Cont: Deeper than primes - Continuation 2

[doronshadmi]

Moreover, given (for example) multiset [1,1,1,...] it is infinite only if the term "the next 1" is its inherent property, as also demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=13121718&postcount=3436 .

So no matter what kind of collection is given, it is infinite only if the term "the next 1" is its inherent property and as a result its accurate amount of objects is undefined (and so is the case with an infinite collection with urelements, for example, ($, #, *, ^, @, 6, :, 234.5678432, ...) ).

More about it is already given in http://www.internationalskeptics.com/forums/showpost.php?p=13028612&postcount=3430.

 

[jsfisher]

|N| is defined (by belief) as fixed amount by traditional mathematicians (the accurate cardinality of any inductive set) where N is such set.

Well, there's your problem right there. No, it's not.

 

[jsfisher]

jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

Sure. Here is the Axiom of Infinity "in terms of Mathematics":

See those first two symbols? Right there.

 

[doronshadmi]

Well, there's your problem right there. No, it's not.

Do you actually say that |N| is not an accurate (transfinite) number > any given N member?

 

[doronshadmi]

Sure. Here is the Axiom of Infinity "in terms of Mathematics":

[qimg]https://latex.codecogs.com/gif.latex?\exists&space;N&space;(&space;\emptyset&space;\in&space;N&space;\land&space;\forall&space;x&space;\in&space;N&space;((x&space;\cup&space;\{x\})&space;\in&space;N))[/qimg]​

See those first two symbols? Right there.

By declare that "there exits N" you can't determine that |N| is an accurate value, exactly because the order of N members does not change the cardinality of N and since N's biggest member does not exist, |N| accurate value is undefined.

So no,

[qimg]https://latex.codecogs.com/gif.latex?\exists&space;N&space;(&space;\emptyset&space;\in&space;N&space;\land&space;\forall&space;x&space;\in&space;N&space;((x&space;\cup&space;\{x\})&space;\in&space;N))[/qimg]​

does not establish an infinite set as a complete whole, even if you call it the Axiom of Infinity "in terms of Mathematics".

In other words, such mathematics is no more than a belief, which is agreed among set of persons called traditional mathematicians.

 

[jsfisher]

Do you actually say that |N| is not an accurate (transfinite) number > any given N member?

No. That is not what I said.

 

[doronshadmi]

No. That is not what I said.

So what you have said by "Well, there's your problem right there. No, it's not."

 

[jsfisher]

By declare that "there exits N" you can't determine that |N| is an accurate value...

That would depend on what you mean by "accurate value". You use the term a lot, but never define it.

For that matter, you need to be clear what you mean by "cardinality". No, wait, better still, I will provide you a definition of cardinality for the purposes of this discussion.

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)​

Note that this definition requires only the introduction of mappings into the set theory.

So no,

[qimg]https://latex.codecogs.com/gif.latex?\exists&space;N&space;(&space;\emptyset&space;\in&space;N&space;\land&space;\forall&space;x&space;\in&space;N&space;((x&space;\cup&space;\{x\})&space;\in&space;N))[/qimg]​

does not establish an infinite set as a complete whole.

I claimed the axiom established a set with two properties, nothing more. It can be shown that it is an infinite set, but that is not a claim of the axiom itself. As for it being "a complete whole", well, you'd have to tell us what you mean by that phrase and why it is relevant.

 

[jsfisher]

So what you have said by "Well, there's your problem right there. No, it's not."

Are you missing the fact my statement was in direct reference to your statement? You made a false statement. I apologize if that wasn't clear.

 

[doronshadmi]

You made a false statement.

Please support your claim, by not skipping on http://www.internationalskeptics.com/forums/newreply.php?do=newreply&p=13123365.

 

[jsfisher]

Please support your claim....

It was your statement that was false. It is your obligation to support it if you believe it to be true.

 

[doronshadmi]

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)​

Note that this definition requires only the introduction of mappings into the set theory.

Note that |A| <= |B| has a meaning only if |A| and |B| are defined as strict (or accurate) values, so "The meanings for strict equality and strict inequality of cardinalities" can't be defined if |A| and |B| strict (or accurate) values are undefined.

Note that the introduction of mappings into the set theory, says nothing about the sets' size unless their cardinalities are strictly defined, which is impossible if, for example, N biggest member does not exist, as already shown in http://www.internationalskeptics.com/forums/showpost.php?p=13028612&postcount=3430.

I claimed the axiom established a set with two properties, nothing more.

Please provide those two properties.

 

[doronshadmi]

It was your statement that was false.

Since you have said that, than please support what you have said.

I already have supported my claim about your beliefs in http://www.internationalskeptics.com/forums/showpost.php?p=13028612&postcount=3430.

 

[jsfisher]

Note that |A| <= |B| has a meaning only if |A| and |B| are defined as strict (or accurate) values

You don't understand relative measures, do you? I do not need "accurate values" (which, by the way, is a term you'd need to define), just a way to evaluate the relationship.

Please provide those two properties.

Remember this?

Everything inside the outermost parentheses.

 

[jsfisher]

I already have supported my claim about your beliefs in http://www.internationalskeptics.com/forums/showpost.php?p=13028612&postcount=3430.

Here's your statement under considerations:

|N| is defined (by belief) as fixed amount by traditional mathematicians (the accurate cardinality of any inductive set) where N is such set.

The statement is false, and you have nowhere supported it.

 

[Little 10 Toes]

|N| is defined (by belief) as fixed amount by traditional mathematicians (the accurate cardinality of any inductive set) where N is such set.

What you say actually supports my argument that |N| accurate value is undefined, unlike the agreed belief among traditional mathematicians (for example, jsfisher's belief, which can't deal (yet) with my question to him about the considered subject).

I've snipped the portion that is not directed to me. I would answer, but jsfisher has already addresses it. I would like to ask, why haven't you answered the request to define "accurate" and/ or "strict" ?

 

[jsfisher]

I've snipped the portion that is not directed to me. I would answer, but jsfisher has already addresses it. I would like to ask, why haven't you answered the request to define "accurate" and/ or "strict" ?

It is probably time to make a list of words and phrases with meanings unknown to the rest of us.

• accurate value
• accurate amount
• strict value
• cardinality
• accurate cardinality
• infinite set
• complete whole
• complete set

Not that I expect Doronshadmi to address any of them, I suspect his personal definition for cardinality would provide the most insight into his confusion.

 

[Little 10 Toes]

I also like how he makes a multiset infinite, and then immediately claims it isn't.

 

[doronshadmi]

Cardinality is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B. (The meanings for strict equality and strict inequality of cardinalities follow directly.)​

Note that this definition requires only the introduction of mappings into the set theory.

In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set A = { 2 , 4 , 6 } contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, allowing to distinguish several stages of infinity, and to perform arithmetic on them. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers.[1] The cardinality of a set is also called its size, when no confusion with other notions of size[2] is possible.

https://en.wikipedia.org/wiki/Cardinality


Let's examine your statement:

You did not show that "The meanings for strict equality and strict inequality of cardinalities follow directly" from the definition of Cardinality, which "is a relative measure of "size" of sets where |A| <= |B| if and only if there exists an injection from A to B" since you did define strict or relative measure
So please define them before we continue.

 

[doronshadmi]

but jsfisher has already addresses it.

jsfisher did not address it.

He used "strict" and "relative measure" and also claimed about "strict equality and strict inequality of cardinalities that follow directly form "a relative measure of "size" of sets", without defining "strict" or "a relative measure of "size" of sets".

Moreover, by looking at wikipadia, "In mathematics, the cardinality of a set is a measure of the "number of elements" of the set." where the word "relative" is not used.