[alancalverd]

There can't be anything outside the Universe. So the Universe is a finite thing.

sadly, an irrelevant non sequitur.

[evan_au]

Is Mathematics Falsified?

It depends on your assumptions.

Unlike physics, mathematics can operate in a purely hypothetical environment. You may say:
(1) 10+3 = 13
(2) 10+3 = 1

And both statements may be true, if you know that:
(1) applies to the natural numbers
(2) applies to a clock face (which is a modification of modulo arithmetic)

Within a particular framework of assumptions, mathematicians strive for:
- Consistency: Eg you can't prove 1=2
- Completeness: To be able to identify whether any particular statement is true or false.

Unfortunately, in 1931, the mathematician Gödel proved mathematically that you can't prove everything mathematically.
- Any (sufficiently complex) mathematical framework cannot be both consistent and complete.
See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

In practice, when mathematicians get "stuck" in proving a statement, they often appeal to a proof in a "larger" context
- So to prove something within a particular mathematical framework, they sometimes reach outside that mathematical framework for help...

[talanum1 (OP)]

Infinite is an adjective which translates roughly to 'without bound'.

Then infinity means ±∞.

[alancalverd]

Depends which way you are going. The term "semi-infinite" turns up in a lot of applied maths, physics, and electrical engineering, meaning "starting from zero and fading into the far distance"

[wolfekeeper]

Dividing an infinite number by infinity gives a result that is undefined; it could be any number at all, so you haven't accounted for the infinite number of other possible solutions to both side of your equation, and that is why you have an impossible result.

[talanum1 (OP)]

It is:

lim_(x -> ∞) x/x = 1

where the same number goes into both x's, and on both sides of the other equation (in the opening post) the same number is substituted. I saw a video where division by infinity is allowed:

√8/∞ = 0.

[wolfekeeper]

You have to be really careful taking things to an infinite limit. There's an infinite number of ways to 'go to infinity':

lim_(x -> ∞) nx/x = n

[talanum1 (OP)]

There is a one to one pairing in the infinities of the left side and right side. Whatever infinity gets substituted in LS also goes into RS.

[chiralSPO]

You can only do a 1-to-1 pairing of "countable" infinities. If you venture into "uncountable" infinities, that doesn't work anymore.

For instance, it is just as reasonable to set up a pairing of all the numbers on the continuous Real number line with all of Real numbers on the continuous interval between any two Real numbers. (ie there are just as many Real numbers between 1 and 2 as there are between 1 and 200,000,000) go figure!

[talanum1 (OP)]

You can only do a 1-to-1 pairing of "countable" infinities.

Specify the four infinities as countable then (and correlated). The range of values that can substitute can be specified as countable, can't it?

[evan_au]

You have to be really careful taking things to an infinite limit. There's an infinite number of ways to 'go to infinity': lim_(x -> ∞) nx/x = n

Yes, you do need to take care with limits.
- But in this example, you don't need much care.
- This equality is true for all values of x≠0, whether x is any real value (or any complex value), any fraction, any integer, any prime number or any Mersenne prime number.
- It applies whether there is a countable or uncountable number of x values tested.

[talanum1 (OP)]

Mathematics is not falsified if we assume ∞ is a foggy range of values so that ∞/∞ != 1.

But then for a countable ∞ we have ∞ != ∞.

[chiralSPO]

There is no shortage of apparent contradictions and paradoxes when dealing with infinity. Many times these can be resolved by making sure that the question and context are properly defined.

For instance, with no context, ∞ – ∞/2 is undefinable. But if we give some context, the answer can be defined in a way that is both meaningful and unambiguously provable to be true.

Imagine a farmer planting a row of seeds, and never stopping. There is a crow that follows behind him and eats the seeds at precisely half the rate that the farmer is planting them.

At any given finite value of time, there will be a non-zero number of seeds in the ground. But if we want to know what the "final" result is, we run into problems without knowing more.

If the crow eats every other seed (let's say it eats the odd ones: 1, 3, 5, 7, 9....), then there will be infinitely many seeds in the ground after infinite time passes, and we can say exactly where they will all be (holes 2, 4, 6, 8, 10....). For any hole, if you know whether it is even or odd, you will know whether there is a seed in it or not.

On the other hand, if the crow eats each seed in sequence (at half the rate that the farmer plants them), then there will be no seeds that can be found in the ground. How can I prove this? Pick a hole, any hole. Did the farmer put a seed in it? yes, at time = t. Did the crow eat the seed? yes, at time = 2t. So there are no holes that you can find a seed in.

Is this useful? It can be if you ask the right questions.

[evan_au]

if we assume ∞ is a foggy range of values so that ∞/∞ != 1

I agree that "a foggy range of values" might be a helpful way of thinking about it.

But if it is a foggy range of values, then "∞/∞ != 1" is trying to define a specific value in this foggy range that ∞ cannot be. That doesn't sound very foggy to me.

In fact, if we take the previous example of limits: lim_(x -> ∞) nx/x = n
- Now, if you set n=1, you have the example lim_(x -> ∞) x/x = 1
- This implies, in this particular limit, ∞/∞ = 1, which violates your assertion

So, "a foggy range of values" may be a helpfully foggy concept, but don't try to put precise values on what it isn't!

[charles1948]

if we assume ∞ is a foggy range of values so that ∞/∞ != 1

I agree that "a foggy range of values" might be a helpful way of thinking about it.

But if it is a foggy range of values, then "∞/∞ != 1" is trying to define a specific value in this foggy range that ∞ cannot be. That doesn't sound very foggy to me.

In fact, if we take the previous example of limits: lim_(x -> ∞) nx/x = n
- Now, if you set n=1, you have the example lim_(x -> ∞) x/x = 1
- This implies, in this particular limit, ∞/∞ = 1, which violates your assertion

So, "a foggy range of values" may be a helpfully foggy concept, but don't try to put precise values on what it isn't!

I wonder whether this "foggy range of values" arises from the use of base-10 decimal arithmetic.  This arithmetic allows a degree of "fogginess".

For example,  the difference  between our numerals  "7", "8" and "9" is quite small.  Their values are close.

Whereas in base-2 binary arithmetic, the difference the numerals "0" and "1" is precise and clear.

Does that mean that we should abandon decimals, and start doing maths in Binary Notation?

[chiralSPO]

There is no number that can be expressed in base 2, which also cannot be expressed in base 10. There are some inherent mathematical advantages to using base e, but really the only special thing about base 2 is that it is easiest to compute using base two and two-state 0/1 switches.

But there is no difference between any base when discussing infinity.

[evan_au]

Does that mean that we should abandon decimals, and start doing maths in Binary Notation?

Most computers already do. Simple calculators may still be using decimal arithmetic.

I agree that teaching binary multiplication tables is much easier than teaching decimal, and lets you focus on the essentials:
0x0 = 0
0x1 = 0
1x0 = 0
1x1 = 1

But there is no need to abandon Decimal arithmetic because it is "inaccurate" or "foggy" in some sense.
- There is an exact 1:1 mapping from every decimal integer to its binary counterpart.

the difference  between our numerals  "7", "8" and "9" is quite small.  Their values are close.

What you are describing here is that many quantities in this world span a great range of values, and it is easier to make sense of them as ratios. So the ratio 2/1 is much bigger than 9/8.

A number of our senses work in this way; the loudness of sounds is represented on a logarithmic scale called "decibels".
See: https://en.wikipedia.org/wiki/Decibel

Many other systems in the real world work this way, even accounting records for a company.This has been used to detect when a fraudster has tried to hide their trail.
See: https://en.wikipedia.org/wiki/Benford%27s_law

Oops! Overlap with chiralSPO...

[alancalverd]

Mathematics is not falsified if we assume ∞ is a foggy range of values so that ∞/∞ != 1.

(a) You haven't demonstrated mathematics to be falsified under any conditions

(b) You shouldn't use ∞ to denote a number x such that x/x! = 1 because ∞ is already reserved  and understood for other purposes.

(c) a range of values, foggy or otherwise, implies that you could write down any real number somewhere in that range. But since ∞ is defined as greater than any number you can think of, x ≠ ∞

[evan_au]

if we assume ∞ is a foggy range of values

The idea of Limits is to provide more information, pinning down a "foggy range of values" to a more precise range of values, and sometimes, to a single value.

Sometimes the Limit is still infinite, but in many useful cases, the limit is finite, and a fixed value that you can use.
- So Limits provide more information than you get by just assuming that "∞ is a foggy range of values"