You can disagree all you like, but it does not give "the wrong results". — jorndoe

It clearly does give the wrong results. There are more numbers than squares in any finite interval. So we can induce this applies to all intervals. But bijection says the same number. It is basically meaningless to try to compare two 'infinite' sets as neither of them can be fully defined and thus neither of them are fully defined.

Although they're equivalent, I've always rather liked Dedekind's description of infinity. I think it's a lot easier to (for want of a better word) picture infinity that way (as a bijection between a set and a proper subset of itself). Probably because it's easier to show it, it's how my professors often spoke about it so maybe that just stuck with me

Wrt physics, there's some evidence that attempts to make discrete models of spacetime might not be feasible:

https://www.nature.com/articles/nature08574

At the very least, the prospect of giving up Lorentz Invariance seems difficult given this. Not a death knell, loop quantum gravity (speculative though it is) is hardly refuted currently. But the standard model treats spacetime as a continuum so that seems to be the best assumption for now.

It clearly does give the wrong results. There are more numbers than squares in any finite interval. So we can induce this applies to all intervals. — Devans99

Explain how this follows. You're using induction to generalize in a way that seems ridiculous. We can always find new squares to map on to so I don't know where you're getting this idea that "it clearly does give the wrong results".

: ∞ ∉ ℝ
Cantor et al has shown there are meaningful ways of going about this, which is taught today in high schools and universities.

, thanks for the article, looks interesting, putting it on the (way too long to-read) queue. :)

Meaningless ways. Bijection is meaningless. How can there be the same number of naturals as rationals? Each natural is clearly composed of a potentially infinite number of rationals. Bijection is just plain wrong.

You need to ask yourself a very simple question: how many numbers are there? Is there a limit?

probably will regret this, but respectfully disagree. Infinity is not a number, it is a concept.

https://en.m.wikibooks.org/wiki/Calculus/Infinite_Limits/Infinity_is_not_a_number

The two are contradictory. Infinity can’t be a number. So it is not maths. — Devans99

Infinity in mathematics is a mathematical concept. Same as mathematical operations. = is not a number, it is a concept, it is still math.

That's talking about limits at infinity in in calculus, not the actual infinite numbers. Limits in calculus are usually defined as something like,

L is the limit of f(x) as 'x' approaches infinity if f(x) becomes arbitrarily close to L whenever 'x' is sufficiently large.

I could just as easily replace the term "Infinity" here with whatever else I need. The point of the limit is that is grows arbitrarily large or small, it's not a definite number. Transfinite numbers are outside the domain and range of real numbers used in calculus, so it's just not the same thing as the infinity under discussion. All real numbers (as in decimal numbers) are finite.

However, the transfinite cardinals and Ordinals are numbers and are universally acknowledged as infinite numbers. They are larger than any finite number and are not limits. They are sizes and order numbers of infinite sets.

It's the commonly used definition. What definition would you give of infinity? — Devans99

There are many different 'infinities'. The one that arguably corresponds most closely to the folk notion of infinity is

"(1) the cardinality of the set of integers".

An alternative definition that is a little closer to what is in the OP because it uses the concept of 'greater than' is

"(2) the smallest ordinal that is greater than all integers".

These two definitions give different mathematical objects, but they are both reasonably close to the folk notion. The second one is denoted by a lower case omega.

Note that neither definition uses the word 'number'.

However, the transfinite cardinals and Ordinals are numbers and are universally acknowledged as infinite numbers. They are larger than any finite number and are not limits. They are sizes and order numbers of infinite sets. — MindForged

I could have pulled from all kinds of other sites, just grabbed that one.

Tranfinite numbers are not, by definition infinity, They are, by definition < infinity, one thing can not, be less than something and be the same thing as that which it is less than.

I could have pulled from all kinds of other sites, just grabbed that one. — Rank Amateur

None of which will tell you that limits make use of infinite numbers, because they don't. The "infinity" mentioned in limits just means "some arbitrary number greater than any yet reached in the sequence". The point of the limit is to avoid infinity, really.

Tranfinite numbers are not, by definition infinity, They are, by definition < infinity, one thing can not, be less than something and be the same thing as that which it is less than. — Rank Amateur

There is no number called infinity. Infinity is a type of number. Transfinite numbers are infinite numbers. The term 'transfinite' is just an old term, you can call them the infinite Cardinals. They are by definition infinite, they can be placed into a bijection with a proper subset of themselves. An infinite set with one less member is not the same set. It lacks the member removed from it. What doesn't change is the cardinality (size) of the set.

ok , sure. Thanks

Perhaps we can qualify your assertion.

Infinity isn't an exact/definite number. Rather it is a number that is indefinite.

Also, if I'm correct, numbers began as a one-to-one correspondence concept. 3 cows - 3 pebbles, 4 sheep - 4 pebbles, and so on.

This concept (1-to-1 correspondence) can be used in infinity, only to find that some infinites are bigger than others.

I'm not a mathematician but I believe infinity can be used in simple math e.g. take the equation y=1÷x

As x approaches infinity, y tends to zero.

I think this is called limit in math. So, you see, infinity is used in math but probably in a very narrow sense.

Math with infinity is difficult I believe. Cantor went mad I hear.

∞ * 10 = ∞
∞ / 4 = ∞
etc...

So an axiom of infinity is effectively 'when you change it, it does not change'. What sort of reasonable system of the world would adopt such an axiom? Where is the evidence for these magic objects that can be changed and remain unchanged?

Every number is inherently finite.

So an axiom of infinity is effectively 'when you change it, it does not change'. What sort of reasonable system of the world would adopt such an axiom? Where is the evidence for these magic objects that can be changed and remain unchanged? — Devans99

"∞" isn't a number. Aleph-null is an infinite number. And again, infinity does change. You just cannot add or remove *finite* amounts of it to change it because it's the definition of finite that it changes by finite modifications to such a value. If you take the Power Set of Aleph-null, it increases and becomes Aleph-One, the size of the continuum, a larger infinity. But that's because I'm adding by infinite amounts, that's what let's it change. And it gets different when you get to the Ordinals too. Adding finite to the infinite Ordinals does change them.

As for what I assume you're asking for (real world examples) we can take space or time. As far as current models go, there's no fundamental unit of space or time, they are continuums. So if I have some slice of space I can always zoom in by some arbitrary amount (even an infinite amount) and there will always be more space or time.

"∞" isn't a number. — MindForged

that's what I said !!!

The infinity symbol isn't a number, but there are infinite numbers. My point was that what one is doing with limits (where you see the infinity symbol) isn't actually about the set theoretic understanding of infinity, it's just an unbounded sequence as opposed to a definite value. But the cardinals and ordinals are definite values and infinite.

are sets of any size numbers ???

The size of sets are numbers called Cardinal numbers. Like some set A with the members {1,2,3} has a size of 3. The size of the set of natural numbers (whole numbers 0 and greater) has a size of Aleph-null, the first infinite number. If it's a set, it has a size.

So that is:

- It's a number
AND
- It's greater than any number

The two are contradictory. — Devans99

I think you've created the contradiction. From the definition you've given, the summation should be:
- It's a number
AND
- It's greater than any other number.

I understand,

but if Aleph-null +1 = Aleph-null
and if Aleph-null + the size of any finite set = Aleph-null
is Aleph-null a number in any way other than as a label ?

not playing get ya - really don't know what operations can or can't be meaningfully done with Aleph-null

The reason the alephs can add any finite number without increasing size is because if how size is defined. As I said, a set A with the members {1,2,3} has a cardinality of 3. Let's take a larger set, an finite set. Take N, the set of Natural Numbers. {0,1,2,3...}. Then let's take E, the set if Even Numbers. Which set is larger, N or E? Well we can do this by trying to make a bijection between the sets, that is, by mapping each element in one set with exactly one element in the other set. In finite sets, it's obvious when a set is larger or smaller because one set will run out of members to map together. Like take set A from above and compare it to set B with the elements {0,1,2} (set A on the left, set B in the right):

0 - 0
1 - 1
2 - 2
3 - (no more elements to map together)

So we know set A is larger, it has more members. But what happens when you try this with the Natural Numbers and the Even Numbers? Intuitively it seems like the natural numbers should be twice as big since the Even Numbers are missing half the numbers (the Odds) which exist in the set of Naturals. But that's not what happens:

0 - 0
1 - 2
2 - 4
3 - 6
4 - 8
Etc.

No matter how far you get into the set of Naturals, you'll always have an Even number to match up a number from set N. This means the Naturals are the same size as the Evens despite lacking the Odd numbers. And that's why adding 1 (or any other finite number) to an Aleph won't change the cardinality. You'll still be able to map elements from the original set to the set + 1 meaning they are the same size. It's still a number, but the way size works means that if the elements of one set can be completely mapped together with another, they are the same size even if your intuition tells you it shouldn't be that way. The logic doesn't show any contradictions arising.

that seems like a whole lot of word that say Aleph - null is a label for a concept to me - What am I missing ??

it's not a concept, it's a number. Remember how I said the size of the set {1,2,3} was the cardinal number 3? The size of the natural numbers is the infinite cardinal number Aleph-null. It's not a concept, it's a number.

ok. seems we are in a do loop. No worries