Cont: Deeper than primes - Continuation 1/3*9

Now here is my question to you:

Please complete the following definition:

|N| > any given n iff ...
 
doronshadmi said:
EDITED:

The bus stands at some station (no matter what station) means D is finite.

The bus does not stand at any station means D is non-finite.

That's all we need in order to distinguish between the finite and the non-finite.

No decision of involved in order to distinguish between stands or does not stand.
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You have provided no means to determine what the bus will do for a given D.

You have therefore provided not means to distinguish finite from non-finite.
 
jsfisher said:
You have provided no means to determine what the bus will do for a given D.

You have therefore provided not means to distinguish finite from non-finite.
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D is a placeholder for sets, therefore D is not any particular set, so I do not have to provide means to determine what the bus will do for a given D.

If the bus stands (|D| is determined by some n) then D is finite.

If the bus does not stand (|D| is not determined by some n) then D is non-finite.
 
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jsfisher said:
Why would I do that? That has no bearing on anything I have offered.
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Because by traditional mathematics |N| > any given n

Since you represent here the reasoning of traditional mathematicians about the discussed subject (where by traditional mathematics |N| > any given n), you are able to complete the following definition:

|N| > any given n iff ...
 
doronshadmi said:
D is a placeholder for sets, therefore D is not any particular set, so I do not have to provide means to determine what the bus will do for a given D.

If the bus stands (|D| is determined by some n) then D is finite.

If the bus does not stand (|D| is not determined by some n) then D is non-finite.
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You have in effect told us you define finiteness in terms of cardinality. Therefore, you need to define cardinality. Part of that will be how to determine whether |D| is some member of N, or not.
 
jsfisher said:
You have in effect told us you define finiteness in terms of cardinality. Therefore, you need to define cardinality. Part of that will be how to determine whether |D| is some member of N, or not.
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Let D be a placeholder for any set.

Let N be the set of natural numbers.

Definition 1: Cardinality is the "size" of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to" means that Cardinality is not necessarily some particular N member, yet it is at the domain of N, where N does not have the greater successor.
 
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doronshadmi said:
Let D be a placeholder for any set.

Let N be the set of natural numbers.

Definition 1: Cardinality is the "size" of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to" means that Cardinality is not necessarily some particular N member, yet it is at the domain of N, where N does not have the greater successor.
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That didn't cut it as a definition that last time you posted it. Reposting it now doesn't improve on that.

The definition body itself is "grammatical" nonsense. You have presented "Cardinality is the 'size' of D" as a true/false proposition. So, in some cases cardinality is the "size" of D, and in other cases it is not. Great, I suppose, but what does cardinality actually mean?

As for "'related to' means that Cardinality is not necessarily some particular N member", you haven't described with any specificity what 'related to' means.

From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of some set, S?

From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of a set, {a,b,c,d}, to be 4 and not be 21?
 
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Please note doronshadmi, I asked the same question three times. Yes, there are ways to determine combinations. Your answer of, "then |D| is at least 1" is nonsense. Why? Because that was not what I asked for. I will ask it again very clearly.

In my example, how do you determine the cardinality of set D?
 
jsfisher said:
You have presented "Cardinality is the 'size' of D" as a true/false proposition. So, in some cases cardinality is the "size" of D, and in other cases it is not.
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Let N be the set of natural numbers.

Cardinality is the 'size' of D, whether it is determined by a particular N member (in this case |D|=particular n and D is called finite set) or it is determined according the fact that N does not have the greatest successor (in this case |D| is not any particular n, yet we stay at N domain and D is called non-finite set).

So being at N domain this is the true state of |D|, whether it is determined by some particular n, or not.

Code: 
p="Cardinality is the 'size' of D" 

q="|D| is at N domain"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T


Definition 1: Cardinality is the 'size' of D (notated as |D|) iff it is related to the members of N, where 0 is a member of N.

"related to the members of N" simply means that |D| is at N domain, whether it is determined by some particular n, or not (in this case the fact that N does not have the greatest successor is taken).

jsfisher said:
From your definitions, and only your definitions, with no extra hand-waving or undefined filler, how would one determine the cardinality of a set, {a,b,c,d}, to be 4 and not be 21?
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The definitions in http://www.internationalskeptics.com/forums/showpost.php?p=13164777&postcount=98 have nothing to do with defining specific values of cardinality, so your question simply show that you basically does not follow after what they are actually define.
 
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jsfisher said:
So, you agree, your "definition" of cardinality is insufficient to determine your concept of cardinality.
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No jsfisher, my definition of cardinality is sufficient to determine the concept of cardinality, which is not limited to your restriction to define some particular cardinality by using the definition.

It is called generalization, which enable to use it in order to distinguish between finite and non-finite sets, without the need for special axioms (which in case of ZF axiom of infinity and the axiom schema of restricted comprehension, they do not provide a non-finite set that its cardinality > any given N member, but such a set is arbitrarily determined. Also the very notion of axiom schema as a non-finite set of axioms, is arbitrarily determined).
 
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The best that the ZF axiom of infinity and the axiom schema of restricted comprehension (with more ZF axioms) establish, is the inductive set known by the name "von Neumann ordinals" that does not have the greatest successor, so the term all in the following equate is arbitrarily plugged in:

jsfisher said:
The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...).

A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element.

So, a set X is a non-finite set if and only if |N| <= |X|, where N is the minimal set established by the Axiom of Infinity.
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Now let's look at this:
jsfisher said:
Since |N| <= |N|, done!
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This is a concrete example of an arbitrary expression.

----------

One may ask if |N| > the cardinality of any given N member (where in our case, the members of N are von Neumann ordinals, as constructed by based on the two properties of ZF axiom of infinity).

Since the cardinality of any given member of N is finite and the term all is arbitrarily plugged in (bye bye to the fact that N does not have the greatest successor, which actually prevents the term all) |N| > cardinality of any given N member, is inevitable by ZF.

Please look at your reply in http://www.internationalskeptics.com/forums/showpost.php?p=13170287&postcount=123 and the following quote:
jsfisher said:
If you want to label the members of the von Neumann ordinals with natural numbers, go right ahead, but that's your doing, not a requirement of mine.
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The cardinality of any given N member (where N is actually the set of von Neumann ordinals) can't be but some natural number, so "not a requirement of mine" has no basis whatsoever.

Therefore please complete the following definition (by using ZF reasoning):

|N| > any given n iff ...
 
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doronshadmi said:
No jsfisher, my definition of cardinality is sufficient to determine the concept of cardinality, which is not limited to your restriction to define some particular cardinality by using the definition.
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Then you haven't defined cardinality at all. You have only told us about some of the characteristics you think it must have.
 
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