## The Definition of Infinity is Contradictory

• 2.7k
“Mathematics
A number greater than any assignable quantity or countable number (symbol ∞)”

https://en.oxforddictionaries.com/definition/infinity

So that is:

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

The two are contradictory. Infinity can’t be a number. So it is not maths.
• 202
Devans you keep making similar threads about infinity. Here is the deal, infinity is used differently in mathematics than how its used in everyday conversation.
Read this and you will see what I mean.
• 7.3k
- It's a number
AND
- It's greater than any number

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

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

It's the commonly used definition. What definition would you give of infinity?
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I've corrected the typos in the OP now. Sorry.
• 1.5k
So that is:
- It's a number
AND
- It's greater than any number
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.
A triangle cannot be a number. Does that mean geometry is not mathematics?
• 243
It's a defective definition, thus not good for anything including posts about it.

I'm not sure the definition is all that bad. Mathematical infinity is certainly not a haircut or a goose, it is discovered while investigating the properties of numbers, series tend towards it, so it must be a number.

It is also greater than any "countable number" i.e. any number that could in principle be counted to.

If I were to improve the definition, I would prefer infinity to be defined as the "the least of the set of numbers" greater than any countable number.
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It's the commonly used definition. What definition would you give of infinity?

That's the issue, isn't it. But apparently what you have repeated here is not quite the "commonly used definition." So you have to go back to that. And it's a rule of mine that when something that should make sense to me doesn't, I think in terms of some fault in my understanding - which with luck is remediable. Or if it isn't I have work to do.
• 2.7k
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

If a number is neither assignable or countable; then what sort of a number is it? It is not a number.
• 763
Why are you quoting Oxford about a mathematics concept? Aside from the fact that the Oxford definition isn't really contradictory, the mathematical definition of infinity has no contradictions. A set has an infinite cardinality if the set can be placed into a one to one correspondence with a proper subset of itself. No contradictions at all. Cardinal numbers are numbers, so infinite Cardinals are numbers as well.
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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.
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So all the stuff about transfinite numbers and one-to-one correspondence is built on a nonsense definition of a nonsense concept.

What are you on about? Do you imagine that your making any sort of point here? Other than the one about yourself, that is.
• 243
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.

The infinity, to which you refer, is not bigger than any number. It is:

greater than any assignable quantity or countable number

• 1.5k
If a number is neither assignable or countable; then what sort of a number is it?
The author of the definition that you quoted would presumably reply: It is an infinite number.

It is not a number.
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.

Infinity can’t be a number. So it is not maths.
A triangle cannot be a number. Does that mean geometry is not mathematics?
• 2.7k
The author of the definition that you quoted would presumably reply: It is an infinite number.

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

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?

Logical concepts only I would argue should be in maths. Triangles are logical. 1+∞ = ∞ implies 1 = 0 is not logical.
• 763
But infinity cant't be bigger than any number because then it would not be a number. That's the mother of all contradictions.

It's not bigger than any number. Stop stop stop using colloquial definitions when talking about a formal discipline. Infinite numbers are larger than any finite number, there is no infinite number larger than all infinite numbers. You don't know what you're talking about.
• 1.5k
An infinite number is a number bigger than any number... same contradiction.
An infinite number is a number bigger than any assignable quantity or countable number ... no contradiction.

Logical concepts only I would argue should be in maths.
Your original statement implied that only numbers belong in mathematics, so this is an improvement.

1+∞ = ∞ implies 1 = 0 is not logical.
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.
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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).
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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

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.
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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.

Math is hard.
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How can a quantity not change when you add another positive quantity to it? Thats impossible so infinity is not a quantity.
Once again, you are smuggling in an additional premise--in this case, that something must be a quantity in order to qualify as a number.

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.
Unfortunately, reading and knowledge do not necessarily translate to understanding. 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.

Math is hard.
So is logic, apparently; among other things, it requires being explicit about one's premises and consistent in one's use of terminology and definitions.
• 763
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.

Infinity is not defined as the largest number. Stop saying that. That is not the mathematical definition of infinity. You keep repeating yourself and ignoring the corrections. If you repeat some crap about infinity being defined as "the largest number" I'm done. Just asserting there are no infinite numbers is a stupid argument.

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

You literally keep repeating that infinity is defined as "the largest number" when no one else here has said so and it's not the mathematical definition of it. Cantor was a celebrated mathematicians (eventually) and unlike you showed his actual proofs and gave rigorous definitions and worked out the consequences to show no contradictions arose. You're doing the mathematical equivalent of shitposting.
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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.

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.
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There are more numbers than there are square numbers yet each number has a square.
How many numbers are there? How many square numbers are there? Unless you can answer those two questions, you cannot assert that one is greater than the other. Note that we are not talking about any finite interval, we are talking about all numbers and all square numbers.

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.
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.

BTW A paradox is usually indicative of an underlying logic error.
Indeed, an underlying logic error by the person who thinks that a paradox entails a contradiction.
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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?

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.

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?
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Infinity is not a number of any kind it is a label for a concept.
• 1.5k
... the rest of maths obeys the arithmetic operators (or appropriate variations of them), infinity should too.
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).

I challenge you to come up with another mathematical 'number' that you can add a non-zero amount to without changing?
Why is another example required to justify the one that we have been discussing? The whole point is that the mathematics of finite quantities are (rather obviously) not applicable to infinity.
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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).

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.
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X+1=X never occurs in maths, apart from when it comes to infinity.
Indeed, infinity is different from any finite quantity. So what? That does not make it illogical or contradictory, just different.
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The two are contradictory. Infinity can’t be a number. So it is not maths.

Again to the op - infinity is not a number- it is a concept
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Indeed, infinity is different from any finite quantity. So what? That does not make it illogical or contradictory, just different

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.
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