Ah, we finally get to the heart of the matter--it is not that the definition of infinity is contradictory, as the thread title asserts, but that you do not like including something in mathematics that does not follow the same rules as finite quantities.

Again to the op - infinity is not a number- it is a concept — Rank Amateur

I agree. I'd also add:

- Its not a quantity either
- Because it's not a quantity, it is not valid for use as a value of various physical quantities like the size or age of the universe
- It's an illogical concept

Ah, we finally get to the heart of the matter--it is not that the definition of infinity is contradictory, as the thread title asserts, but that you do not like including something in mathematics that does not follow the same rules as finite quantities — aletheist

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'.

Thats such as basic axiom, taken as reality by most people...

so how would one answer questions like how many real numbers are there between say 1 and 2, how many lines can pass through a point, without a concept of infinity?

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'. Thats such as basic axiom, taken as reality by most people... — Devans99

Because most people only ever deal with and think about finite quantities, which is the domain in which that axiom applies.

The width of a number is 0, so the number of real numbers between 1 and 2 is 1 / 0 = UNDEFINED.

An actual infinity of numbers may appear to exist in our minds but that's not the case mathematically. Actual infinity is bigger than any number so not a number, so concepts build on it like continua, infinite regress, eternity are not valid mathematically.

How many numbers there are in an interval? or lines can be drawn through a point? are examples of potential infinity which I don't have a problem with.

Because most people only ever deal with and think about finite quantities, which is the domain in which that axiom applies. — aletheist

A more basic form of the axiom: 'if I change something, it changes' could be adopted. Infinity runs contrary to this axiom (which applies to the domain of 'stuff').

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'. Thats such as basic axiom, taken as reality by most people... — Devans99

The problem is that relativity does not follow the axiom: "no matter how fast something is traveling, mass, length, and time are constant." That is such a basic axiom, taken as a reality by most people ...

Infinity is not a number of any kind it is a label for a concept. — Rank Amateur

It's a number and a concept. Of note is that the concept is best understood by working out the properties of the numbers. Namely, the transfinite Cardinals and Ordinals, which are infinite numbers.

The problem is that relativity does not follow the axiom: "no matter how fast something is traveling, mass, length, and time are constant." That is such a basic axiom, taken as a reality by most people ... — aletheist

But we have good evidence that the speed of light is constant and the rest follows. We have no evidence for 'stuff that we change that does not change' and it makes no logical sense anyway.

The problem is infinity does not follow the axiom: 'if I add (non-zero) to something, it changes'. — Devans99

It does change. The set has an element it did not have before. But the cardinality does not change. Derive the contradiction or just admit that you cannot. This is not a logic error, you're just using a dumb definition of infinity and wondering why people aren't using it. It should be obvious why mathematics does not use your definition, while you pretend to have found an area where maths is 'illogical'.

But we have good evidence that the speed of light is constant and the rest follows. We have no evidence for 'stuff that we change that does not change' and it makes no logical sense anyway. — Devans99

That's false though. We have both mathematical evidence and current scientific evidence suggests that space and time can be arbitrarily subdivided without reaching a discrete unit.

It does change. The set has an element it did not have before. But the cardinality does not change. — MindForged

OK so thats equivalent to saying 'I have this set to which I can add to and the size does not change'. Thats nonsense - anything you add (non-zero) to the size changes. That should be an indisputable axiom of mathematics or at least derivable from simpler axioms.

Folks, its clear that on this topic that Devans99 is just plain willfully and repetitively stupid. Dunno why. But he clearly is not interested in any information or understanding. He is ignorant, which in itself is the general condition of mankind, so no sin there. But he insists on the conclusions of his ignorance, while in possession of plenty of information and references that would inform him of his errors, and that is the very definitition of stupid.

So, Devans, recant this stupidity; study and learn the errors of your thinking, and then rejoin us. I say rejoin because while I respect folks who post here, I have little patience with stupidity (as above defined), including my own, and its stupid displays of itself. Go, then, away and return when you're not so much stupider than I am, at least.

You are very closed minded and rude.

Not a fraction as closed minded, rude, and crude as persistent ignorance, that most of us call stupidity. Everything you want is just a few keystrokes away, why do you not access it and learn?

I have spent years looking at infinity, I am entitled to my opinions.

OK so thats equivalent to saying 'I have this set to which I can add to and the size does not change'. Thats nonsense - anything you add (non-zero) to the size changes. That should be an indisputable axiom of mathematics or at least derivable from simpler axioms. — Devans99

And? That's not a contradiction. Size is understood by the theory of cardinalities, not the intuitive idea you're working from. It's not a contradiction at all. Anything FINITE that you add to has it's size change. In fact, that's the very definition of something which is finite: it changes size when additions or subtractions are made to it. And so too with infinity, it's very definition entails it does not change when some finite amount is added to it.

I am entitled to my opinions. — Devans99

Indeed you are! Nothing wrong with opinions in their proper place and time. Nor pigs nor parlors, but there is with pigs in parlors. Your opinion is a pig that has slipped its pen and is playing in the parlor.

To the point: what you have adduced here is half-baked nonsense. Others in this and other threads seemingly with more patience and good will than sense, ultimately, have addressed the problems with this nonsense of yours and given you good direction for corrective study and understanding. The which you apparently disregard in favor of your own ignorance. This is stupid, and the person who insists on being stupid just is stupid, at least to that extent. I'm not name calling, here. I am calling out what is.

You want argument? Show me how my characterization of your thinking as stupid is wrong!

You are arguing contrary to a basic axiom I believe in:

'when you change something, it is changed'.

Infinity in mathematics does not follow this axiom.

So that is:
- It's a number
AND
- It's greater than any number
The two are contradictory.
— Devans99

Read the definition that you quoted more carefully. It does not state, "A number greater than any number," which would indeed be contradictory. Instead, it states, "A number greater than any assignable quantity or countable number," which is not contradictory at all.

Infinity can’t be a number. So it is not maths.
— Devans99

A triangle cannot be a number. Does that mean geometry is not mathematics? — aletheist

/thread

'when you change something, it is changed'. — Devans99

This is characteristic of many of your posts. You lasso perfectly good thought, hog-tie and rebrand it - change it - make your argument, and then try to pretend we're not adrift on your private understanding and reworking of well-established thinking, but rather that your conclusions follow from the main line of the thought you have hi-jacked. And this is more than a twice-told story - we've all - many of us - been here several times. My invitation you is, you can do better, so do better! So when I ask what you're about, it's a serious question.

That is just waffle. Address the axiom:

'when you change something, it is changed'

Do you buy it or not? If you buy it, you don't buy infinity.

/thread — S

If only.

'when you change something, it is changed'.
Infinity in mathematics does not follow this axiom. — Devans99

and already explain how this is false. The axiom that you are really following is, "When you add a finite quantity to another finite quantity, you get a different finite quantity." That axiom straightforwardly does not apply to infinity, which does not entail that infinity is somehow contradictory--only that it is different from a finite quantity, and must accordingly be treated differently than a finite quantity.

'when you change something, it is changed' — Devans99

How do you change a number, e.g. 13?

'when you change something, it is changed'

Do you buy it or not? If you buy it, you don't buy infinity. — Devans99

Fallacy of both false and too few alternatives - and too little understanding. But as you said, this is something you believe in. You're entitled. I merely point out that there is no accounting for some people's beliefs. But mere belief in itself does not produce anything - except perhaps further belief and the implications of the beliefs. I think you're stuck in this, and you need to break out. Where you are isn't worth being. You're entitled to question, but you've been answered.

Infinity requires 'when you change something it is never changed' as an axiom. If you apply that axiom to real life you find it inductively in error.

Sorry, your game isn't worth any candle at all.

@Devans99, FYI, some details about the real numbers, ℝ, that we use for modeling the world:

Zero is the additive identity:

• 0 ∈ ℝ
• ∀ x ∈ ℝ [ x + 0 = 0 + x = x ]

ℝ is closed under addition and subtraction (for example):

• ∀ x, y ∈ ℝ [ x ± y ∈ ℝ ]
• all "distances" are also reals

ℝ an Archimedean set:

• ∞ ∉ ℝ
• ε ∉ ℝ
• infinites (∞) and infinitesimals (ε) are not reals,
  do not involve them in addition and subtraction (for example)

Colloquially, ∞ could be thought of as a quantity that's not a (real) number.

Two more concise definitions of infinite:

• Dedekind:
  
  • |S| = ∞ ⇔ ∃ ƒ (bijection): S → T ⊂ S
    a set is infinite if and only if there is a bijection between the set and a proper subset of itself
• Tarski:
  
  • S is a set
  • P(S) is the set of all subsets of S including ∅ and S itself
    the power set, Weierstraß, Cantor
  • F ⊆ P(S) is a family of subsets of S
  • m ∈ F is a minimal element of F ⇔ ∀ x ∈ F [ x ⊄ m ]
    no smaller subset
  • M(F) = { m ∈ F | x ∈ F ⇒ x ⊄ m }
    the set of minimal elements
  • S is finite ⇔ ∀ F ⊆ P(S) [ F ≠ ∅ ⇒ M(F) ≠ ∅ ]
    a set is finite if and only if every non-empty family of its subsets has a minimal element, Tarski
  • S is infinite ⇔ S is not finite

They can be shown identical.

We understand plenty about infinites (cf the continuum hypothesis). Yes, ℝ is an infinite set, and any numbers therein are separated by another such (real) number. There's a lot more to say, including that ℝ being an infinite set is not contradictory. In fact, had it been, some rather significant problems would have come about. Archaic (Aristotelian) verbiage like "potential" and "actual" aren't of any use here. The standard mathematical modeling we use today is the best we know of as yet.

Let me just quote Eric Schechter:

Prior to Cantor's time, ∞ was
• mainly a metaphor used by theologians
• not a precisely understood mathematical concept
• a source of paradoxes, disagreement, and confusion — Eric Schechter

And that first bullet there is indeed an outdated tradition. Fortunately we know more these days. Cantor showed that there are infinite different infinites, no less; in a concise context, ∞ is ambiguous.

On the physics side we have general relativity, the Friedmann-Lemaître-Robertson-Walker model, all that, and the evidence, all of which seems consistent per se. Well, we have no established unification with quantum mechanics, that is, we already know that there are shortcomings, limits of applicability, things we don't know.

You'll have to understand at least some of this stuff to comment.

a set is infinite if and only if there is a bijection between the set and a proper subset of itself — jorndoe

I do not agree with the bijection procedure; it gives the wrong results; see Galileo's paradox.

Colloquially, ∞ could be thought of as a quantity that's not a (real) number. — jorndoe

No, it can't be thought of as a quantity; its defined as greater than any quantity therefore its is not a quantity.

I do not agree with the bijection procedure; it gives the wrong results; see Galileo's paradox. — Devans99

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

Galileo concluded that the ideas of less, equal, and greater apply to (what we would now call) finite sets, but not to infinite sets. In the nineteenth century Cantor found a framework in which this restriction is not necessary; it is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others. — https://en.wikipedia.org/wiki/Galileo%27s_paradox

No, it can't be thought of as a quantity; its defined as greater than any quantity therefore its is not a quantity. — Devans99

You switched to a different definition from a (less technical) dictionary that's quite informal. The colloquial definition above is somewhat better, and the two more concise definitions better still. If you just wish to show some sort of inconsistency with informal dictionary definitions, then have at it. Has no bearing on the mathematics. Sorry, there's more to it than what you suggest.