May 28th, 2017, 09:12 PM  #41 
Member Joined: May 2017 From: USA Posts: 31 Thanks: 0 
Is it possible, that N=P(N) for the domain [∞,∞], as both N and P(N) representing infinite "quantitative parameters" sets, along a line?

May 29th, 2017, 06:17 AM  #42  
Senior Member Joined: Jun 2014 From: USA Posts: 574 Thanks: 44  Quote:
Simply, let's assume there is a function $f$ that is bijective (or surjective) from $\mathbb{N}$ onto $P( \,\mathbb{N}) \,$. We denote it like this: $$\exists f : \mathbb{N} \xrightarrow{\text{1to1, onto}} P( \,\mathbb{N}) \,$$ Now, remember my 'sneaky' set $A$?: $$\text{Let } A = \{ n \in \mathbb{N} : n \notin f( \,n) \, \}$$ In English, the above definition says "let $A$ be the set of natural numbers $n$ such that $n$ is not an element of $f( \,n) \,$." For example, if 1 was an element of $f( \,1) \,$, then 1 would not be an element of $A$ by definition. The same goes for 2, 3, 4, ... (all natural numbers). We know that $A$ is a set of natural numbers (or the empty set, denoted $\emptyset$), therefore, $A$ is an element of $P( \,\mathbb{N}) \,$. Because $A$ is in $P( \,\mathbb{N}) \,$, there must be a natural number $k$ such that $f( \,k) \, = A$ should our assumption that $f$ is bijective be correct. No natural number $k$ can exist, however: $k \in A \implies k \notin f( \,k) \, = A$ (a contradiction: "$k$ is in $A$ implies $k$ is not an element of $f( \,k) \, = A$) $k \notin A \implies k \in f( \,k) \, = A$ (again, a contradiction: "$k$ is not in $A$ implies that $k$ is an element of $f( \,k) \, = A$). Therefore, there does not exist a natural number $k$ such that $f( \,k) \, = A$. Where no natural number $k$ can map to $A$, the function $f$ is not bijective (or, likewise, surjective). Where no bijective or surjective function can exist from $\mathbb{N}$ onto $P( \,\mathbb{N}) \,$, we have proven: $$\mathbb{N} < P( \,\mathbb{N}) \,$$  
May 29th, 2017, 06:30 AM  #43  
Member Joined: May 2017 From: USA Posts: 31 Thanks: 0  Quote:
I have to work on it. I am missing a lot. Can you suggest any literature I can pull up to read and study.  
May 29th, 2017, 07:36 AM  #44  
Senior Member Joined: Jun 2014 From: USA Posts: 574 Thanks: 44  Quote:
http://www.math.toronto.edu/weiss/set_theory.pdf Jech does a wonderful job too, but can be a little more difficult... check out this one that he coauthored: http://www.unalmed.edu.co/~jmramirez...setTheory.pdf  
May 29th, 2017, 07:39 AM  #45  
Member Joined: May 2017 From: USA Posts: 31 Thanks: 0  Quote:
 
May 29th, 2017, 07:57 AM  #46 
Senior Member Joined: Jun 2014 From: USA Posts: 574 Thanks: 44  
May 29th, 2017, 11:09 AM  #47 
Member Joined: May 2017 From: USA Posts: 31 Thanks: 0 
Does anyone has any idea? How its possible to endlessly splitting PRECISED matter into any P(N) size particles that will never make same size matter again? Look like I am trying to fill 10 gallons tank with light. 
May 29th, 2017, 11:12 AM  #48 
Senior Member Joined: Aug 2012 Posts: 2,395 Thanks: 750  
May 29th, 2017, 11:28 AM  #49 
Member Joined: May 2017 From: USA Posts: 31 Thanks: 0  
May 29th, 2017, 02:38 PM  #50 
Member Joined: May 2017 From: USA Posts: 31 Thanks: 0  What about coordinates along of precised line   P ( ab ) ? Where (a)  starting point and ( b )  ending. This is a math! Last edited by Microlab; May 29th, 2017 at 02:43 PM. 

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