Boolean Logic cannot deal with infinitely many objects

[moshek]

Hi Organic,

I did not understand until now,
That you wan't axiomatic set theory to hold as it is today?

Are you please with P.Choen forcing method
to solve Hilbert first problem CH ?

please explain that to me.

Moshek

 

[master_coda]

{} is the empty set. Emptiness is a word that has a great deal of philosophical baggage that adds confusion to the issue.

For example, you seem content to define fullness as the opposite of emptiness. After all, fullness is clearly the opposite of emptiness.

Except that it isn't rigorous at all. What is fullness? The set of all sets? The set of all things?

 

[Organic]

Hi moshek,

CH problem has some meaning if 2^aleph0 > aleph0, but in my first post in this thread i show that (2^aleph0 >= aleph0) = {}.

 

[Organic]

Master_coda

Fullness is the highest limit of any form of information.

Emptiness is the lowest limit of any form of information.

 

[master_coda]

Originally posted by Organic
Master_coda

Fullness is the highest limit of any form of information.

Emptiness is the lowest limit of any form of information.

This is exactly the problem with you saying emptiness is {}, and saying that axiomatic set theory depends on it. The mathematical definition of the empty set is the set A such that x\notin A is always true, regardless of what x is.

Your definition talks about "lowest limits of information" which is just something else you haven't defined.

Then you seem to apply your non-mathematical definition to the standard definition in set theory.

 

[Organic]

Master_coda,

At this point i need your help.


Zf set theory defines the empty set as:

the set A such that x not in A is always true, regardless of what x is.

If you look at this definition, then x must be some existing input form of information.

My idea go deeper then that, and start its research by including the limits of any form of information.


Can you help me to define this idea in a formal way?

Thank you.

Organic

 

[master_coda]

Originally posted by Organic

the set A such that x not in A is always true, regardless of what x is.

If you look at this definition, then x must be some existing form of information.

Why must x be an existing form of information?

 

[Organic]

And what if x is Emptiness?

 

[master_coda]

Originally posted by Organic
And what if x is Emptiness?

Then x is not in the empty set.

 

[Organic]

And what if x is Fullness?

 

[master_coda]

Originally posted by Organic
And what if x is Fullness?

Then x is not in the empty set.

No matter what x is, x is not in the empty set.

 

[Organic]

x is the input A is the set.

the set A such that x not in A is always true, regardless of what x is.

Whet is A if x is Emptiness?

What is A if x is Fullness?

What is A if x is not Epmtiness nor Fullness?

 

[master_coda]

Originally posted by Organic
x is the input A is the set.

the set A such that x not in A is always true, regardless of what x is.

Whet is A if x is Emptiness?

What is A if x is Fullness?

What is A if x is not Epmtiness nor Fullness?

A is the same thing in all three cases. A is the empty set. It doesn't matter what x is.

 

[Organic]

A is the same thing in all three cases. A is the empty set. It doesn't matter what x is.

Really ?

x=Eemptiness

the set A such that Emptiness not in A is always true.

 

[master_coda]

Originally posted by Organic
Really ?

x=Eemptiness

the set A such that Emptiness not in A is always true.

Of course, I'm still waiting for a mathematical definition of Emptiness from you. But however you define it, it isn't contained in the empty set.

 

[Hurkyl]

Your post is poorly written, my best guess is that this sentence is supposed to be a definition of A:

the set A such that x not in A is always true, regardless of what x is.

And if that is correct, then A has been defined; it doesn't become something different. master_coda would thus be correct.

 

[Organic]

No, you used a formal definition and learned through our last posts
that this formal definition has logical holes in it that you did not close.

Please show me why do i have to define intuiative concepts like 'Emptiness' and 'Fullness'?

 

[master_coda]

Originally posted by Organic
No, you used a formal definition and learned through our last posts
that this formal definition has logical holes in it that you did not close.

Please show me why do i have to define an intuiative concepts like 'Emptiness' and 'Fullness'?

Because in math, you have to define everything. Intuitive concepts have no mathematical value until they've been formally defined.


If you use the definition Emptiness=Empty Set then it is still true that \mathrm{Emptiness}\notin\mathrm{Empty Set}.

If you define Emptiness as "the thing that is contained in the empty set" than your formal system is inconsitent.

If you provide a non-mathematical definition than there's no point in even talking about what your definition has to do with math.

 

[Hurkyl]

Please show me why do i have to define intuiative concepts like 'Emptiness' and 'Fullness'?

You wanted help expressing your ideas in a formal way. That entails defining everything.

 

[Organic]

There is no such a thing like "Empty set".

All we have is the set concept, and its name is given by its content.

We cannot separate between a set's name and its content's property,
as you wrongly show in your example.

 

[Organic]

Hi Hurkyl,

Then what is the definition of the set concept?

What is the definition of the content concept?

What is the definition of the number concept?

What is the definition of belonging?

 

[master_coda]

Originally posted by Organic
There is no such a thing like "Empty set".

All we have is the set concept, and its name is given by its content.

We cannot separate between a set's name and its content's property,
as you wrongly show in your example.

Ultimately, this is why your math is worthless. The name of the set is arbitrary. Call it the empty set. Call it \varnothing. Call it xerfniernisetjilsegtilnerilsneirk. It doesn't matter.

Saying "the empty set should contain emptiness because it's name is the empty set" doesn't mean anything. Math isn't about what you think should be true.

Math is about taking definitions and applying logic to them to see what conclusions you can try. If you aren't willing to define things, than you can't do math. It's as simple as that.

 

[Hurkyl]

A "set" is an object in ZFC. (or substitute your favorite set theory)

I have no idea what "content" is because that's your idea and you haven't defined it.

As for "number" you're going to have to be more specific; e.g. do you mean real number?

 

[Organic]

master_coda

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

 

[Organic]

Hurkyl,

A "set" is an object in ZFC.

x='set'

y='member'

A x is an object in ZFC.
A y is an object in ZFC.

How can i distinguish between them by your definition?

 

[Hurkyl]

Everything in ZFC is a set, including members of other sets.

(There are other set theories, including some very similar to ZFC, where sets can contain things that aren't sets)

 

[master_coda]

Originally posted by Organic

x='set'

y='member'

A x is an object in ZFC.
A y is an object in ZFC.

How can i distinguish between them by your definition?

x is a set if it can be constructed using the axioms of ZFC. y is a set if it can be constructed using the axioms of ZFC.

If y is not a set, then x is not equal to y.

If y is a set, then you can determine if x and y are equal using the axiom of extensionality.

Of course, in ZFC everything is a set, so the case of "y is not a set" doesn't actually matter.

 

[Organic]

So, set is an object in ZFC.

Then what is an object?

 

[Hurkyl]

Then what is an object?

As used here, it's just a descriptive English word, and not a mathematical term. (Actually, so is "set" in this case, though in, say, Category Theory or NBG "set" is actually a mathematical term)

 

[Organic]

Hukyl,

Also "set" is just an Enlgish word.

Therefore we are in a circular definition like:

... a set is an object is a set is an object is ...

 

[Hurkyl]

However, the axioms of ZFC are not just english words; they clearly define what one may do with sets.

 

[Organic]

Hurkyl,

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

Please tell me what is A?

 

[master_coda]

Originally posted by Organic
Hurkyl,

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

Please tell me what is A?

A is the empty set. We don't need to know the properties of x, since the definition of A doesn't mention any of the properties of x. If the definition said somewhere "x must have property y" then we would need to know something about x. But the definition doesn't depend on what x is.

 

[Organic]

master_coda,

The definition is fine, but A's name depends on x property.

So, here is my question again:

Emptiness=

x=Emptiness

the set A such that x not in A is always true, regardless of what x is.

A cannot be defined without x property.

Please tell me what is A?

 

[Hurkyl]

\varnothing is defined by:

\forall x: x \notin \varnothing

This is well defined, because one can prove that:

<br /> \forall y: \left(<br /> ( \forall x: x \notin y ) \Rightarrow y = \varnothing<br /> \right)<br />


IOW if A is a set such that for all x, x \notin A, then A = \varnothing.

 

[master_coda]

You can name A whatever you want. The name is just an arbitrary label. If you don't like calling it the empty set, then call it whatever you want. Just make it clear that your name for it is a label for the thing mathematicians call the empty set.

There is only one set that satisifies the definition
\forall x\colon(x\notin\varnothing)

 

[Organic]

So to get A as an Empty set we have to define it like that:

if A is a set such that for all x,x not in A, then A={}(=Empty set) .

But:

Emptiness=

All x=Emptiness

What is set A?

 

[master_coda]

Originally posted by Organic
But:

Emptiness=

All x=Emptiness

What is set A?

I don't understand what you're asking.

 

[Organic]

to get A as an Empty set we have to define it like that:

COND='all x' Or COND='any x'

if A is a set such that for COND,x not in A, then A={}(=Empty set) .


Do you agree with both COND?

 

[master_coda]

Originally posted by Organic
to get A as an Empty set we have to define it like that:

COND='all x' Or COND='any x'

if A is a set such that for COND,x not in A, then A={}(=Empty set) .


Do you agree with both COND?

Yes.

 

[Organic]

Ok,

So to get A as an Empty set we have to define it like that:

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

Emptiness=

Any x=Emptiness

Then what is set A?

 

[master_coda]

Ah, I see what you're saying.

If you have no x, then you have nothing. Not even the empty set. There is no A.

That's why the ZFC axioms include the axiom of the empty set. It asserts that the empty set exists.

 

[Organic]

Bravoooo !


And the opposite concept of Emptiness is Fullness.

 

[master_coda]

But we don't have to worry about not having any x. Because we know that empty set exists.

And I believe its been mentioned before...you can't define things with "opposite". The idea of opposite depends a great deal on context, so without supplying one, you can't use it.

 

[Organic]

But there is another point of view on x.

Like in a computer program x can be a variable of some value.

Therefore if x= , then x as a variable exists, but without any content.

Emptiness=

Any x=No Emptiness

Now, to get A as an Empty set we have to define it like that:

if A is a set such that for any x-content,x-content not in A, then A={}(=Empty set) .

But:

Emptiness=

Any x=Emptiness

Then what is set A?

Answer: Set A is a non-empty set.

 

[master_coda]

Why would we want to define the set A in such a way?

Besides, what does it mean for something to exist, but have no content?


I certainly hope you aren't thinking that variables in math are anything like variables on a computer. They're two very different concepts that unfortunately use the same name.

 

[Organic]

{} exists but has no content.

 

[master_coda]

Then what is x-content supposed to be?

 

[Organic]

In the x computer-model x is like a temporary container that delivers its content to the final destination, which is some set.

The deliverd thing is called x-content, which defines set's property.

 

[master_coda]

You can't really use such a computer model to describe math. Computers have a concept of time. Math doesn't.

Variables in math do not change over time.