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...

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

Indeed, infinity is different from any finite quantity. So what? That does not make it illogical or contradictory, just different — aletheist

Maths should not run contrary to logic. Where in logic do we find objects that behave like X+1=X. Things that we change that do not change? Where in reality? Nowhere. So I think it's downright wrong to incorporate such illogical concepts into an important field like maths.

Why? The fact of the matter is that it does not, so we can either throw up our hands (like you do) or find and develop meaningful alternatives (like mathematicians have). — aletheist

All the core mathematical quantities (integers, reals, complex, vector, matrix, etc...) obey the arithmetic operators or common sense extensions of them. X+1=X never occurs in maths, apart from when it comes to infinity.

Maths has made up magic 'numbers' that you cannot assign to any variable and you cannot use with the arithmetic operators. What are normal people meant to make of that? Just that maths does not follow logic in this area.

How many numbers are there? How many square numbers are there? — aletheist

Thats a fundamentally unanswerable question; we can never realise an infinite set so we can never answer. Maths tries to answer and gives patently the wrong answer.

See, the only thing contradictory in this entire discussion is your childish insistence on repeatedly applying the axioms of finite mathematics to infinity. Your "induction" here is straightforwardly false. — aletheist

You are again in violent disagreement with common sense, the rest of maths obeys the arithmetic operators (or appropriate variations of them), infinity should too. I challenge you to come up with another mathematical 'number' that you can add a non-zero amount to without changing?

You have yet to identify a single contradiction when the relevant terms are defined consistently, and a paradox is simply an opportunity to think more carefully. — aletheist

OK Galileo's paradox:

https://en.wikipedia.org/wiki/Galileo%27s_paradox

There are more numbers than there are square numbers yet each number has a square. We know by induction that there are more numbers than square numbers in all finite intervals so we can induce this implies to infinity as a whole. Cantor's one-to-one correspondence procedure produces the non-sensical answer that there are the same number of numbers as there are squares. That is easily disproved by examining any finite interval.

So the current 'resolution' to the paradox gives a nonsense result. You can't compare infinite sets because they are not fully defined is probably closer to the actual resolution to the paradox.

BTW A paradox is usually indicative of an underlying logic error.

What is not logical is the claim that "1+∞ = ∞ implies 1 = 0"; it reveals an utter lack of understanding about the mathematics of infinity, which at this point is clearly willful — aletheist

Your statement above runs completely opposite of any logic. How can a quantity not change when you add another positive quantity to it? Thats impossible so infinity is not a quantity.

I know plenty about the maths of infinity thank you. I have spent much time reading up on it. It's shot through with contradictions and paradoxes.

But there are no infinite numbers. There is no greatest number (because X+1>X), so there can be no number larger than any finite number.

I am not using colloquial definitions; I'm doing my best to be logical about it (unlike Cantor).

The author of the definition that you quoted would presumably reply: It is an infinite number. — aletheist

An infinite number is a number bigger than any number... same contradiction.

Once again, you are smuggling in an additional premise--in this case, that something must be assignable or countable in order to qualify as a number — aletheist

If I can't assign it to a variable or count with it? No other number behalves like that.

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

Logical concepts only I would argue should be in maths. Triangles are logical. 1+∞ = ∞ implies 1 = 0 is not logical.

But infinity cant't be bigger than any number because then it would not be a number. That's the mother of all contradictions.

So all the stuff about transfinite numbers and one-to-one correspondence is built on a nonsense definition of a nonsense concept.

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 — aletheist

If a number is neither assignable or countable; then what sort of a number is it? It is not a number.

I've corrected the typos in the OP now. Sorry.

t's a defective definition, thus not good for anything including posts about it. — tim wood

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

Current science can't tell us about an alleged start to time. — S

We know that time slows in the presence of gravity. The intense gravity of the big bang would have caused time to run very slowly, maybe even stopping at the singularity.

people will readily admit that the real between 0 and 1 are infinite despite — MindForged

Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED.

Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity).

When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua.

If its not a quantity, which it is not by definition, we should not assign it to physical quantities like, time, size, mass etc...

I make that case here:

https://thephilosophyforum.com/discussion/comment/241257

But we would agree that the Trinity is in conflict with reason so should not be an act of faith?

Faith is 100% belief in something without proof.

Maybe it's better to say 'I induce the axioms of arithmetic are true from volumes of existing evidence' rather than 'I have faith in the axioms of arithmetic'? Or does that transfer faith to induction? I'm not sure thats quite right; its more I induce induction is reliable that I have faith in induction.

you even chose not to believe the mathematical definition of a point - because it didn't fit your position — Rank Amateur

It's rather that it does not fit maths (leads to a divide by zero error) rather than it fits my position. So my argument stems from a believe in logic and the axioms of arithmetic. So I have faith in the axioms of arithmetic.

Faith - can not be in conflict with fact or reason — Rank Amateur

Faith can be in conflict with reason: people have had and do have faith in all sorts of different Gods. Some of that faith must be misplaced.

I don't think it is intended to, and logic has nothing at all to do with it. At least in Catholicism - it is referred to as the mystery of the Trinity - it is outside reason, it is a matter of faith — Rank Amateur

I am a faithless heathen :grin: God quite probably exists sums up how I feel.

Oh dear. I've noticed a whole in my argument. It allows you to get something from nothing timelessly.

I would like to withdraw this argument for further consideration...

I don’t think it hangs together logically:

‘The Christian doctrine of the Trinity (Latin: Trinitas, lit. 'triad', from Latin: trinus "threefold")[1] holds that God is one God, but three coeternal consubstantial persons[2] or hypostases[3]—the Father, the Son (Jesus Christ), and the Holy Spirit—as "one God in three Divine Persons". The three Persons are distinct, yet are one "substance, essence or nature”’

https://en.wikipedia.org/wiki/Trinity

So they are distinct, means they were distinct, means they have always been distinct. So how can they be the same substance?

Or where they once all one? But that is not consistent with them being coeternal.

So coeternal and of one substance are at odds.

Potential infinity I will leave to one side.

You can view actual infinity as the set of natural numbers { 1, 2, 3, 4, ... }. The '...' indicates it is not completely defined in a logical sense. If it's not completely defined, it's not defined at all and it does not exist logically or mathematically.

"The number of events in an infinite regress is greater than any finite number," but that is not self-contradictory at all; in fact, it is trivially true. — aletheist

But there are only finite numbers:

1. If actual infinity is a number, there must be a number larger than any given number.
2. But that’s contradictory.
3. Can’t be a number AND larger than any number.
4. So actual infinity is not a number
5. Invention of magic numbers runs contrary to the spirit of science.

Or

Actual infinity, if it existed, would be a quantity greater than all other quantities, but:
There is no quantity X such that X > all other quantities because X +1 > X

"A magician pulling a rabbit from a hat without using a trick of some sort.
— Devans99
That is actually impossible, but not logically impossible. — aletheist

Then your axiom system contains magic. I'm trying to stay on the scientific side by avoiding magic.

Again, that is indeed actually impossible; but it is not logically impossible, because we can imagine it. — aletheist

1. The number of events in an infinite regress is > any number
2. Thats a contradiction (can’t be both a number and > any number)
3. Making up magic numbers is not allowed (can break any theory if magic is admissible)
4. There must be a first event.

So just because we can imagine an mathematical infinite regress; it does exist mathematically.

I would say that that's optimistic. I seem to have thrown a spanner in the works. But I wish you the best of luck in that endeavour. It has certainly been interesting so far to examine what you've been coming up and to subject it to scrutiny. — S

Not sure I entirely agree but thank you for your time also... very thought provoking.

Please give me an example of something that you can imagine, yet is logically (not just actually) impossible. — aletheist

A magician pulling a rabbit from a hat without using a trick of some sort.

Please tell me exactly how you can imagine squaring the circle. — aletheist

Well the square is similar to a circle but an octagon is more similar but I can construct an octagon so maybe I can get there. So superficially it seems possible but logically it is in fact impossible. So many things we can conceive of can never be in reality. An infinite regress in time for example as demonstrated earlier cannot exist in reality.

A real circle is truly continuous, and its mere possibility is sufficient for its reality. — aletheist

But just because we can imagine something it does not mean it is possible. I can imagine squaring the circle all I like but its still impossible.

it logically follows that your model is impossible. — S

I would say it's a work in progress. All models of this nature have problems. This model has less problems than most IMO.

I disagree with the conventional definition. It leads to a divide by zero error.

a point is a specific place in space, it has no size — Rank Amateur

So it would not exist, how can something have no size and exist?. A point is purely in our minds. How many things do you know that exist and have length zero?

Well I can understand not wanting to depart from the Presentist view point; it is after all our gut feeling for the way the world works. But presentism is logically impossible by the reasonable axiom 'events are caused by events'. And it causes paradoxes, some of which I pointed out earlier. It is fair to say I can't make my mind up between the two.

And it would regress in that way infinitely. — S

But on a fixed set of finite co-ordinates. So it could depending on how time works perhaps be a potential rather than actual infinity. It would also be more logically consistent that a linear infinite regress (which fails mathematically).

just for some mathematical clarity - between any two points- whether it is on a circle or a line are an infinite number of points. — Rank Amateur

I would dispute that. I mathematical point is defined to have length=0. How many points in an interval length 1? 1 / 0 = UNDEFINED.

Any sensible definition of a point requires a non zero length... things with zero length just don't exist.

What we can draw on paper is a representation of a circle, and we can mark as many points on it as we like--up to any finite number. However, a real circle--note, not an actual circle, since there is no such thing--does not consist of any number of discrete points, finite or infinite; it is a continuous curve, infinitely divisible into smaller and smaller arcs — aletheist

A continuous circle is only possible in the mind. In reality circles are made of molecules of material which are discrete all the way down.

Sigh. I give up. You are very set in your ways.

hierarchy of infinities — MindForged

Infinity is defined to be bigger than anything else. That means there can only be one infinity by definition.

I think you have not addressed my argument... you have just introduced magic numbers that obey different, mad rules to all other numbers.

Aristotle believed actual infinities were impossible but they are not — MindForged

They are so!

Actual infinity is not a number:

1. If actual infinity is a number, there must be a number larger than any given number.
2. But that’s contradictory.
3. Can’t be a number AND larger than any number.
4. So actual infinity is not a number
5. Invention of magic numbers runs contrary to the spirit of science.