1) Actual Infinity is larger than any other number
2) Actual infinity plus one is larger than actual infinity
3) Hence there is no number larger than all other numbers
4) Actual Infinity does not exist — Devans99

#1 is false because there are lots of infinities, with some being larger and smaller relative to others. The infinity of the natural numbers has a smalelr cardinality than the infinity of the real numbers. There's no one singular "actual infinity".

#2 is wrong as well. Adding one member to an infinite set does not increase its cardinality (and neither does removing one element). This is bound up in the very definition of infinity, as it's finite things which change in size when things are added or removed.

#3 is just a complete misunderstanding of infinity in modern mathematics.

#4 is just what you've been assuming in every premise. The very misguided definition and misunderstandings of infinity are where the errors lay. Please just go study some set theory, stop this pseudo-philosophy of mathematics.

There are less squares than numbers because not all numbers are squares. Yet each number has a square so the number of numbers and squares must be the same.

He is trying to compare two actually infinite sets, IE comparing two undefined things. A set definition is not complete until all its members are interated. — Devans99

Again, a total misunderstanding. Stop linking things you don't understand. There's a one-to-one correspondence between the squares and the positive integers so we know those sets have the same cardinality. It only seems weird because you don't understand the properties of infinity. No contradictions arises here, and further infinite sets are not undefined. They can be given perfectly clear intensional definitions. Hell, you just gave an intensional definition of two infinite sets: the square numbers and the positive integers.

#1 is false because there are lots of infinities, with some being larger and smaller relative to others. — MindForged

- Yes but my argument shows none of them exist because they are not constructable

Adding one member to an infinite set does not increase its cardinality — MindForged

- So you are saying there is a number (cardinality) with the property that number plus one equals same number? X+1=X !!!

Objects quantified over are not assumed to exist. — MindForged

If it's an object then it exists. To be an object is to exist. There is no non-existent object, that's contradiction. You're just trying to find a semantic loophole, but you are really digging yourself deeper into a hole of contradiction.

A set is a well-defined collection, often characterized by sharing some property in common or holding to some specified rule. — MindForged

You agree with me that a set is a collection, so we have no disagreement over the definition of a set. Our disagreement is over what constitutes a collection. I think that things must be collected to be a collection. You seem to think that things which are by some principle "collectible" are a collection. Clearly you are wrong, a collection must be collected, and collectible things do not constitute a collection. "Collection" often refers to the act of collecting. So it is quite clear that a collection does not exist until the things are collected in the act of collection. That is why an infinite set is utter nonsense, because it is absolutely impossible to collect an infinity of things.

"My" definition (in actuality, the mathematical definition) of sets are clear and they allow for infinity. — MindForged

Absolutely not. Until you demonstrate how an infinity of things may be collected into a collection, your definition of sets does not allow for an infinite set. You are in denial, refusing to understand the words of your own definition.

You are confusing determining if an object belongs to a set with whether or not the object does in fact belong to a set. — MindForged

Obviously, if the object has not yet been collected it does not belong to the collection. What could constitute the act of collecting other than determining that the object belongs to the set? It's nonsense for you to think that an object could belong to a collection without having been collected. Therefore there is no such thing as an object belonging to a set without having been determined as belonging to that set. It's your nonsensical way of thinking which is making you believe that belonging to a set is something other than having been determined as belonging to that set.

You are making up definitions of sets, I'm literally using the standard mathematical definition which in fact captures many of our intuitions about collections and does so without any contradictions. — MindForged

No, clearly we agree on the definition of a set, it is a collection. But nothing can belong to a collection without the act of collecting (collection), by which the thing is collected. And it's also very clear that an infinity of things cannot be collected. Therefore, according to the definition of 'set" which we both agree on, an infinite set is absolutely impossible.

Yes but my argument shows none of them exist because they are not constructable — Devans99

They are constructable. I gave an intensional definition of the sets. If you literally mean that "construct" means to individually gather each element and put them in a real box, well there's your problem. Your definition of set membership is inadequate for any useful mathematics beyond simple combinatoric reasoning.

- So you are saying there is a number (cardinality) with the property that number plus one equals same number? X+1=X !!! — Devans99

It's provably the case. Add one to the cardinality of the natural numbers. It's still able to be put into a function with the original set of natural numbers, and is still not uncountably infinite, as it cannot be put into a function with the set of real numbers. Arguments from incredulity don't impress anyone.

Add one to the cardinality of the natural numbers. — MindForged

I can’t because cardinality is an ill-defined concept - it includes cardinality of actually infinite sets - which are not numbers so I can’t add one to it.

Cardinality should be defined as the NUMBER of elements in a set. Actual infinity is not a NUMBER.

I can’t because cardinality is an undefined concept - it includes cardinality of actually infinite sets - which are not numbers so I can’t add one to it. — Devans99

It's not an undefined concept, you are out of your mind. Transfinite cardinal numbers are numbers which are infinite. Since wiki wish apparently good enough for you, here: http://en.wikipedia.org/wiki/Transfinite_number

Cardinality should be defined as the NUMBER of elements in a set. Actual infinity is not a NUMBER.

The cardinality of a set is determined by the number of elements of a set. The cardinality of the set {apple, banana, orange} is three because there are three elements. The cardinality of the set of natural numbers {0,1,2...} is aleph-null, the smallest infinite number. You have absolutely no idea what you're talking about.

Sorry I meant I’ll-defined concept.

‘Aleph-null is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers’.

This definition is self contradictory; the cardinality of the natural numbers is not a number as it claims.

Speed is always relative, and if it is defined by distance/ time ( our human time, that alows us to coinside events within metrics of our human existence), then time and distance are also relative concepts. Let me say it like this: "Universe is not under obligation" to come up with a concept of time for it's own use. People are. People consider electromagnetic changes in Universe to be time, while this is a way in which Universe saves and compresses multitude of material to fit inside of self for the reason that it can not get rid of any space ( whatever it is made of) or material it consists of. It can only change one kind of matter particles, waves, etc. into others in ongoing, chaotic and neverending manner and eventually turn on a loop, restarting the proceses that it already went through long before. One thing Universe does not do is to waist space. Yet in our human understanding there is more wasted, empty space than we can fathom. Human understanding of spaces and distances is granduerly, inesplicably off, accentuating our minisculeness and insignificance in universal affairs.

If it's an object then it exists. To be an object is to exist. There is no non-existent object, that's contradiction. You're just trying to find a semantic loophole, but you are really digging yourself deeper into a hole of contradiction. — Metaphysician Undercover

What one quantifies over in logic or places in a set does not commit one to the existence of the thing. If you think modern logic assumes existential import, then, well you just believe something false. It's not a "semantic loophole", you're just wrong man.

Our disagreement is over what constitutes a collection. I think that things must be collected to be a collection. You seem to think that things which are by some principle "collectible" are a collection. Clearly you are wrong, a collection must be collected, and collectible things do not constitute a collection. "Collection" often refers to the act of collecting. So it is quite clear that a collection does not exist until the things are collected in the act of collection. That is why an infinite set is utter nonsense, because it is absolutely impossible to collect an infinity of things. — Metaphysician Undercover

"Collection" does not refer the process of collecting things. If I talk about the collection of stars in the sky and I call that a set, no one thinks I've literally gathered the stars in the sky. They readily understand I'm mean that there's a condition each of those objects share (that is, "being in the sky") and that I'm grouping them into a collection.

And again, this is literally just an argument by definition, easily defeatable. Let's say I'm not talking about "collections" because you're somehow just obviously using the only coherent, sensible definition of that word. Here's a new word: "schmollections". Schmollections are like collections, except they don't refer to the process of collecting things. They refer to well defined groups of objects related by some common property, condition or rule and are referred to as a whole as a "Schmet" because OBVIOUSLY that's not a "set", supposedly. Well great then, it looks like infinite "schmollections" and "schmets" are possible since they don't make the same assumptions as "collections" and "sets" according to you. So modern mathematics (which you are dispensing with by making these objections, funnily enough) use "schmollections" and "schmets".

You're argument is trivial and presumptive.

Absolutely not. Until you demonstrate how an infinity of things may be collected into a collection, your definition of sets does not allow for an infinite set. You are in denial, refusing to understand the words of your own definition. — Metaphysician Undercover

You are confusing your own definitions with the definitions used in mathematics. I an not. I've stated my definitions, you're only response was to equivocate by making objections from your own separate definitions. I've already shown how easy intensional definitions of sets allows for perfectly obvious infinite sets to be created.

No, clearly we agree on the definition of a set, it is a collection. But nothing can belong to a collection without the act of collecting (collection), by which the thing is collected. And it's also very clear that an infinity of things cannot be collected. Therefore, according to the definition of 'set" which we both agree on, an infinite set is absolutely impossible — Metaphysician Undercover

No we clearly do not agree. You think collections are by definition finite, I do not. And unlike you, my definitions are actually used by virtually all modern mathematicians.

This definition is self contradictory; the cardinality of the natural numbers is not a number as it claims. — Devans99

It's not contradictory. How many times are you going to make wild claims with no explanation?

How is it not a number? Cardinal numbers are numbers. Transfinite cardinals are cardinals. Therefore transfinite cardinals are numbers. Seriously man, go read some actual foundational mathematics stuff. You're wasting everyone's time by ignoring the actual learning needed to even understand the fundamental terminology at play. Linking Wikipedia articles and providing simplistic arguments which ignore the actual definitions actual mathematicians use for these words is lazy and deceptive. You're not doing philosophy, you're being an ideologue.

Universe is not under obligation" to come up with a concept of time for it's own use — Victoria Nova

Don’t forget the speed of light - the universe has a built-in speed limit and speed = distance / time. A speed limit is needed for consistency of any universe. Hence time is a fundamental concept of this universe.

Therefore transfinite cardinals are numbers — MindForged

Well transfinite cardinals have strange properties like:
X+1=X
X-1=X
What sort of number behaves like this?

Well transfinite cardinals have strange properties like:
X+1=X
X-1=X
What sort of number behaves like this? — Devans99

Obviously they have different properties. Finite numbers are finite, transfinite numbers are infinite. It's like complaining that odd numbers cannot exist because you can't divide them by 2 with no remainder like you can with even numbers.

And which numbers behave like that? I've already answered this: Transfinite cardinals and transfinite ordinals behave like that. It's only finite numbers which change in size by removing finite amounts from them.

These properties are nonsensical compared to the properties of any normal number.

X+1=X

No other number, complex, vector, matrix, whatever, has this nonsensical property.

These properties are nonsensical compared to the properties of any normal number.

X+1=X

No other number, complex, vector, matrix, whatever, has this nonsensical property. — Devans99

I repeat, you can say that about ANY other kind of number that has a unique property others don't have. It's not a contradiction so the repeated claims you made that it was contradictory are just false. Weird is not the same as false, neither is unexpected, nor is counterintuitive.

But maths has defined a number that behaves non-numerically... we can at least agree that such ‘numbers’ are not to be used in the physical sciences?

It doesn't behave non-numerically, that doesn't make sense. The normal operations can be performed with such numbers, but that doesn't mean you'll get the results you would expect with finite numbers. And the reason is clear: Because you're dealing with a different type of number.

Whether such numbers have any place in science is an empirical matter, not an a priori one.

But maths has a responsibility to make sure it clearly communicates concepts to its end users.

Actual Infinity need to come with a health warning:

- This is a conceptual concept only
- Applying it to the real world is nonsense
- It is logically inconsistent with the rest of maths and common sense (see Hilberts Hotel)

But maths has a responsibility to make sure it clearly communicates concepts to its end users. — Devans99

It does. Literally every mathematician and any maths student paying attention knows the things I'm saying.

Actual Infinity need to come with a health warning:

1) This is a conceptual concept only
2) Applying it to the real world is nonsense
3) It is logically inconsistent with the rest of maths and common sense (see Hilberts Hotel) — Devans99

#1 is not known to be true and previous examples were given of it possible being false.
#2 is the same as before.

#3 is just stupid. It's not logically inconsistent at all, stop saying things you've been rebutted on and refuse to give real evidence of. I don't care about common sense, we don't deal with infinity in common matters so why should I apply it to the mathematics of infinity? Hilbert's Hotel is not a contradiction. No one thinks the scenario described can actually occur. The process of moving hotel guests around is a temporal process that operates at finite speeds. We only have a finite amount of material, we can only create finite structures at finite rates over a period of time. Obviously it's never going to terminate for those reasons. None of those things are true of mathematical operations involving infinite sets.

A hotel which is completely full, an infinite number of new guests show up and they are all accommodated by the magic of infinity.

Magic is the key word. How did such a concept find its way into maths? I think it’s historical and relates to our original concept of God - God is omnipotent so must be able to do anything, including the Actually Infinite, so they were thinking.

However it happened we are left with pure and applied math containing spiritualism.

Whatever man, I've said my piece in this thread as many times as I feel like doing anymore. Responses that it's magic or that it's contradictory or that it's nonsense aren't good responses when those declarations are only at best backed up by using definitions besides the actual definitions mathematicians use.

So... I'll leave you to it.

It doesn't behave non-numerically, that doesn't make sense. The normal operations can be performed with such numbers, but that doesn't mean you'll get the results you would expect with finite numbers. And the reason is clear: Because you're dealing with a different type of number.

It's not true that the "normal operations" can be performed with transfinite numbers. Analogous operations can be defined, but the are not the SAME operation. The fact that transfinite numbers have mathematical properties has no bearing on whether or not they have a referent in the real world - mathematics deals with lots of things that are pure abstraction with no actual referent (look into abstract algebra).

So... I'll leave you to it. — MindForged

A wise decision!

- Actual Infinity is larger than any other number — Devans99

Only in the sense that infinity is larger than any finite number. Otherwise, it is not true.

- Actual infinity plus one is larger than actual infinity

Only in the sense that some specific infinite number plus one is larger than that specific infinite number.

For example, the set of even natural numbers S = {2, 4, 6, ...} plus 1 equals {2, 4, 6, ..., 1}. Clearly, these two sets aren't equal. The resulting set has a greater number of elements than the set S.

- Hence there is no number larger than all other numbers

Which does not follow. An infinite number is a number that is larger than any finite number. An infinite number is not larger than any infinite number.

infinite number is a number that is larger than any finite number. An infinite number is not larger than any infinite number — Magnus Anderson

- So I have infinity X and a copy X’.
- I add one to X
- then X > X’ by common sense

Not sure this is much better:

- There is an number X such that X > all N
- X+1 > X
- There is no such number

The problem, as you pointed out above, with the preceding argument is that there are two sorts of numbers involved; finate numbers following the normal rules and infinite numbers in an illogical world of their own.

I think the math is frankly nonsense, how can we operate with two types of different numbers one of which is defined only axiomatically, does not exist in reality, obeys different counter intuitive rules and leads to contradictions?

Cantor's Paradox
‘The set of all sets is its own power set. Therefore, the cardinal number of the set of all sets must be bigger than itself.’

The set of all sets is an ACTUAL INFINITY so not a completly described set. You cannot soundly reason with it. Leads to the paradox.

"Collection" does not refer the process of collecting things. If I talk about the collection of stars in the sky and I call that a set, no one thinks I've literally gathered the stars in the sky. They readily understand I'm mean that there's a condition each of those objects share (that is, "being in the sky") and that I'm grouping them into a collection. — MindForged

You haven't actually grouped those stars into a collection though. That collection is completely imaginary, in the mind only . That's the point of the thread, such a collection is not an "actual" collection it's an imaginary collection.

Now the problem with an infinite collection is that it is impossible to actually collect an infinite number of things. So not only is that collection imaginary, but it is impossible due to contradiction. It's very easy to name impossible collections. The difficult thing is to determine whether such a collection is actually possible or not.

They refer to well defined groups of objects related by some common property, condition or rule and are referred to as a whole as a "Schmet" because OBVIOUSLY that's not a "set", supposedly. — MindForged

The problem with your analogy here is that you are concentrating on the defined "common property", and claiming that this constitutes a "group", "a whole", but neglecting what in reality are the criteria for "a group", or "a whole". You seem to think that you can define an object (a whole) into existence. Your "Schmet" has existence as a group, a whole, an object, because you say that it does.

That's fine, I have no problems with that, as that's the way that concepts exist as objects, they exist as definitions. So we can give intelligible objects existence in that way. The problem is that with "infinite set" you are attempting to create a contradictory concept, and this must be disallowed as unacceptable. To have "a group" or "a whole" which is infinite is contradictory, so the existence of that concept, as an intelligible object, must be disallowed as actually unintelligible.

And unlike you, my definitions are actually used by virtually all modern mathematicians. — MindForged

That's an appeal to authority. Do you think that just because mathematicians accept and use this concept, therefore it is not contradictory. You're only fooling yourself, as modern mathematics is full of contradiction. In mathematics there is no real principle by which an axiom is judged as acceptable or not. They are generally accepted on pragmatic principles. So when they are well disguised, as is the case with "infinite set", contradictions are accepted by mathematicians quite readily. It seems that mathematicians do not subject their axioms to the same scrutiny that philosophers do, and that's how such mistakes occur.

From what I am able to comprehend there has to be something that is infinite. Our universe, for example, is always expanding. It must expand into something. Whatever this something is has to be in something else as well. This is true over and over until finally some space or thing is infinite. There will always be the issue of what one thing is in until that something is infinite.