Deeper than primes

[jsfisher]

See what? that a mathematician thinks that he/she avoids circular reasoning just because he/she ignores it?

If I recall correctly, the last time you hyper-ventilated about the empty set, it wasn't over this latest lunacy, circular reasoning. It was because of the word, 'the', instead of 'an', as if the name used to refer to an axiom somehow negated it.

 

[doronshadmi]

By the way:

Emptiness is that has no predecessor. [sic]​

In other words:

Emptiness is a thing that has no thing less than it.​

Must not emptiness be one of the things that isn't less than emptiness? Is this circular reasoning? OMG!!!!

Emptiness is no-thing, and no-thing does not have predecessor.

You are still get things at the level of the empty set, which is an existing empty thing, which its predecessor is Emptiness.

 

[jsfisher]

Emptiness is nothing.

That speaks to its existence, not whether it is a thing.

 

[doronshadmi]

Here is the argument in an even more basic form, with (1) the definition, (2) the axiom, and (3) the conclusion:

(1) {} is the set such that if the set X exists then X is not its member.
(2) {} exists.
(3) Therefore, {} is not a member of {}.

According (1) X is any set, including {}, and {} exists also by (1), otherwise set X can't be "not its member". So right at the level of the definition {} is used to define its own property by not being one of the members, which is a circular reasoning because {} is defined by using {} as a part of its own definition.

Since this is the case then (2) and (3) do not hold.

 

[doronshadmi]

That speaks to its existence, not whether it is a thing.

Emptiness is the absence of existence.

 

[jsfisher]

Emptiness is the absence of existence.

Yes, we agree. Emptiness does not exist. You pseudo-axioms ensure that to be the case. Nonetheless, non-existent things are still things.

 

[zooterkin]

The answer goes like this:

I wish to share with you my reasoning about the concept of Complexity, and how it is related to Ethics and Logic.
...

I know you didn't just compose this off the cuff; where have you quoted it from?

 

[doronshadmi]

Yes, we agree. Emptiness does not exist. You pseudo-axioms ensure that to be the case. Nonetheless, non-existent things are still things.

No "they are" no-thing.

 

[doronshadmi]

EDIT:

jsfisher please reply to:

The axiom of the empty set uses the definition of being a set with no members.

Correct.

"X exists" certainly does provide the "terms of X's existence" because I defined X, the empty set, before I asserted its existence.

Here is the argument in an even more basic form, with (1) the definition, (2) the axiom, and (3) the conclusion:

(1) {} is the set such that if the set X exists then X is not its member.
(2) {} exists.
(3) Therefore, {} is not a member of {}.

According (1) X is any set, including {}, and {} exists also by (1), otherwise set X can't be "not its member". So right at the level of the definition {} is used to define its own property by not being one of the members, which is a circular reasoning because {} is defined by using {} as a part of its own definition.

Since this is the case then (2) and (3) do not hold.

 

[jsfisher]

jsfisher please reply to:

Reply to what? It doesn't pose a question; it just makes faulty statements amongst a strawman.

 

[doronshadmi]

Reply to what? It doesn't pose a question; it just makes faulty statements amongst a strawman.

What strawman?

The axiom of the empty set is based on the definition of the empty set.

I show that this definition (which is step 1) is based on circular reasoning.

As a result the axiom of the empty set (step 2) and the conclusion (step 3) do not hold.

It is all in http://www.internationalskeptics.com/forums/showpost.php?p=6600274&postcount=12729 .

 

[jsfisher]

What strawman?

The axiom of the empty set is based on the definition of the empty set.

No, it isn't.

Perhaps you should actually consider the real axiom:

[latex]$$$\exists x\, \forall y\, \lnot (y \in x)$$$[/latex]​

The first part of the axiom stipulates the existence of something, identified only by the letter x, and the remainder of the formula establishes a property of x.

I show that this definition (which is step 1) is based on circular reasoning.

No, you didn't.

As zooterkin has already observed, you haven't a clue what circular reasoning actually is, but if you did you might also realize it is not possible within first-order predicate calculus. However, since you haven't, I'm sure you won't.

 

[doronshadmi]

A definition gives a name to something.

Utter nonesense.

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

A definition is a passage that explains the meaning of a term (a word, phrase or other set of symbols), or a type of thing.

http://www.merriam-webster.com/dictionary/definition

 

[jsfisher]

Utter nonesense.

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


http://www.merriam-webster.com/dictionary/definition

So, lacking any sort of valid points for discussion, you pursue hyper-parsing and meaningless quibbles?

 

[doronshadmi]

As zooterkin has already observed, you haven't a clue what circular reasoning actually is, but if you did you might also realize it is not possible within first-order predicate calculus. However, since you haven't, I'm sure you won't.

Cut the nonesense.

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

Circular reasoning is a formal logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in one of the premises.

The first part of the axiom stipulates the existence of something, identified only by the letter x, and the remainder of the formula establishes a property of x.

It establishes a property of x by using x as one of the elements that establish a property of x, by not being a member of x, which is a circular reasoning.

 

[doronshadmi]

So, lacking any sort of valid points for discussion, you pursue hyper-parsing and meaningless quibbles?

Why do you care about meaning, after all

A definition gives a name to something.

If we wish to define Mathematics, then by jsfisher "Mathematics" is all we need.

 

[laca]

Doron, you're the Black Hole all Black Knights go when they give up.

 

[jsfisher]

It establishes a property of x by using x as one of the elements that establish a property of x, by not being a member of x, which is a circular reasoning.

More confirmation you haven't a clue what circular reasoning means. Be that as it may, let's stick to the actual axiom, please:

[latex]$$$\exists x\, \forall y\, \lnot (y \in x)$$$[/latex]​

What premise are you claiming is implicitly or explicitly used in the proposition? By the way, few axioms have premises. If they did, we would probably call them theorems instead of axioms.


Oh, and how is that definition of "at the domain of" coming along?

 

[The Man]

Cut the nonesense.

By all means, please, you first.

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

Circular reasoning is a formal logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in one of the premises.

That is exactly why people have been telling you “you haven't a clue what circular reasoning actually is”. That the empty set is empty is not a “proposition to be proved” it is explicitly what defines the empty set as being, well, empty.

It establishes a property of x by using x as one of the elements that establish a property of x, by not being a member of x, which is a circular reasoning.

So tell us Doron does your “empty set” have itself as a member? If so then how is it empty? If not then you are just using “a circular reasoning” that you claim is invalid. You have simply painted yourself into a corner which is evidently why you think you need the contradiction of your “non-locality” to extricate you. Too bad though, as that very contradiction which you claim “non-locally” is not a contradiction makes your “non-locality” “at AND not at the domain of” your locality. So at “the domain of" your locality it still remains a contradiction while “not at the domain of” your locality it simply has no relation to the corner you have painted yourself into and thus can’t extricate you. After all this time you still fail to relize that your assertions and notions just fail when applied to themselves.

Doron if one “establishes a property of x by” not “using x as one of the elements that establish a property of x” then one has simply not established a property of x. x is always an element of the properties of x, specifically being the element that has said properties of x.


This exchange over the past couple of days seems centered around your usual misinterpretations. As noted above That the empty set is empty is not a “proposition to be proved” as you seem to want to consider it.

Also in this assertion before…

It asserts about sets that are not its members. If one of the sets that are not the members of the empty set is the empty set, then a circular reasoning is used, because we cannot determine something about the empty set by using the empty set.

…you seem to think we need to “determine something about the empty set” before one defines the empty set.

Please tell use Doron how one can “determine something about the empty set by” not “using the empty set” or without at least having a definition of the empty set?

 

[HatRack]

According (1) X is any set, including {}, and {} exists also by (1), otherwise set X can't be "not its member".

Incorrect. I did not declare {} to be an existent set in step 1, that happened in step 2. At the point of step 1, we do not know any set to be an existent set. Therefore, you cannot conclude that {} is not a member of {} at this point. That cannot be done until after step 2, when the existence of {} is asserted.

Your incessant claims of this argument being incorrect boils down to your unrelenting misunderstanding of definitions and existence. It's definitely possible to state this axiom without using a definition at all, as I have done in previous posts informally and jsfisher has done formally. And then, the definition can be given after. Although I personally find it to be in better style to give definitions first, I apparently can't do that because you are having a really hard time grasping the difference between a definition and a statement of existence.

The Axiom of the Empty Set
There exists a set X such that if Y is a set then Y is not a member of X.

Let X in the previous step be known as the empty set, denoted by {}.

For the last time: An axiom is a statement that you accept as true without question, there is no need to prove them. Given that you can only use circular reasoning in a proof, there can be no circular reasoning here.