Is that which is not equal to itself..equal to itself or not?

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Reply Sun 28 Feb, 2010 07:15 am
In virtue of meaning, that which is not equal to itself (the x:~(x=x)), is not equal to itself. We can show that (the x:~(x=x)) = (the x:~(x=x)), is a contradiction, because it (the x:~(x=x)) does not exist.

But, (the x:~(x=x)) = (the x:~(x=x)) <-> (All F)(F(the x:~(x=x)) <-> F(the x:~(x=x))) is a theorem. And, (All F)(F(the x:~(x=x)) <-> F(the x:~(x=x)) is tautologous. Therefore, (the x:~(x=x)) = (the x:~(x=x)), is a tautology.

See:
Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

What is the solution to this puzzle?
 
ughaibu
 
Reply Sun 28 Feb, 2010 08:03 am
@Owen phil,
Owen;133532 wrote:
What is the solution to this puzzle?
How about this: dfnul2 - Metamath Proof Explorer
 
Owen phil
 
Reply Sun 28 Feb, 2010 08:31 am
@ughaibu,
ughaibu;133538 wrote:


The null set, {the x's: ~(x=x)} does exist by axiom.
That is {x:~(x=x} = {x:~(x=x)} is a theorem.
{x:~(x=x)} is a value of a variable.

My question is..Does the null object (the x:~(x=x)) equal itself?
 
ughaibu
 
Reply Sun 28 Feb, 2010 09:01 am
@Owen phil,
Owen;133543 wrote:
Does the null object (the x:~(x=x)) equal itself?
I dont see the problem with it equaling itself, as it has no elements.
 
Owen phil
 
Reply Sun 28 Feb, 2010 09:31 am
@ughaibu,
ughaibu;133547 wrote:
I dont see the problem with it equaling itself, as it has no elements.


Individuals do not have elements.


D1. G(the x:Fx) =df (Some y)((All x)(x=y <-> Fx) & Gy).

~((the x:~(x=x)) = (the x:~(x=x))).

Proof:

(the x:~(x=x))=(the x:~(x=x)) <-> (Some y)((All x)(x=y <-> ~(x=x)) & y=y). By D1.

~(x=x) is contradictory for all x.

(the x:~(x=x)) = (the x:~(x=x)) <-> (Some y)((All x)(x=y <-> (contradiction)) & y=y).

[x=y <-> (contradiction)] <-> ~(x=y).

ie.
(the x:~(x=x))=(the x:~(x=x)) <-> (Some y)((All x)(~(x=y)).
(the x:~(x=x))=(the x:~(x=x)) <-> ~(All y)((Some x)(x=y)).

But, (All y)(Some x)(x=y) is a theorem.

That is to say, ~(All y)(Some x)(x=y) is a contradiction.

therefore, ~((the x:~(x=x))=(the x:~(x=x))) is a logical necessity.
 
ughaibu
 
Reply Sun 28 Feb, 2010 09:59 am
@Owen phil,
Owen;133551 wrote:
therefore, ~((the x:~(x=x))=(the x:~(x=x))) is a logical necessity.
If there is such an x, but there isn't such an x because the empty set is the set of all such x. I still dont see the problem, if you've proved that all x not equal to themselves aren't equal to themselves, there's no conflict with the empty set equaling itself. Is there?
 
Owen phil
 
Reply Sun 28 Feb, 2010 10:14 am
@ughaibu,
ughaibu;133556 wrote:
If there is such an x, but there isn't such an x because the empty set is the set of all such x. I still dont see the problem, if you've proved that all x not equal to themselves aren't equal to themselves, there's no conflict with the empty set equaling itself. Is there?


The empty set is defined as that set which has no elements...{x:~(x=x)}.

ughaibu,
"I still dont see the problem, if you've proved that all x not equal to themselves aren't equal to themselves, there's no conflict with the empty set equaling itself."

I did not prove that 'all x not equal to themselves aren't equal to themselves'.

I did prove that the described individual object (the x:~(x=x)), does not equal itself.

(the x:~(x=x)) is a member of {x:~(x=x)}, is false!
It is false to say that the null set contains non-existent objects.
There are no non-existent objects at all...they do not exist.

ughaibu,
"there's no conflict with the empty set equaling itself. Is there?"

As I have already said, the null set is equal to itself and it does exist.

The null object is not the same as the null set.
 
ughaibu
 
Reply Sun 28 Feb, 2010 07:45 pm
@Owen phil,
Owen;133560 wrote:
I did prove that the described individual object (the x:~(x=x)), does not equal itself.
As far as I can see, the thing doesn't equal itself, by definition, but I suspect that I'm not getting what you mean by an "individual object". What is "x"? If it's the object itself, then the object includes this definition, and if the object includes this definition, then the object doesn't exist. But, if the statement is not a definition, what is it?
 
Reconstructo
 
Reply Sun 28 Feb, 2010 11:08 pm
@Owen phil,
This sentence is false.
 
Owen phil
 
Reply Mon 1 Mar, 2010 02:49 am
@Reconstructo,
Reconstructo;133866 wrote:
This sentence is false.


That which is not equal to itself is equal to itself, or, it is false that (that which is not equal to itself is equal to itself)..is a tautology of the form (p v ~p).

This sentence cannot be false.
 
Reconstructo
 
Reply Mon 1 Mar, 2010 03:02 am
@Owen phil,
Owen;133971 wrote:
That which is not equal to itself is equal to itself, or, it is false that (that which is not equal to itself is equal to itself)..is a tautology of the form (p v ~p).

This sentence cannot be false.


"This sentence is false." Doesn't that seem like a self-referential paradox?
 
Owen phil
 
Reply Mon 1 Mar, 2010 07:41 am
@Reconstructo,
Reconstructo;133974 wrote:
"This sentence is false." Doesn't that seem like a self-referential paradox?


Yes, if 'this sentence' has no reference.

This sentence is false, is gibberish..because 'this sentence' has no reference, and cannot be true or false.

Truth or falsity apply only to declaritive statements which have truth or falsity ..ie. to propositions.
 
Reconstructo
 
Reply Mon 1 Mar, 2010 06:15 pm
@Owen phil,
There is only one number, it seems. This is implicit in Wittgenstein's quote here, and goes all the way back to Parmenides & Pythagoras.

Quote:


6.1271 It is clear that the number of the 'primitive propositions
of logic' is arbitrary, since one could derive logic from a single
primitive proposition, e.g. by simply constructing the logical product
of Frege's primitive propositions. (Frege would perhaps say that
we should then no longer have an immediately self-evident primitive
proposition. But it is remarkable that a thinker as rigorous as Frege
appealed to the degree of self-evidence as the criterion of a logical
proposition.)


6.13 Logic is not a body of doctrine, but a mirror-image of the world.
Logic is transcendental.


6.2 Mathematics is a logical method. The propositions of mathematics are
equations, and therefore pseudo-propositions.


6.21 A proposition of mathematics does not express a thought.


6.211 Indeed in real life a mathematical proposition is never what
we want. Rather, we make use of mathematical propositions only in
inferences from propositions that do not belong to mathematics to others
that likewise do not belong to mathematics. (In philosophy the question,
'What do we actually use this word or this proposition for?' repeatedly
leads to valuable insights.)


6.22 The logic of the world, which is shown in tautologies by the
propositions of logic, is shown in equations by mathematics.


6.23 If two expressions are combined by means of the sign of equality,
that means that they can be substituted for one another. But it must be
manifest in the two expressions themselves whether this is the case
or not. When two expressions can be substituted for one another, that
characterizes their logical form.
But negation is also necessary, and yet negation is not a number. Or is it?
 
Owen phil
 
Reply Tue 2 Mar, 2010 07:00 am
@Reconstructo,
Reconstructo;134233 wrote:
There is only one number, it seems. This is implicit in Wittgenstein's quote here, and goes all the way back to Parmenides & Pythagoras.

But negation is also necessary, and yet negation is not a number. Or is it?


What the hell are you talking about?
Where in Wittgenstein do you believe he said or implied "there is only one number"???

Negation is a number??? Are you serious, surely nobody could make this stupid claim.
 
Reconstructo
 
Reply Tue 2 Mar, 2010 03:44 pm
@Owen phil,
Owen;134549 wrote:
What the hell are you talking about?
Where in Wittgenstein do you believe he said or implied "there is only one number"???

Negation is a number??? Are you serious, surely nobody could make this stupid claim.


There is only one number, yes. But negation is not this number...Number is the maximum negation of logos, or word.. It's a brilliant statement.. That's the truth, friend, and it's clear as day....once you really think on it..
 
 

 
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