[talanum1 (OP)]

One has:

m∞ = n∞ for any ∞ > m, n ∈ Real numbers, since ∞ is the biggest number. Now we can divide through by infinity to get:

m = n for any n, m.

Doesn't this falsify Mathematics?

[alancalverd]

since ∞ is the biggest number.

No, it is defined as greater than the largest number.

[chiralSPO]

No, you have successfully falsified your own misconception.

There are some holes in mathematics (many involving infinities), but this is not one of them.
https://en.wikipedia.org/wiki/P_versus_NP_problem
https://en.wikipedia.org/wiki/Axiom_of_choice
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
(I don't understand 90% of this, but it makes for interesting reading)

[talanum1 (OP)]

No, it is defined as greater than the largest number.

That makes no sense: there is nothing larger than the largest number.

[Bored chemist]

No, it is defined as greater than the largest number.

That makes no sense: there is nothing larger than the largest number.

That's OK, infinity is an abstract concept.

[evan_au]

Now we can divide through by infinity

No, we can't just divide by infinity, and hope to get sensible answers (any more than we can divide by zero and hope to get a sensible answer).

This  "divide by infinity" problem frequently comes up in mathematics, and the way mathematicians often get around it is by using the idea of "Limits".
- Instead of saying "infinity", say "take a large number x". Now what happens as x approaches infinity?
- This is written as Lim (for Limit) as x→∞.
- The same trick often works for dividing by 0

Now let's take your example (adjusted a bit to read better):

m∞ = n∞ for any n > m; n, m ∈ Real numbers

- instead of infinity, take a large number x.
- Now what happens as x →∞?
- nx > mx for all values of x (since n>m)
- Now we can divide through by x to get:
- n > m
- This applies for any n, m satisfying n > m, and applies for any value of x, even x→∞.

This proves mathematics (if you do it properly).

Limits are extremely useful for many common mathematical operations like integration and differentiation.
- These produce important relations like Kinetic Energy E=½mv² (where the ½ and the squared are related)
- And they involve getting sensible answers from limits that effectively approach 0 x ∞

[alancalverd]

No, it is defined as greater than the largest number.

That makes no sense: there is nothing larger than the largest number.

Let N be the largest real number.
What is N+1? 2N? N^2? N/0.3?  These are all valid operations on any real number. However large you make N, you can always define and manipulate a larger number N'. So ∞ must be larger than the largest number if it has any meaning. 

[Kryptid]

there is nothing larger than the largest number.

There is no largest number.

[charles1948]

(I don't understand 90% of this, but it makes for interesting reading)

I wish posters on this site were allowed to put "mottos" under their posts.  Like they can on other sites.

If they were, your brilliant aphorism: "I don't understand 90% of this, but it makes for interesting reading", would be perfect!

[Halc]

There is no largest number.

https://www.smbc-comics.com/comic/the-largest-number
https://www.smbc-comics.com/comic/the-largest-number-2

I wish posters on this site were allowed to put "mottos" under their posts.  Like they can on other sites.

members->forum profile
which gets you to:
https://www.thenakedscientists.com/forum/index.php?action=profile;area=forumprofile

Put your motto under the signature section, just above the map.

[talanum1 (OP)]

There is no largest number.

I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.

[charles1948]

I

Put your motto under the signature section, just above the map.

Er,  I may be dumb,  but what "map"?

[Halc]

I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.

Non-sequiter

The logic follows only if infinity is defined as the largest number (as you have in the OP), or something greater than such a nonexistent number (As Alan has in the 1st reply).

Perhaps it would be best to use the actual definition instead of all the nonsense in this thread. Infinite is an adjective which translates roughly to 'without bound'.  Infinitiy is something like 'unbounded' and isn't a number with which one can meaningfully do arithmetic. Hence most of the logic put forth is meaningless, and mathematics is safely not falsified.

Er,  I may be dumb,  but what "map"?

There's a map on that page to allow you to indicate where you're from if you so choose. The signature section is just above it.  Scroll down a ways, since the page does not fit on a screen.

[charles1948]

There is no largest number.

I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.

Although I agree with your mathematical idea, could there be an actual physical " largest number"?

For example, suppose you could count the numbers of electrons in the Universe.  That might  turn  out to be, say 99 billion trillion trillion trillion. Or whatever.

Whatever it is, it's the largest number you can physically count.  You've counted every single electron in existence, throughout the entire Universe.  Wouldn't that be the largest "real, physical, number"?

[alancalverd]

When you have counted the N electrons, you can count the protons. I'll save you the trouble - it's N  again, so we now have a number twice as large as your largest physical count, and we still haven't considered photons, which are being produced all the time. Or neutrons. Or quarks.

[evan_au]

count the numbers of electrons in the Universe.  That might  turn  out to be, say 99 billion trillion trillion trillion

The current estimate is around 10⁸⁰ (give or take a factor of maybe a thousand).
- There are more neutrinos.
- There are far more (hypothetical) gravitons
- And physicists are just guessing when it comes to the number of Dark Matter particles.
See: https://en.wikipedia.org/wiki/Eddington_number

[charles1948]

When you have counted the N electrons, you can count the protons. I'll save you the trouble - it's N  again, so we now have a number twice as large as your largest physical count, and we still haven't considered photons, which are being produced all the time. Or neutrons. Or quarks.

Isn't this mere quibbling.  About photons, quarks and stuff.  The practical  point is this:

The Universe is everything that exists.  There can't be anything outside the Universe. So the Universe is a finite thing.

A finite thing can only contain a finite amount of information.  So when the information is written in numbers, there will only be a finite amount of  actual physical numbers to write it in.

[Bored chemist]

There is no largest number.

I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.

Unless, of course, infinity isn't actually a number.
If it isn't, there's no problem.

[charles1948]

There is no largest number.

I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.

Unless, of course, infinity isn't actually a number.
If it isn't, there's no problem.

You have hit the nail on the head.  "Infinity" isn't a number. Nor does it represent a physical location, which we could reach if we travelled far enough. There's no such thing as "infinity".  It doesn't exist.

The only reason we think it might exist, is purely linguistic. "Infinity" is a noun.  And a noun is supposed to represent a "Thing".  Therefore we're lead to think that "infinity" must be a "Thing".

But suppose we replace the noun "Infinity" by a functionally  equivalent adverb: "endlessly".
So instead of saying:  "Space extends to Infinity", we say "Space goes on endlessly".

Then the concept of "Infinity"  as a thing, disappears.

[Halc]

The current estimate is around 10⁸⁰ (give or take a factor of maybe a thousand)

This might be an estimate for the visible universe, a finite volume. The visible universe is not all that exists by at least some definitions of existence.

To say that there is some finite number of Electrons in the universe seems contradictory to most models currently accepted (per cosmological principle) that don't support some kind of edge to it, like the flat Earther's have.

The Universe is everything that exists.  There can't be anything outside the Universe. So the Universe is a finite thing.

Non-sequitur. The universe is quite capable of being unbounded in size and still being everything that exists.