Do you believe there can be an Actual Infinite

Devans99

The set of reals between 0 and 1 is provably infinite, and clearly bounded — MindForged

The Reals between 0 and 1 are unbounded in terms of precision. Imagine writing out all such reals to 1 decimal place (0.1, 0.2, etc...), then to 2 decimal places, then 3 etc... This is an example of potential infinity.

When you’re working out how many things compose another you take the overall length and divide by length of the constituent parts. So to work out how many points there are in the interval 0,1 you divide 1 by point size.

The problem with the number line example is that numbers have no length. They are labels that have no length. They don’t exist. So the number of numbers between 0 and 1 is 1 / 0 = undefined which is what you’d expect.

Magnus Anderson

A "sphere" (or "ideal sphere") is an abstraction, not an actually existing thing. — Relativist

We use abstractions, i.e. symbols, in order to represent reality. For example, the term "human being" is a symbol -- a written or a spoken word -- that can be used to represent certain portions of reality. We don't say human beings don't exist merely because the term "human being" is an abstraction. We only say that human beings don't exist if there is no portion of reality that can be represented by the term "human being".

You bring up another abstraction: the number of possible paths being infinite. This is hypothetical; in the real world, you cannot actually trace an infinite number of paths. So in the real world you cannot actually COLLECT an infinity. All you can do is to conceptualize. — Relativist

You don't need to be able to count an infinite number of things in order for that infinite quantity of things to exist. Things exist whether or not we are conscious/aware of them. Similarly, our beliefs are true or false regardless of whether we can justify them. Just because we cannot make an infinite number of observations does not mean we cannot come up with a theory that we can use to make an infinite number of predictions each one of which is true.

MindForged

The Reals between 0 and 1 are unbounded in terms of precision. Imagine writing out all such reals to 1 decimal place (0.1, 0.2, etc...), then to 2 decimal places, then 3 etc... This is an example of potential infinity. — Devans99

Utterly irrelevant. They have a specific magnitude, they are demonstrably greater than 0 and lesser than 1. The length of their decimal expansion has no bearing on the boundedness of the set. It's bounded. It's not a potential infinity, it's an actual, literal infinity.

When you’re working out how many things compose another you take the overall length and divide by length of the constituent parts. So to work out how many points there are in the interval 0,1 you divide 1 by point size.

The problem with the number line example is that numbers have no length. They are labels that have no length. They don’t exist. So the number of numbers between 0 and 1 is 1 / 0 = undefined which is what you’d expect.

What? You just count them, and counting is well understood mathematically. The set of numbers between 0 and 1 has the cardinality of the continuum. It's clearly bounded, there are numbers larger and smaller than any element in the set.

Devans99

You just count them — MindForged

But you’d never finish counting the reals between 0 and 1 so you can’t completely define the set.

And no way is the set bounded in terms of precision; that stretches to infinity so it’s unbounded.

But my main point you ignore - numbers have size zero so they do not exist - so talking about how many you can get on a number line between 0 and 1 is nonsense.

MindForged

But you’d never finish counting the reals between 0 and 1 so you can’t completely define the set. — Devans99

In math, counting in understood rigorously, e.g. one-to-one correspondence. I'm not talking about the temporal process that people do.

And no way is the set bounded in terms of precision; that stretches to infinity so it’s unbounded. — Devans99

That has absolutely nothing to do with being bounded or not. There are numbers greater than all those in the referenced set, and numbers lesser than them. That's a bound, you're grasping at straws my dude.

But my main point you ignore - numbers have size zero so they do not exist - so talking about how many you can get on a number line between 0 and 1 is nonsense. — Devans99

I really don't think you understand the purpose of a number line.

Magnus Anderson

If a point has no length it does not exist so the definition is contradictory. — Devans99

So if a symbol has no property called length assigned to it, it follows that there is no portion of reality that can be represented by it?

The word "point" is just that -- a word. It is a symbol we use to represent reality. And that symbol has no property called length and that's simply because we didn't define one. If you want, you can do so. But that won't change the fact that it does not follow that just because some symbol has no property called length assigned to it that there is no portion of reality it can represent.

Relativist

Relativist: 'A "sphere" (or "ideal sphere") is an abstraction, not an actually existing thing.'

We use abstractions, i.e. symbols, in order to represent reality. For example, the term "human being" is a symbol -- a written or a spoken word -- that can be used to represent certain portions of reality. We don't say human beings don't exist merely because the term "human being" is an abstraction. We only say that human beings don't exist if there is no portion of reality that can be represented by the term "human being". — Magnus Anderson

Agree, but note that what exists is an instantiation of the abstraction: a real world object that has the properties described by the abstraction. I'm just rejecting the argument that an abstracted X implies there are necessarily real-world X. We are more justified in beliefing X if there are clearly instantiations of X.

Relativist: 'You bring up another abstraction: the number of possible paths being infinite. This is hypothetical; in the real world, you cannot actually trace an infinite number of paths. So in the real world you cannot actually COLLECT an infinity. All you can do is to conceptualize.'

You don't need to be able to count an infinite number of things in order for that infinite quantity of things to exist. — Magnus Anderson

Sure, but you need some reason to think the abstracted infinity is instantiated in the real world, otherwise your justification is the mere fact that we can abstractly conceptualize infinity.

Magnus Anderson

I'm just rejecting the argument that an abstracted infinity implies there are real-world infinities. — Relativist

Well, just because we can imagine something, it does not mean it exists. Just because we can come up with a symbol such as "unicorn" does not mean there is a portion of reality it can represent. But I thought that your argument is that we need to count an infinite number of things in order for there to be an infinite number of things, or at the very least, in order for us to prove or justify that an infinite number of things exists. I don't think any of these two beliefs is true. We don't need to observe every human being dying in order to prove or justify our belief that all human beings are mortal.

Metaphysician Undercover

Good luck doing that without the rigorous mathematical understanding of infinity as opposed to the vague colloquial understanding. — MindForged

"Rigorous mathematical understanding of infinity". Lol. But if your not joking, you have my sympathy.

The set of reals between 0 and 1 is provably infinite, and clearly bounded. After all, every element in that infinite set is larger than 0 and yet smaller than 1; — MindForged

Does anyone even know what it means to be larger then zero? .So let's leave zero out of this. That there is an infinity of real numbers between any two real numbers is the assumption of infinite divisibility. The possibility for division is assumed to extend infinitely, just like the possibility for adding another natural number is assumed to extend infinitely. That the thing being divided is bounded, is irrelevant to the infinity which involves the act of dividing. So the infinite thing itself, divisibility, is not bounded. Likewise, in the case of the natural numbers, that the one unity being added at each increment of increase is bounded and indivisible, is irrelevant to the infinity which involves the act of increase. That the increasable amount is bounded, restricted to exclude fractions, is not a limit to the infinity itself. Nor is the fact that a divisible unit is bounded a limit or restriction to divisibility.

But whether or not sets are bounded or not really has nothing to do with infinity. A set whose members are ever increasing due to some iterative calculation is clearly unbounded, but it's not infinite. Just loop a program which adds new members to an array every iteration; at every iteration the number of members of the array are obviously going to be finite. — MindForged

That's incorrect. Whether or not something is infinite has everything to do with whether or not it is unbounded, because "infinite" is defined as unbounded. Where is your rigorous understanding of infinitiy? And no, an iterative calculation is not unbounded. It is limited by the physical conditions, and the capacity of the thing performing the iteration. That it is so bounded is the reason why it is not infinite.

Relativist

I thought that your argument is that we need to count an infinite number of things in order for there to be an infinite number of things, or at the very least, in order for us to prove or justify that an infinite number of things exists. I don't think any of these two beliefs is true.

We don't need to observe every human being dying in order to prove or justify our belief that all human beings are mortal. — Magnus Anderson

I agree about humans, but this has nothing to do with my position. My issue is that we can't assume some set of properties is instantiated in a real world object solely because we can coherently define the properties.

The abstraction "human being" is derived from things we know exist: we abstract out the properties that we observe in human beings, so there's no question about these abstractions being instantiated in the real world.

The concept of infinity is not formed by abstracting out properties of known existents. The concept is formed by extrapolation of other abstractions. One such extrapolation is the infinity of natural numbers. 4 doesn't exist in the real world; 4-ness is a property of certain states of affairs - those consisting of 4 objects. So we know 4-ness is instantiated. Is infinity-ness instantiated? We can't point to anything that has this property.

The infinity of natural numbers can be conceptualized by contemplating an unending count, but that isn't a process that can be instantiated - that was my point with counting. No, this isn't a proof, because there may be other ways an infinity might be instantiated. But without one to point to, we have no basis to assume it CAN be instantiated.

Again, this reasoning isn't a proof. Rather, it's a justification for me to believe it more likely there are no instantiated infinities in the real world, than that there are.

MindForged

"Rigorous mathematical understanding of infinity". Lol. But if your not joking, you have my sympathy. — Metaphysician Undercover

Great argument, about your usual standard in this thread.

Does anyone even know what it means to be larger then zero? — Metaphysician Undercover

I continue to be amazed by the questions asked here.

That there is an infinity of real numbers between any two real numbers is the assumption of infinite divisibility — Metaphysician Undercover

It's not an assumption if you can prove it. Seriously, assume there is some limit to how many reals there are between any two naturals. A simple expansion can be done to yield a new natural. Ergo on pain of contradiction the initial supposition must be false. There is no smallest real.

So the infinite thing itself, divisibility, is not bounded. Likewise, in the case of the natural numbers, that the one unity being added at each increment of increase is bounded and indivisible, is irrelevant to the infinity which involves the act of increase. That the increasable amount is bounded, restricted to exclude fractions, is not a limit to the infinity itself. Nor is the fact that a divisible unit is bounded a limit or restriction to divisibility. — Metaphysician Undercover

This has absolutely nothing to do with what I responded to.

That's incorrect. Whether or not something is infinite has everything to do with whether or not it is unbounded, because "infinite" is defined as unbounded. Where is your rigorous understanding of infinitiy?. — Metaphysician Undercover

"Infinite" is not defined as unbounded. Seriously, show me two mathematics textbooks that define infinite that way. Stop stop stop. The set of naturals has a smaller cardinality than the reals; the former is countable and the latter is uncountable ("unlistable" is probably a better word). So the naturals are bounded, we know numbers which are larger than it so there's a very obvious boundary: The cardinality of the naturals, no matter how far you go, is always smaller than the cardinality of the reals. Ergo your nonsensical claim that "infinite is defined as unbounded" is just false. You're not really adding more members to the set of natural numbers as you go further, you're just discovering more numbers that were already part of the set. No one that isn't brain dead is trying to create an extensionally defined infinite set.

And no, an iterative calculation is not unbounded. It is limited by the physical conditions, and the capacity of the thing performing the iteration. That it is so bounded is the reason why it is not infinite

You either have no reading comprehension or you don't know what I'm talking about. Just some simple (picking a language...) JavaScript.

let arr = [], i;

for(i = 0; arr.length < Infinity; i++){
     arr.push(i);
}

// result: arr = [0,1,2,3,....] (it actually never completes, for obvious reasons)

The set of numbers in the array is obviously ever increasing. I didn't have to put infinity in the loop comparison, I could have just put some tautology. The set is obviously unbounded, it's members keep increasing with every iteration. That's unboundedness .An infinity can very well be bounded unless you're just using some idiosyncratic definition of "bounded".

Devans99

A computing array is obviously bounded by memory limitations as you found out when your program hung.

The naturals {1,2,3,...} are unbounded on the right as denoted by the ...

The reals between 0 and 1 {.1, .01, .001, ... } are unbounded ‘below’.

Both are an example of potential not actual infinity in that it is an iterative process that generates an infinity of numbers.

The number of reals between 0 and 1 is undefined: a number has ‘length’ 0 and 1/0 = undefined. If you let number have length>0 you get a finite number of reals between 0 and 1. So there is no way to realise actual infinity...

Metaphysician Undercover

It's not an assumption if you can prove it. Seriously, assume there is some limit to how many reals there are between any two naturals. A simple expansion can be done to yield a new natural. Ergo on pain of contradiction the initial supposition must be false. There is no smallest real. — MindForged

I agree, but as I explained, the thing which is infinite is not the same thing as the thing which is bounded. Therefore the limits expressed are irrelevant to the infinity expressed, and the infinity is unbounded. Therefore your argument that there can be a bounded infinity is not sound.

The set of naturals has a smaller cardinality than the reals; the former is countable and the latter is uncountable ("unlistable" is probably a better word). So the naturals are bounded, we know numbers which are larger than it so there's a very obvious boundary: — MindForged

I don't believe this. Both the naturals and the reals are infinite, so I believe it is false to say that one is larger than the other. This is where I believe that set theory misleads you with a false premise. I would need some evidence, a demonstration of proof, before I would accept this, what I presently believe to be false. Show me for example, that there are more numbers between 1 and 2, and between 2 and 3, than there are natural numbers. The natural numbers are infinite. So no matter how many real numbers you claim that there are, they will always be countable by the natural numbers.

This is what I've been telling you over and over again. To stipulate that the cardinality of the natural numbers is less than something else, and to also say that the natural numbers are infinite, is contradictory. To say that the number of something is infinite and that there is less of these than something else, is blatant contradiction. If you truly believe (as you appear to), that the natural numbers are infinite, yet there are more real numbers than natural numbers, then you ought to be able to show me the limits, restrictions which have been placed on the natural numbers to allow that there are more reals than naturals. After showing me these limits, explain to me how the natural numbers can be limited in this way and still be infinite.

SteveKlinko

But how can you know that Naturalism holds before the Beginning? — SteveKlinko

Science (or natural philosophy as it used to be called) is based on naturalistic explanations. Science, for example, excludes god and magic as valid explanation for natural phenomena.

If the early universe does not follow naturalistic rules then we have little hope of ever understanding it.

Rather than giving up, why not assume the universe behaves in a naturalistic ways and proceed to argue from there? — Devans99

We don't give up but we look for the extended Naturalistic rules that probably apply before the Beginning. What Naturalistic rules can cause the Inflation of the early Universe? The Inflation is a theoretical expectation based on observations of the Universe. The phenomenon of Inflation could not be deduced from known Naturalistic rules. It requires tremendously faster than Light speeds. There is a new Naturalistic rule lurking here.

Devans99

Well I would class FTL travel as potentially naturalistic; it’s certainly not a magical proposition.

‘Something from nothing’ is however magical so I’d rule it out. Returning to the argument:

1. Something can’t come from nothing
2. So base reality must have always existed
3. If base reality is permanent it must be timeless
4. So base reality must be timeless (to avoid the infinities) and permanent
5. Time was created and exists within this permanent, timeless, base reality
6. So time must be real, permanent and finite

Do you buy the argument as far as 2 now or do you still have objections?

tim wood

1. Something can’t come from nothing — Devans99

You need to go to the trouble of defining your "nothing." The philosopher's nothing is very different from the nothing "out there" that scientists deal with. For them, something coming from nothing is routine, simply part of the way it all works.

Devans99

I define nothing as no matter, energy or dimensions. Complete nothing. Not a quantum fluctuation in sight.

SteveKlinko

Well I would class FTL travel as potentially naturalistic; it’s certainly not a magical proposition.

‘Something from nothing’ is however magical so I’d rule it out. Returning to the argument:

1. Something can’t come from nothing
2. So base reality must have always existed
3. If base reality is permanent it must be timeless
4. So base reality must be timeless (to avoid the infinities) and permanent
5. Time was created and exists within this permanent, timeless, base reality
6. So time must be real, permanent and finite

Do you buy the argument as far as 2 now or do you still have objections? — Devans99

1 and 2 certainly hold in the manifested Physical Universe that exists today. But these are unproven theories when it comes to the reality before the Big Bang. As for 3 to 6 which seem to be about Time, all I can say is that a lot of people interpret Relativity as having shown that there is no such thing as Time.

tim wood

I define nothing as no matter, energy or dimensions. Complete nothing. Not a quantum fluctuation in sight. — Devans99

That seems a philosopher's definition. Nothing like it actually exists, or, no one's found it yet.

MindForged

I agree, but as I explained, the thing which is infinite is not the same thing as the thing which is bounded. Therefore the limits expressed are irrelevant to the infinity expressed, and the infinity is unbounded. Therefore your argument that there can be a bounded infinity is not sound. — Metaphysician Undercover

You did not give any counterargument here that the real between zero and one are either finite or unbounded. I gave an argument for why it was both, and thus why something can be finite and bounded. So I don't know what you're trying to say here. The "thing" which is infinite is the number of reals, the thing which is bounded is the number of reals. Ergo "infinite" and "bounded" can be possessed by one and the same thing.

I don't believe this. Both the naturals and the reals are infinite, so I believe it is false to say that one is larger than the other. This is where I believe that set theory misleads you with a false premise. I would need some evidence, a demonstration of proof, before I would accept this, what I presently believe to be false. Show me for example, that there are more numbers between 1 and 2, and between 2 and 3, than there are natural numbers. The natural numbers are infinite. So no matter how many real numbers you claim that there are, they will always be countable by the natural numbers. — Metaphysician Undercover

If you ignore the last 150 years of math you can believe this, but Cantor's diagonal argument is pretty clearly proof of this. On pain of needless contradiction, one can show that the naturals are smaller in size than the reals. A real can always be constructed such that the set of reals cannot be put into a one-to-one correspondence with the naturals. If they were the same size, this correspondence provably hold but we know it doesn't via the diagonal argument.

This is what I've been telling you over and over again. To stipulate that the cardinality of the natural numbers is less than something else, and to also say that the natural numbers are infinite, is contradictory. — Metaphysician Undercover

It's not stipulated, it's proven. Again, just read about Cantor's diagonal argument,

MindForged

A computing array is obviously bounded by memory limitations as you found out when your program hung. — Devans99

By unbounded I meant there was not in principle a final member which the array could reach. In practice is irrelevant, we're talking about abstract objects not the limitations of finite state machines.

The naturals {1,2,3,...} are unbounded on the right as denoted by the ...

The reals between 0 and 1 {.1, .01, .001, ... } are unbounded ‘below’.

Both are an example of potential not actual infinity in that it is an iterative process that generates an infinity of numbers.

Sets are already whole, they aren't iterative calculations. The point of of the loop I posted before was to show that finite things which increase over time are clearly unbounded despite there finitude. In other words, MU's assumption that infinity is to unbounded, and finitude is to bounded, is just false. Infinite sets can bounded, and finite sets constructed over time can be unbounded.

The number of reals between 0 and 1 is undefined: a number has ‘length’ 0 and 1/0 = undefined. If you let number have length>0 you get a finite number of reals between 0 and 1. So there is no way to realise actual infinity...

That is not the length of the set, what are you talking about? You don't divide to determine the number of members in a set, you count them (counting as understood in math, not finger counting).

Metaphysician Undercover

You did not give any counterargument here that the real between zero and one are either finite or unbounded. I gave an argument for why it was both, and thus why something can be finite and bounded. — MindForged

It is the number of intervals between zero and one which is unbounded and infinite. You gave no argument that this number is bounded. You have stated an arbitrary boundary of zero and one, but this does not bound the infinite. You could have set your boundaries as 10 and 20, or 200 and 600, or zero and the highest natural number. These boundaries do not bound the infinite itself. So you have provided no argument that the thing which is infinite, the numbers between the boundaries, is bounded. There is an unbounded number of possible places between any two designated real numbers

The "thing" which is infinite is the number of reals, the thing which is bounded is the number of reals. — MindForged

No, clearly the number of reals is not bounded, so get your facts straight. By no means is that number bounded. You appear to be confusing the symbols, the numerals 0 and 1, with the numbers which are assumed to lie between them. The "thing" which is infinite is the number of real numbers between 0 and 1,and this is not bounded, just like the number of reals between 2 and 1000, or whateve,isnot bounded. The number of reals is in no way bounded, just like the number of naturals is in no way bounded. The boundaries are in the definitions by which they are produced, but the definitions are made such that the numbers themselves are not bounded. The two systems, the naturals and the reals, are just two distinct ways of expressing the same infinite numbers. Remember, separate the numerals (as part of the description) from the numbers which are signified by the description. The description, "reals between 0 and 1" signifies infinite numbers without boundary.

If you ignore the last 150 years of math you can believe this, but Cantor's diagonal argument is pretty clearly proof of this. — MindForged

There is a real problem with this so-called proof. It's called begging the question. By assuming that the natural numbers are a countable "set", it is implied that the naturals are not infinite. It is impossible to count that which is infinite. By definition, that which is infinite is uncountable, and that's why I've argued that the natural numbers cannot be a set. What Cantor needed to do was prove that the naturals are a set. But this would be impossible, because as I've explained to you (many times, in many ways) "infinite set" is self-contradictory.

So, as I've explained already, to say that the natural numbers are infinite, and to say that they are a countable set, is contradictory. Therefore we must give up one or the other. If we accept Cantor's proof, then we must accept Cantor's premise, which denies the infinity of the natural numbers. We cannot have both, the infinity of the natural numbers, and Cantor's proof, because Cantor assumes the finitude of the natural numbers as a countable set.

Now, as I explained to andrewk, there is good reason to maintain the infinity of the natural numbers, because this allows that every object is countable. This allows us to measure every object. But if we allow that the natural numbers are an object (set), then all we have done is created an object which cannot be counted (measured), because despite the fact that we might claim that the natural numbers are countable, they remain uncountable. So Cantor has created an object, the countable set of natural numbers, which cannot be counted, or measured in any way, because it is a fictitious object, because the natural numbers are really infinite and cannot be counted, nor can they be an object. That is why set theory ought to be dismissed so that we can go back to a true infinity of natural numbers, and allow that every object may be counted and measured, instead of allowing the existence of objects which cannot be counted or measured, this renders the world as unintelligible.

tim wood

I don't believe this. Both the naturals and the reals are infinite, so I believe it is false to say that one is larger than the other. — Metaphysician Undercover

It's just your ignorance, MU. To call either set infinite is to presuppose you know what exactly you mean by "infinite" in context. You neither know, nor as you have made completely clear, do you want to know or care to know. You merely want to continue to blow the smoke of you ignorance.

And to be sure, whether right or wrong, what you believe does not matter.

Metaphysician Undercover

To call either set infinite is to presuppose you know what exactly you mean by "infinite" in context. — tim wood

Have you read any of my posts? I insist that it is contradictory to say that a set is infinite. So no, I am not saying any set is infinite.

↪MindForged

↪tim wood

Here are the two distinct reasons why the natural numbers and the real numbers are considered to be infinite. In the case of the natural numbers, we can start counting, one, two, three, four, etc,, and never reach completion. Therefore we say that the natural numbers are infinite because they can never be completed. In the case of the real numbers we cannot even start to count, because if we start at one we have already missed an infinity of numbers. And no matter what number we put as the first number after zero, we have already missed an infinity of numbers. Therefore we say that the reals are infinite because we cannot even start to count them.

So the two, the naturals, and the reals, are both said to be infinite because they are uncountable. One is uncountable because we can never reach the end to counting them, the other because there can be no beginning to counting them. Yes, they are different infinities, for this very reason, but it is incorrect to say that one is larger than the other because they are both uncountable and therefore both immeasurable.

tim wood

Have you read any of my posts? I insist that it is contradictory to say that a set is infinite. — Metaphysician Undercover

This is you, MU.

Both the naturals and the reals are infinite, — Metaphysician Undercover

And this is you, MU.

Enough and done. Unfortunately, you're not worth the candle.

Metaphysician Undercover

↪tim wood

As each is infinite they cannot be consider as "sets", because "set" implies finitude. We went through all this days ago, about the time you left the discussion in disgust. I hope you do so again, because you seem to have no serious input in this matter

Devans99

That is not the length of the set, what are you talking about? You don't divide to determine the number of members in a set, you count them (counting as understood in math, not finger counting). — MindForged

Yes you do divide:

- the number line between 0 and 1 has length 1
- to find out how many things fit on the line
- divide line length by the thing length
- a number has length 0
- so the number of number between 0 and 1 is 1/0=UNDEFINED
- if you let number have non-zero length then there is a finate number of numbers in the interval but a potential infinity as number length tends to zero

I can’t believe you; we’ve been talking about this for ages and you have learned nothing. You are still not even using the proper language to discuss this is (actual/potential infinity).

You need to realise that you were told the wrong things about infinity at school and free your mind of Cantor’s muddled dogma.

Metaphysician Undercover

↪tim wood

↪Devans99

MindForged's problem is in the assumption that there is such a thing as an infinite set.

A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals. — MindForged

Clearly, for any set of natural numbers, a proper subset is always smaller. That is always the case, and there is never an exception. In order for the proper subset to have an equal cardinality, it must either be the original set, or contain numbers which are outside the original set. Then it is not a proper subset. Therefore an infinite set, under that definition is impossible.

So for example, the set of even numbers must contain numbers outside the set of natural numbers in order for it to have an equal cardinality. Counting by twos requires that you count twice as high as counting by ones, in order to have the same number of members in your set. But this set of even numbers, which has numbers higher than the set of natural numbers, in order to have an equal cardinality, is not a subset of that set of natural numbers.

You need to realise that you were told the wrong things about infinity at school and free your mind of Cantor’s muddled dogma. — Devans99

Exactly, with smoke and mirrors Cantor created the illusion of coherency, but he was really a master of deception. A mathematical magician is nonetheless, a magician. We need to see through the smoke and mirrors to root out the contradictions which lie within his fundamental assertions.

tim wood

↪Devans99

↪Metaphysician Undercover

What follows are grabbed from various parts of the internet. If one looks, one also finds your sources of nonsense, someone named Steve Patterson, et al. Near as I can tell, his paper is regarded as a joke.

The substance of it all is that if you use idiosyncratic and inappropriate definitions, you can prove anything you want. Happens with religion all the time. I can prove you're on Mars, viz:

If MU is alive then MU is on Mars definition
MU is alive
-----------------
MU is on Mars.

But who needs nonsense like this?

Btw, you wouldn't mind if I refer to prime numbers as being members of a well-defined set (the set of prime numbers), would you?

From https://en.wikipedia.org/wiki/Set_(mathematics)

A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.[2]

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.

Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

Russell's paradox—It shows that the "set of all sets that do not contain themselves," i.e. the "set" { x : x is a set and x ∉ x } does not exist.
Cantor's paradox—It shows that "the set of all sets" cannot exist.
The reason is that the phrase well-defined is not very well-defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born.

For most purposes, however, naive set theory is still useful.

And this from https://math.stackexchange.com/questions/356264/infinite-sets-dont-exist

an infinite set doesn't need to "exist" any more than the number 3 needs to "exist", they are all imaginary concepts and there is nothing wrong with that for the purposes of doing useful/interesting mathematics.

It is certainly true that infinite sets do not necessarily "exist" in most uses of the word other than the mathematical one. It is not, however, true that accepting set theory as foundations forces one to believe in such existence in any sense beyond the mathematical. Furthermore, the existence of "infinite sets" is no more contentious than the existence of "finite sets."

Mathematicians at the end of the day deal with certain systems of rules on how to manipulate symbols on a piece of paper. Such systems are composed of two parts: a language which consists of the rules that say which strings of symbols are valid (i.e. are sentences or formulas), and the transformation (inference) rules which say how to transform certain (collections of) sentences and formulas into other sentences and formulas. Formally, this is all we do as mathematicians: we come up with languages and inference rules, pick some sentences or formulas in the language that seem interesting and then we go on and try to obtain certain other interesting sentences and formulas (you get at mathematical logic if you ask yourself whether you can obtain certain interesting sentences and formulas at all).

There is a set X having the property that ∅ is an element of X, and whenever x is an element of X, then x∪{x} is also an element of X.

This is a very precise formulation which one can show yields a set which is not finite (hence infinite):

As ∅ is in X, then ∅∪{∅}={∅} is an element of X.
As {∅} is in X, then {∅}∪{{∅}}={∅,{∅}} is in X.
As {∅,{∅}} is in X, then {∅,{∅}}∪{{∅,{∅}}}={∅,{∅},{∅,{∅}}} is in X.
...
You see that these elements of X get larger and larger without (finite) bound, and so it stands to reason that such an X must be infinite.

And so on.

MindForged

You have stated an arbitrary boundary of zero and one, but this does not bound the infinite. You could have set your boundaries as 10 and 20, or 200 and 600, or zero and the highest natural number. These boundaries do not bound the infinite itself. — Metaphysician Undercover

How? It doesn't matter if the numbers I chose arbitrary, we're talking about whether it's infinite or not. Because there are numbers greater and smaller than those in the real between zero and one, they are bounded under any standard definition of "bounded" (having a limit). Nothing you've said here even attempts to address this because otherwise you'd be forced to admit that "infinite" is not defined as "unbounded" in mathematics. You're whole approach requires ignoring the definitions mathematicians use, it's disingenuous.

The boundaries are in the definitions by which they are produced, but the definitions are made such that the numbers themselves are not bounded. The two systems, the naturals and the reals, are just two distinct ways of expressing the same infinite numbers. — Metaphysician Undercover

You have lost the plot. You're just question begging again. You think I'm saying the number of reals between 0 and 1 are finite because I'm saying they're bounded, because you're conflating the two terms. The set of reals between 0 and 1 is uncountably infinite (because it can't be put into a function with the naturals) but it is bounded because there exists a lower bound demarcating where the set begins and an upper bound demarcating where the set ends. Those are boundaries, that doesn't entail the cardinality is finite.

There is a real problem with this so-called proof. It's called begging the question. By assuming that the natural numbers are a countable "set", it is implied that the naturals are not infinite. It is impossible to count that which is infinite. — Metaphysician Undercover

You are just ignorant man. Counting here is the *mathematical* notion of counting, not finger counting. I've mentioned it repeatedly: It's the one-to-one correspondence. That in no way entails finitude because this method actually gives us a way of defining "infinite" in a way which is actually useful in math. An infinite set in no way is a contradiction and anyone saying it does literally has no knowledge of modern mathematical foundations and the definitions of the terms used in standard mathematics formalisms.

Do you believe there can be an Actual Infinite

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