Deeper than primes

[doronshadmi]

No, it cannot be. The power set of a set is the set of all that set's subsets. That is the definition that everyone who does mathematics agrees upon, and it will never change. If you wish to specify some other type of set, then give it a name that's not already taken.

Again, instead of open your mind to generalization, you follow after particular agreements.

, and it will never change

Again a live demonstration of mind.

 

[HatRack]

The cardinality of a set is not a set.

Sorting through your gibberish is always worth it in the end, because I can always extract a blatantly wrong statement out of you to prove just how clueless you really are.

 

[HatRack]

Again, instead of open your mind to generalization, you follow after particular agreements.


Again a live demonstration of mind.

Again, a live demonstration of your inability to comprehend the simple linguistic tool known as a definition. When sane people come up with something new, they assign a new definition to it, rather than changing the meaning of old things. What a strange world it would be if we arbitrarily changed definitions around whenever we felt like it.

 

[doronshadmi]

Again, a live demonstration of your inability to comprehend the simple linguistic tool known as a definition. When sane people come up with something new, they assign a new definition to it, rather than changing the meaning of old things. What a strange world it would be if we arbitrarily changed definitions around whenever we felt like it.

HatRack, something like a development of notions (where fundamental changes like mutations of already existing terms, are allowed) is beyond your dogmatic comprehension.

 

[doronshadmi]

Sorting through your gibberish is always worth it in the end, because I can always extract a blatantly wrong statement out of you to prove just how clueless you really are.

|{0}|={0} by HatRack

 

[HatRack]

|{0}|={0} by HatRack

Nope, that's not what I said. You said "The cardinality of a set is not A set". Cardinal numbers, if you bother to learn how they are formally defined, are in fact sets themselves. More misrepresentation on your part to cover your own blunders.

 

[HatRack]

HatRack, something like a development of notions (where fundamental changes like mutations of already existing terms, are allowed) is beyond your dogmatic comprehension.

It's not about development of anything, it's about communication. You can develop whatever notions you like w/o having to "mutate" any terms. The only purpose that mutation of the meaning behind terms serves is to confuse others. Of course, someone who has been too many times may have a hard time understanding that.

 

[doronshadmi]

The only purpose that mutation of the meaning behind terms serves is to confuse others.

A typical notion of a dogmatic mind.

 

[doronshadmi]

Cardinal numbers, if you bother to learn how they are formally defined, are in fact sets themselves.

Wrong, cardinal numbers are the size of given sets.

 

[HatRack]

Wrong, cardinal numbers are the size of given sets.

And the size of a set is itself defined in terms of sets. Is that so hard to understand?

From http://en.wikipedia.org/wiki/Cardinal_number#Formal_definition:

...the cardinality of a set X is the least ordinal α such that there is a bijection between X and α.

Hmm, okay. So cardinality is an ordinal. Let's see how ordinals are defined. From http://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals:

Rather than defining an ordinal as an equivalence class of well-ordered sets, we will define it as a particular well-ordered set which (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.

The cardinality of a set is an ordinal, and an ordinal is a set. Hence, cardinality is a set. Oh look at that, Doron was completely wrong again.

 

[jsfisher]

You have missed this:

No, I didn't. You said something grossly false, and now you are trying to cover it up by writing about something else. You said sets share members with their power set. Sets and their power sets fall under no such restriction.

Actually the power set of some set is 2^(the number of the distinct objects of that set)

This is also an untrue statement. But you have no concept of what you said, do you Doron? Maybe it is not just a reading comprehension issue you continually manifest, but a more general cognition failure.

You statement quite clearly states that the power set of, for example, the set {A,B,C} is 2^|{A,B,C}| = 8. That is clearly wrong.

I swear, Doron, you have the talent to mess up a pig's sty.

 

[zooterkin]

No, that's not correct either. Circular reasoning is when you assume a proposition A to prove A. Every set uses itself as a factor in its definition. For example, the set {1} is defined as the set such that 1 is its member and every other set X (including {1} itself) is not a member. This type of "self-reference" is inescapable in defining sets.

You have a simple mistake, 1 is not a set, but can be a member of a set.

The mistake is yours. HatRack's subsequent explanation notwithstanding, in the quoted piece '1' is not used to refer to a set, only to a member of a set.

 

[The Man]

Emptiness denials the existence of -1, 1 and 0.

No one asked you what your “Emptiness denials the existence of”.


Again, as you seem to keep missing it, the question was and still is…

So now your “magnitude of existence” can have a negative value?

Would you care to actually address the question that was asked as opposed to whatever questions in you head that you continue to address instead?

ETA:
So as your “Emptiness denials the existence of” a denial of the existence of the value 1 (represented as -1) it then asserts the existence of the value 1. Why how unsurprisingly and characteristically self-contradictory of you

 

[epix]

Originally Posted by HatRack
No, that's not correct. <0,1>^X is not only countable, but finite (since X is a finite number ...
Wrong, X can be finite or infinite number.

Wrong, X can be finite or infinite number.

I think HatRack referred to the whole expression <0,1>^X which seems to be the power set of some sort. That's why he used the words "countable and "finite." It's very rare to see set members being enclosed in symbols "greater than" and "lesser than" even by folks who saw once math pass by in a '75 Chevy. That's why Egbert's mind went and the same is awaiting Haldegard, if he doesn't depart your head soon enough.

You can call Cantor's Theorem the "Duh Theorem." See, Cantor lived in the time when expressions, such as n = ∞, were populating the pages of math books. So he had to resort to the unusual kind of descriptive rigor to make all the bad habits and misconceptions regarding infinity go away. Hence the power set.

Sure, you can construct a power set of a power set and so on. But what purpose would it serve? To drive Haldegard crazy, like crazy^X? LOL.

 

[doronshadmi]

And the size of a set is itself defined in terms of sets. Is that so hard to understand?

From http://en.wikipedia.org/wiki/Cardinal_number#Formal_definition:



Hmm, okay. So cardinality is an ordinal. Let's see how ordinals are defined. From http://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals:



The cardinality of a set is an ordinal, and an ordinal is a set. Hence, cardinality is a set. Oh look at that, Doron was completely wrong again.

The simple fact is this:

|{}| = 0

|{{}}| = |{0}| = 1

|{{},{{}}}| = |{0,1}| = 2

|{{},{{}},{{},{{}}}}| = |{0,1,2}| = 3

...

but

{} ≠ 0

{{}} ≠ {0} ≠ 1

{{},{{}}} ≠ {0,1} ≠ 2

{{},{{}},{{},{{}}}} ≠ {0,1,2} ≠ 3

...

In other words HatRack, you have no case.

 

[doronshadmi]

No one asked you what your “Emptiness denials the existence of”.

But I am talking about Emptiness, where negative existence (like -1) is not Emptiness, which is something that you can't comprehend by your relative-only view of this subject.

So as your “Emptiness denials the existence of” a denial of the existence of the value 1 (represented as -1) it then asserts the existence of the value 1. Why how unsurprisingly and characteristically self-contradictory of you

How relative-only view of this subject you have. Emptiness (which is total denial of existence) denials the existence of 1, -1 and 0, and your relative-only view of this subject simply can't comprehend it, exactly as it can't comprehend the totality of fullness, which is beyond collections AND also appears as the non-local property that bridges between localities of a given collection.

 

[HatRack]

The simple fact is this:

|{}| = 0

|{{}}| = |{0}| = 1

|{{},{{}}}| = |{0,1}| = 2

|{{},{{}},{{},{{}}}}| = |{0,1,2}| = 3

...

but

{} ≠ 0

{{}} ≠ {0} ≠ 1

{{},{{}}} ≠ {0,1} ≠ 2

{{},{{}},{{},{{}}}} ≠ {0,1,2} ≠ 3

...

In other words HatRack, you have no case.

So, in the top half of your post, {} = 0, but in the bottom half, {} ≠ 0. What a strange world Doronetics is. The meaning and properties of objects shifts around from sentence to sentence. It's almost as if the author of Doronetics hasn't the slightest idea what he's doing.

 

[doronshadmi]

The mistake is yours. HatRack's subsequent explanation notwithstanding, in the quoted piece '1' is not used to refer to a set, only to a member of a set.

You are wrong, look:

The cardinality of a set is an ordinal, and an ordinal is a set. Hence, cardinality is a set.

 

[doronshadmi]

So, in the top half of your post, {} = 0, but in the bottom half, {} ≠ 0. What a strange world Doronetics is. The meaning and properties of objects shifts around from sentence to sentence. It's almost as if the author of Doronetics hasn't the slightest idea what he's doing.

At the top of the half post, there is no comparison between the members of sets (there is only a comparison between cardinality's value).

EDIT: At the bottom of the half post, there is a comparison between the members of sets and also between sets and cardinals (which are not sets).

 

[jsfisher]

So, in the top half of your post, {} = 0, but in the bottom half, {} ≠ 0. What a strange world Doronetics is. The meaning and properties of objects shifts around from sentence to sentence. It's almost as if the author of Doronetics hasn't the slightest idea what he's doing.

It is also fun to compare and contrast the past with the present. Consider this not so recent example:

1={{}}

2={{},{{}}}

3={{},{{}},{{},{{}}}}

...

in comparison to this:

{} ≠ 0

{{}} ≠ {0} ≠ 1

{{},{{}}} ≠ {0,1} ≠ 2

{{},{{}},{{},{{}}}} ≠ {0,1,2} ≠ 3