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September 19th, 2018, 06:56 PM   #11
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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.
How do you explain the logical fallacy of Hilberts Hotel? Surely such a Hotel could never exist in reality?

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
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September 19th, 2018, 07:51 PM   #12
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How do you explain the logical fallacy of Hilberts Hotel? Surely such a Hotel could never exist in reality?

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
My examples of infinity were half facetious. But you never answered the question: Mathematics is not based on the "real world." It's based in a logic system (I've heard there are several different ones that can do the job) with no bearing on what happens in our Universe. So why are you insisting that if we can't physically construct an infinity that the notion can't be used?

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 non-Abelian groups, Algebra, Differential Geometry, Matrices, and the like. (The Grassmann Algebra is my favorite. Non-commuting complex numbers? Genius!) Physics has wound up using just about all of them.
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September 19th, 2018, 08:05 PM   #13
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How do you explain the logical fallacy of Hilberts Hotel? Surely such a Hotel could never exist in reality?
What logical fallacy? Of course such a hotel couldn't exist in the real world.
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We have logical paradoxes like that because we have a flawed concept in maths - the actually infinite set.
So, without jumping into technicalities of what a set actually is, your position is that it doesn't make any sense to talk about "the natural numbers". That's extremely and needlessly limiting.
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A set is only fully defined if you iterate all members.
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.
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September 19th, 2018, 08:05 PM   #14
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‘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 non-existent and illogical concept.
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September 19th, 2018, 08:12 PM   #15
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So there is serious research going on built on a non-existent and illogical concept.
How can it be an illogical concept? Mathematics is pure logic.

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.
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September 19th, 2018, 08:12 PM   #16
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So, without jumping into technicalities of what a set actually is, your position is that it doesn't make any sense to talk about "the natural numbers". That's extremely and needlessly limiting.
People can still take about the natural numbers, they just can’t form a completed set of all the natural numbers, which is as it should be because no such set exists.

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.
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September 19th, 2018, 08:17 PM   #17
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How can it be an illogical concept? Mathematics is pure logic.
The axiom of infinity is wrong so everything deduced using it is wrong.
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September 19th, 2018, 08:22 PM   #18
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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.
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September 19th, 2018, 08:30 PM   #19
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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.
There is an old fashioned but still relevant definition of an axiom as a self evident truth. The axiom of infinity is however provably false: No matter how many times I add one to a set I never reach an infinite set.

Maths does not do itself any favour by incorporating logic errors in its foundation.
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September 19th, 2018, 08:44 PM   #20
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There is an old fashioned but still relevant definition of an axiom as a self evident truth.
I would argue that it's not relevant at all. It's only relevant if you are trying to construct a model - mathematics left that idea a long time ago.

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The axiom of infinity is however provably false
That's nonsense. If you could prove it to be either true or false, it wouldn't be an axiom.

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Maths does not do itself any favour by incorporating logic errors in its foundation.
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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