite-number has a FrontView,
a BackView
and a Middle region. So in infinity, we just conveniently tuck the
infinity into the middle
and in this manner is able to perform alot more work on sets and
numbers and ideas.

Since an infinite set or infinite number has both a FrontView and
BackView with infinity
in the middle ground, depending on which end one takes we can call one
end the start
and the other end the finish. Now the proof is that with Finite
numbers or Finite Sets,
we can always add or multiply and achieve a new and larger number. But
with an Infinite
Set or infinite-number we come to the impossible where we have the
largest infinite number
and unable to multiply two such large numbers or the largest by itself
multiplied cannot achieve a newer larger infinite number. With an
infinite-set, we cannot do a powerset
because the cardinality of the powerset is the same as the orginal
infinite set.

A bit uneasy about the above, but it makes sense.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
From: Marshall on 16 Jan 2010 11:41
On Jan 15, 11:43 pm, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
>
> I want to prove that the only way to well-define or
> precisely define Finite is to pick a large number and
> say that is the end of Finite.

Anyone can define anything to be anything. The idea that
there is only one right definition of something is a failure
to understand what definitions are.


Marshall
From: Nam Nguyen on 16 Jan 2010 14:23
Marshall wrote:
> On Jan 15, 11:43 pm, Archimedes Plutonium
> <plutonium.archime...(a)gmail.com> wrote:
>> I want to prove that the only way to well-define or
>> precisely define Finite is to pick a large number and
>> say that is the end of Finite.
>
> Anyone can define anything to be anything. The idea that
> there is only one right definition of something is a failure
> to understand what definitions are.

Totally agreed with you on this. (Not that AP's "precise" definition
of "Finite" would make a lot of mathematical sense anyway).

So are you with me that the currently widely accepted definition of
the "natural numbers" is *not* the only right definition?

For instance, the following 2 definitions would be equally the right
ones (as well as the current one):

Let F be the formula "There are infinite counter examples of GC"

Def 1: The natural numbers = the current definition + that F is true.
Def 2: The natural numbers = the current definition + that F is false.

Right?
From: Nam Nguyen on 16 Jan 2010 15:01
Jesse F. Hughes wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Marshall wrote:
>>> On Jan 15, 11:43 pm, Archimedes Plutonium
>>> <plutonium.archime...(a)gmail.com> wrote:
>>>> I want to prove that the only way to well-define or
>>>> precisely define Finite is to pick a large number and
>>>> say that is the end of Finite.
>>> Anyone can define anything to be anything. The idea that
>>> there is only one right definition of something is a failure
>>> to understand what definitions are.
>> Totally agreed with you on this. (Not that AP's "precise" definition
>> of "Finite" would make a lot of mathematical sense anyway).
>>
>> So are you with me that the currently widely accepted definition of
>> the "natural numbers" is *not* the only right definition?
>>
>> For instance, the following 2 definitions would be equally the right
>> ones (as well as the current one):
>>
>> Let F be the formula "There are infinite counter examples of GC"
>
> Can you specify that formula in the language of PA?

Sure. Assuming we have a P(x), the statement F = "There are infinite examples
of P" can be formally written [or "translated"] in L(PA) as:

F = Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))]

Naturally x is a counter example of GC iff ~GC(x), and GC(x) iff x satisfies
GC.

> For simplicity's
> sake, let's assume that P(x) is a first order formula in PA such that
>
> P(x) <=> x is a counterexample to Goldbach's conjecture.
>
> So your formula F is essentially
>
> (Ex)( x is infinite & P(x) ).
>
> How do you plan on expressing "x is infinite" in the language of PA?
>
>> Def 1: The natural numbers = the current definition + that F is true.
>> Def 2: The natural numbers = the current definition + that F is false.
>>
>> Right?
>
From: Nam Nguyen on 16 Jan 2010 16:14
Nam Nguyen wrote:
> Jesse F. Hughes wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> Marshall wrote:
>>>> On Jan 15, 11:43 pm, Archimedes Plutonium
>>>> <plutonium.archime...(a)gmail.com> wrote:
>>>>> I want to prove that the only way to well-define or
>>>>> precisely define Finite is to pick a large number and
>>>>> say that is the end of Finite.
>>>> Anyone can define anything to be anything. The idea that
>>>> there is only one right definition of something is a failure
>>>> to understand what definitions are.
>>> Totally agreed with you on this. (Not that AP's "precise" definition
>>> of "Finite" would make a lot of mathematical sense anyway).
>>>
>>> So are you with me that the currently widely accepted definition of
>>> the "natural numbers" is *not* the only right definition?
>>>
>>> For instance, the following 2 definitions would be equally the right
>>> ones (as well as the current one):
>>>
>>> Let F be the formula "There are infinite counter examples of GC"
>>
>> Can you specify that formula in the language of PA?
>
> Sure. Assuming we have a P(x), the statement F = "There are infinite
> examples
> of P" can be formally written [or "translated"] in L(PA) as:
>
> F = Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))]
>
> Naturally x is a counter example of GC iff ~GC(x), and GC(x) iff x
> satisfies
> GC.
>
>> For simplicity's
>> sake, let's assume that P(x) is a first order formula in PA such that
>>
>> P(x) <=> x is a counterexample to Goldbach's conjecture.
>>
>> So your formula F is essentially
>>
>> (Ex)( x is infinite & P(x) ).
>>
>> How do you plan on expressing "x is infinite" in the language of PA?

Of course by "There are infinite counter examples of GC" I meant the set
of such counter example would be infinite, not each number is an infinite
number. (A natural number is actually neither finite nor infinite!)

>>
>>> Def 1: The natural numbers = the current definition + that F is true.
>>> Def 2: The natural numbers = the current definition + that F is false.
>>>
>>> Right?
>>
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