May 9th, 2017, 03:12 PM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Real Numbers and Natural Numbers
A real number is a pair of natural numbers. The natural numbers are expressed as a unique countably infinite sequence of digits (binary, decimal, octagonal, etc.). For example: ......0000347.125000...... There is no conceptual difference between the left and right of the decimal point. Leading and trailing zeroes are customarily omitted. The conventional uses of algebra and analysis apply. "Countable infinity" is used instead of infinity to accommodate current misconceptions. Last edited by skipjack; May 10th, 2017 at 12:06 AM. 
May 9th, 2017, 03:25 PM  #2  
Senior Member Joined: Sep 2015 From: USA Posts: 2,202 Thanks: 1157  Quote:
Take the integers, they are ordered and countable. Map $i \to 2i$ Now put the 2nd integer of the pair into all the new slots. I think this was called the hotel room problem. At any rate, you've now put all pairs of integers into an isomorphism with the integers which are countable. The reals are not countable and thus they cannot be pairs of integers. Last edited by skipjack; May 10th, 2017 at 12:08 AM.  
May 9th, 2017, 03:28 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra 
Blue is green. The world is flat. Nothing that anybody else says matters because I am the sole arbiter of truth. You are wong and I am right.

May 9th, 2017, 03:52 PM  #4  
Senior Member Joined: Apr 2015 From: Planet Earth Posts: 140 Thanks: 25  Quote:
Every natural numbers can be represented in binary notation as a unique, FINITE sequence of bits. Every FINITE sequence of bits also represents a unique natural number. This is call a bijection. By padding with infinite 0's, every natural number can also be represented in binary notation as a unique, INFINITE sequence of bits. But the reverse isn't true; there are infinite sequences of bits (010101... and 11111..., for example) that do not represent natural numbers. This is called an injection. There is a conceptual difference between the left and right of the radix point. You can have infinite trailing ones, or repeated groups that include at least one one. But you cannot have infinite leading ones, or infinite leading groups with at least one one. Neophytes often confuse the concept of "potential infinity" with the infinity Cantor uses. "Potential infinity" is the imaginary goal in a summation like 1/2+1/4+1/8+... . It can never be reached, but sometimes certain properties, like the sum in my example, can be inferred as if it could. Just keep in mind that it is the property, not the number of elements in the sequence, or any ending to the sequence, that is inferred. Cantor uses what is sometimes called "actual," or "realized," infinity. IT IS NOT SIMILAR TO A NATURAL NUMBER IN ANY WAY. It is not the goal at the end of sequences like my example. It doesn't mean that you found an end, it still means you can't. Last edited by skipjack; May 10th, 2017 at 12:13 AM.  
May 9th, 2017, 04:04 PM  #5 
Senior Member Joined: Aug 2012 Posts: 2,099 Thanks: 604  True in the sense that $\mathbb N^{\mathbb N}$ has the same cardinality as $\mathbb R$. So we could encode each real as a pair of naturals. It might be amusing to think of specific ways. Off the top of my head, cute problem. Is there a closedform or understandable or in some way nice or natural bijection between these two sets? Beyond that, starting a new thread to flog the same deceased and cranky equine seems pointless. Last edited by Maschke; May 9th, 2017 at 04:36 PM. 
May 9th, 2017, 07:25 PM  #6 
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations 
Hey Zylo. Somewhere I heard about space filling fractal curves. Those can be used as functions to represent an infinite number of digits. I wasn't really interested in proving anything against Cantor initially (this is probably extremely confusing for some), but now I am interested in piecing together some of these concepts to see if they can challenge our popular notion of infinities. I don't know what I'm talking about; but maybe, or at least I have an entertaining thought. I'm a javascript developer, I'm not coming from the arts, but I'm a big fan. ~~~~~~Crazy Zone~~~~~~ //since we're already there. Let = 0.000....0001 Why I think your intuition about the significance of may actually be for a very valid area of math, just not for "physical" space defined with all the math we know about geometry and topology and other crazier forms of space. I think there is a "conjecture" space. The topology of which you can only estimate with conjectures. Can this space be orthogonal? Yes, not always, probably rarely in post singularity practicality. My ideas branch from crazy, I don't want to debate that, so I'll be concise & deal with the fact that I may not be entertaining you (logic & humor wise) the way I want to. It's discrete and continuous estimations (conjectures of theorems) of units & containers of logic. We need to represent all logic, but we don't have to do so "perfectly" in conjecture space, but it still needs to be represented & physical to the system. could be the size of a certain container of logic. "Conjecture Space" is the space describing the nature of math. Primarily in geometry and topology for excessively high dimensions(for the geometrical concepts "relevant" at those dimensions). We come up with conjectures to estimate higher properties of math sort of like you can estimate a structure with a chaotic fractal; let be the "Conjecture Space" Barnsley Fern. But instead of 3D space, this logical space maps to the nature of Mathematics and numbers. You could physically represent that space as a chain of associated conjecture nodes. Potentially using partial dimensions. is only physical informationally. All this would imply that infinities behave in two distinct ways, one for physical space, one for conjecture space. Again it is this space that, if expressed physically, would attempt to roughly map to the logical systems representing the informational structure of higher concepts in math. The only chance I can disprove our whole notion of infinite sets is that I can produce a function the we can prove generates the full density of all possible combinations of infinitely repeating digits. I think that can be done with the infinitely dense fractal thing I heard about; and I can't find it but it was somewhere on here, I think. I was explaining my challenge trying to do something else and it was brought up, and around the same time I learned you can use a function to define a sequence. Last edited by skipjack; May 10th, 2017 at 12:15 AM. 
May 9th, 2017, 07:30 PM  #7  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,738 Thanks: 609 Math Focus: Yet to find out.  Quote:
It will hopefully provide you with a rough idea on what mathematics is exactly, and what it aims to achieve (generally speaking). Last edited by skipjack; May 10th, 2017 at 12:16 AM.  
May 9th, 2017, 07:54 PM  #8 
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations  We make conjectures about the nature of math; are those by definition philosophical because they aren't proven?
Last edited by skipjack; May 10th, 2017 at 12:19 AM. Reason: accident. 
May 9th, 2017, 08:03 PM  #9 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,738 Thanks: 609 Math Focus: Yet to find out.  Maybe not crazy.. But what's your point? You are crossing philosophical ideas with mathematical ones. I can't read your post and make any sense of what you are trying to say... Maybe that's just me.
Last edited by skipjack; May 10th, 2017 at 12:17 AM. 
May 9th, 2017, 08:31 PM  #10 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond 
The fact remains that there are fundamental disagreements, "trolling" or some sort of chicanery going on here. It's all the usual effluvium. Why I'm wasting my time replying to any of this is beyond the limited confines of my mind ...


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