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 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: 6,806 Thanks: 716 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
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 Originally Posted by Devans99 I am not making too much of it: Actual Infinity does not exist implies time has a start which is interesting...
Just because we can't physically construct something does not mean it can't exist. Mathematics constructions are not physical so why should we base the existence of infinity (which, yes, is not a number) on a physical basis? We can't construct the exact value of $\displaystyle \pi$ by measuring the circumference and diameter of a circle. So do we say that $\displaystyle \pi$ does not exist?

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
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 Originally Posted by Devans99 - 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 - It is impossible to construct a line segment with the property that it is longer than all other line segments
You are confusing all manner of different "infinities". In fact, mathematics doesn't define "infinity". Instead there are several different types of infinite objects. Infinite sets are one such object - although they are actually sets having transfinite cardinality and there are various transfinite numbers. These are not natural numbers, and so no, you can't subtract 1 from them any number of times to get zero.

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
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 Originally Posted by Devans99 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...
Almost every single thing in your OP is completely wrong, often in multiple ways. However, this comment is the most egregious.

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
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 Originally Posted by Devans99 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...
Who says Mathematics has to be taken as the image of the real world?

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
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 Originally Posted by SDK Almost every single thing in your OP is completely wrong, often in multiple ways. However, this comment is the most egregious. 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.
- You can remove every molecule theoretically and approximately.
- 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
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 Originally Posted by topsquark Who says Mathematics has to be taken as the image of the real world? 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
But mathematics is used to model the real world and so is Actual Infinity.

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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