September 19th, 2018, 06:56 PM  #11  
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Quote:
We have logical paradoxes like that because we have a flawed concept in maths  the actually infinite set. A set is only fully defined if you iterate all members. Specifying selection criteria like ‘must be yellow’ is not sufficient to define a set. All the paradoxes of infinity stem from operations on undefined sets  
September 19th, 2018, 07:51 PM  #12  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,914 Thanks: 774 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
I'm a Physicist and I can still see a point to constructions that don't belong to the real world. Dan Addendum: There was something of a competition between Physicists and Mathematicians post 1850 or so where Mathematicians were creating Mathematics that would have no bearing on Physics. Things like nonAbelian groups, Algebra, Differential Geometry, Matrices, and the like. (The Grassmann Algebra is my favorite. Noncommuting complex numbers? Genius!) Physics has wound up using just about all of them.  
September 19th, 2018, 08:05 PM  #13  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
Quote:
That is exactly how the infinite set of the naturals numbers is defined. As topsquark keeps telling you, the mathematical world has only a tenuous link to the real world at best. It started off as a simplification of the real world and has been heavily abstracted from there. The concept of infinite objects exists because its useful in mathematics, not because they exist in the real world. Actually, it would probably exist even if it weren't useful, but in that case it might not be part of mainstream mathematics.  
September 19th, 2018, 08:05 PM  #14 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1 
‘Who says Mathematics has to be taken as the image of the real world?’ It is a true point you make but the fact is mathematics is our best tool for modelling the real world. Infinity is useful as you say but only as an approximation to the very large/small. The problem is we have models from cosmology that use the actually infinite literally rather than as an approximation. So some folks literally claim time and space to be actually infinite. So there is serious research going on built on a nonexistent and illogical concept. 
September 19th, 2018, 08:12 PM  #15  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
And, yes. You do research based on models that give good results. But they are models and therefore don't exist as reality. Actually, pretty much all research is done on the places where the models don't work  and people modify them so that they still don't work, but slightly better. If the models weren't simplified, you'd need knowledge of every particle in the universe to make use of them. So we have simplified models that are guaranteed to be wrong, but at least they have a manageable number of inputs. On the other hand, mathematics is never wrong. It follows as night follows day  only more reliably.  
September 19th, 2018, 08:12 PM  #16  
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Quote:
What you call the set of natural numbers is not a set, it’s a selection criteria for populating a set. A set is a distinct collection of objects. Spiritual ideas like Actual Infinity should not be involved. Then we would not have so many paradoxes.  
September 19th, 2018, 08:17 PM  #17 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  
September 19th, 2018, 08:22 PM  #18 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra 
No, what I call the set of natural numbers is a set by the axioms of the mathematical system I work under. You are welcome to choose a different set of axioms to work under, but then anything you say about your system is not definitive for my system. Unfortunately for you, my system is the one that most people work with, because it's useful. There's nothing spiritual about infinite sets. I think infinities are great and useful and I'm going to keep involving them in my system, whatever you say. Paradoxes are even better. I especially like Bertrand's Probability Paradox, which only includes the infinite in the fairly standard idea that there are an infinite number of points on a circle. Last edited by v8archie; September 19th, 2018 at 08:40 PM. 
September 19th, 2018, 08:30 PM  #19  
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Quote:
Maths does not do itself any favour by incorporating logic errors in its foundation.  
September 19th, 2018, 08:44 PM  #20  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
That's nonsense. If you could prove it to be either true or false, it wouldn't be an axiom. There is no logical error is stating the assumptions that you are going to incorporate into your system. Maths does itself significant favours by incorporating infinite objects because they are very useful.  

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