September 19th, 2018, 01:17 PM  #1 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Infinity
Set theory uses the ‘axiom of infinity’ which is equivalent to ‘actual infinity exists’. I believe this axiom is wrong  Actual Infinity cannot exist: Numerically:  There is no natural number with the property that repeatedly subtracting one from it never reaches zero  Hence Actual Infinity is not a number. Or  Actual Infinity is larger than any other number  Actual infinity plus one is larger than infinity  Hence there is no number larger than all other numbers Geometrically:  It is impossible to construct a line segment with the property that it is longer than all other line segments Temporally:  Assume infinite time  so anything than can possibly happen will eventually happen  If it happens once it will eventually happen again  So it will eventually happen an infinite number of times  no matter how unlikely it was in the first place  But is this not absurdity? Paradoxes are solved:  Zeno’s paradoxes. It’s impossible to travel any distance as movement requires an infinite number of steps. We can move. Reductio ad absurdum. Space must be discrete  Galileo's paradox is solved:  There are less squares than numbers because not all numbers are squares. Yet each number has a square so the number of numbers and squares must be the same.  He is trying to compare two actually infinite sets, IE comparing two undefined things. A set definition is not complete until all its members are iterated.  Hilbert’s infinite hotel paradox is solved; such a hotel cannot exist. Common sense View: The Actually Infinite exists. Reductio ad absurdum. No it doesn’t. This is surely the only possible conclusion for those with a materialistic view of the world; materialism and the Actually Infinite do not mix. 
September 19th, 2018, 01:39 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,756 Thanks: 696 
You are making too much of it. Mathematicians prefer to use the term becomes infinite, rather than goes to infinity, so that for example, to resolve Zeno's paradox, you don't stop the process after a finite number of steps.

September 19th, 2018, 01:55 PM  #3 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1 
I am not making too much of it: Actual Infinity does not exist implies time has a start which is interesting...

September 19th, 2018, 03:00 PM  #4  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,156 Thanks: 876 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Besides, the Axiom of Infinity is an axiom... Axioms are assumed statements that are useful for the basis of a construction. They are neither provable nor unprovable. Dan  
September 19th, 2018, 03:09 PM  #5 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1 
But you can’t numerically or geometrically construct Actual Infinity. If it can’t be constructed physically or mathematically and does not exist in nature then it does not exist. Got to be the worst axiom in history... 
September 19th, 2018, 04:02 PM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,655 Thanks: 2633 Math Focus: Mainly analysis and algebra  Quote:
However, there are many types of number that you can't subtract 1 from to get zero: "most" of the rationals for example. The axiom of infinity effectively only says (to me) that it is reasonable to talk about "the natural numbers" as a set. It doesn't make any claims about either natural numbers themselves or transfinite numbers.  
September 19th, 2018, 04:12 PM  #7  
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
I also can't remove every molecule of gasoline from my car's gas tank. Thus by your logic I should stop gassing it up because I can't run out. If you want to do math, I suggest you start by spending some time reading about it.  
September 19th, 2018, 04:50 PM  #8  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,156 Thanks: 876 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Infinity: The distance from the origin to where two parallel lines meet. Infinity: The number of ways to circle the globe and get back to your original position. Infinity: The time spent trying to get the pretty girl at the bar to notice you. Infinity: The number of possible ways that two electrons can interact in QED. I'm sure we can come up with more. To infinity! And beyond! Dan  
September 19th, 2018, 05:40 PM  #9  
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Quote:
 But you can’t construct actual infinity either theoretically or approximately (in a finite number of steps) Surely this tells us something; you’re gas tank is real and Actual Infinity is not  
September 19th, 2018, 05:47 PM  #10  
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Quote:
You do not have enough graph paper for your parallel lines example to work in the real world. You lack a truely continuous surface on which to realise your globe example. There are no examples of Actually Infinite in the natural world.  

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