Argument Against an Actually Infinite Quantity

[PseuTonym]

What is the first mathematical evident proposition that serves as the foundation of the theorem “There is an actual infinity”?

I will not answer that question, but will simply provide a train of thought, for whatever it may be worth.

First, there would need to be some mathematical concept of “actual” before it would be appropriate to refer to the statement “there is an actual infinity” as a theorem. The sticky point here is your choice of the word “theorem.”

Second, it strikes me as strange to describe the existence of infinitely many primes as an elementary theorem, while simultaneously suggesting that the question of whether or not there are infinitely many positive integers is a challenging and interesting problem in mathematics. Now, you didn’t actually say that it (the theorem that there are infinitely many primes) is elementary, but in the context of this thread anybody could make that claim, and it would be strange if any debate arose on that question.

Third, if you want to use variables in mathematics …

(i.e., to expand your language beyond what was available to ancient Babylonian mathematicians who basically could solve the quadratic equation, but had to express their solution in terms of specific examples, because they had a very restricted system of mathematical language)

… then you need to also have what are known as “quantifiers” such as “for every” and “there exists.”

However, quantifiers may not be enough. If you accept that zero is a number, then you can consider the following two alternative attempts to state something that seems to be generally accepted as true:

(1) For all values of the variable r that are elements of the set of non-zero real numbers, (r/r)=1.

(2) For all values of the variable r that are elements of the set of real numbers, the following statement is true: if r is not equal to zero, then (r/r)=1.

In statement (1), we use a set of values for a variable in order to refrain from asserting something for values of the variable that don’t belong to our set. For that purpose, we rely upon the assumption that there exists the set of non-zero real numbers, which is simply a set obtained from the set of all real numbers by “removing” the one element that is the real number zero.

In statement (2), there is an attempt to impose a prohibition, but the attempt will fail. It is not obvious that there is any way to control the value of the variable r when it may be inside some part or fragment of some statement.

 

[GEddie]

If you can conceive an “actual infinity” (I don’t think you really can), then it is not an infinity.

And yet, we can imagine and even long for an everlasting life.

What would that say about us, our minds and our ideas?

ICXC NIKA

 

[GEddie]

Yes, it gives something to work with. For example, if you start thinking that infinity is a proper mathematical object, you can get to the conclusion, for instance, that the total sum of the natural numbers (which are all positive, and non fractions) is a negative fraction, which is crazy. Obviously, there is something wrong with the first assumption.

I’m not sure how you get from one to the other?

 

[rossum]

What is the first mathematical evident proposition that serves as the foundation of the theorem “There is an actual infinity”?

That is not a mathematical theorem, it is more of a philosophical statement. The closest you will get in mathematics will be something like: “infinity is defined as the cardinal number of the set of integers.”

That is a definition, not a proof of existence. It is perfectly possible for mathematics to define something that may, or may not, exist. For example perfect numbers are well defined, but all that perfect numbers we have found so far are even. An odd perfect number is well defined – we can easily recognise one should we find it. However, to date none have been found, nor do we have a proof of either their existence or non-existence.

rossum

 

[rossum]

Second, it strikes me as strange to describe the existence of infinitely many primes as an elementary theorem, while simultaneously suggesting that the question of whether or not there are infinitely many positive integers is a challenging and interesting problem in mathematics.

Strictly, the mathematical proof relative to prime numbers says, “given any set of prime numbers it is always possible to find another prime number that is not in the set”. It does not actually mention infinity as such. That proof does show that we can always find a new prime number, no matter how many we have already found, but does not explicitly use the word “infinite”.

rossum

 

[PseuTonym]

Strictly, the mathematical proof relative to prime numbers says, “given any set of prime numbers it is always possible to find another prime number that is not in the set”. It does not actually mention infinity as such. That proof does show that we can always find a new prime number, no matter how many we have already found, but does not explicitly use the word “infinite”.

I agree that the proof in Euclid’s Elements doesn’t explicitly refer to an infinite set. However, there are other proofs. For example, if I am not mistaken, there is a proof that uses fairly elementary concepts from topology, and that does explicitly show that the set is infinite.

In books and lectures about set theory itself, it is common for attention to be drawn to theorems that can be proven without relying upon the axiom of choice. This might be related to the fact that Euclid does more than merely prove the existence of a prime number larger than any in some hypothetical finite list of all primes, but actually provides a recipe for constructing a number that has a larger prime divisor.

Now, what is the difference between saying that there doesn’t exist a finite list of all primes, and saying that there do exist infinitely many primes? To most observers, I imagine that it would seem to be unmotivated nitpicking. Perhaps, if it were presented in the context of a survey of the history of ideas and ideologies, with reference to a list of specific, now-obscure movements, then I can see a motive.

 

[IWantGod]

I agree that the proof in Euclid’s Elements doesn’t explicitly refer to an infinite set. However, there are other proofs. For example, if I am not mistaken, there is a proof that uses fairly elementary concepts from topology, and that does explicitly show that the set is infinite.

In books and lectures about set theory itself, it is common for attention to be drawn to theorems that can be proven without relying upon the axiom of choice. This might be related to the fact that Euclid does more than merely prove the existence of a prime number larger than any in some hypothetical finite list of all primes, but actually provides a recipe for constructing a number that has a larger prime divisor.

Now, what is the difference between saying that there doesn’t exist a finite list of all primes, and saying that there do exist infinitely many primes? To most observers, I imagine that it would seem to be unmotivated nitpicking. Perhaps, if it were presented in the context of a survey of the history of ideas and ideologies, with reference to a list of specific, now-obscure movements, then I can see a motive.

Well, there is not a list of all numbers counting from zero, but the list is always finite non the less. There is not an actually infinite number.

 

[JuanFlorencio]

I will not answer that question, but will simply provide a train of thought, for whatever it may be worth…]

I selected the term “theorem” because of what Rossum wrote in post # 15. As I suggest in one of my previous posts it is not possible for us to conceive an actual infinity, so much the less to express it using symbols, either mathematical, philosophical or whatever.

What we mean when we say “infinity” is a process that we can start and find no reason to stop. What would it mean, for example, that the set of real numbers is infinite and that you conceive it?

 

[JuanFlorencio]

And yet, we can imagine and even long for an everlasting life.

What would that say about us, our minds and our ideas?
ICXC NIKA

I find it very hard even to conceive “a year of my life” as a whole set of events. Trying to think about an everlasting life as a whole (as an actual infinity) seems contradictory to me, because my life will always be a succession.

(However, that does not mean I don’t believe on an everlasting life).

 

[JuanFlorencio]

I’m not sure how you get from one to the other?

I don’t! But you can have a look at this video:

youtu.be/jcKRGpMiVTw

 

[JuanFlorencio]

That is not a mathematical theorem, it is more of a philosophical statement. The closest you will get in mathematics will be something like: “infinity is defined as the cardinal number of the set of integers.”

That is a definition, not a proof of existence. It is perfectly possible for mathematics to define something that may, or may not, exist. For example perfect numbers are well defined, but all that perfect numbers we have found so far are even. An odd perfect number is well defined – we can easily recognise one should we find it. However, to date none have been found, nor do we have a proof of either their existence or non-existence.

rossum

That happens not only when you develop mathematical thoughts. You can define whatever you like in whatever field of thought, and once you do it, it may or may not exist.

What do you mean by “a philosophical statement”? Just “verbal concepts”?

 

[PseuTonym]

Quoting the relevant part of what PseuTonym posted:

Now, what is the difference between saying that there doesn’t exist a finite list of all primes, and saying that there do exist infinitely many primes? To most observers, I imagine that it would seem to be unmotivated nitpicking. Perhaps, if it were presented in the context of a survey of the history of ideas and ideologies, with reference to a list of specific, now-obscure movements, then I can see a motive.

I agree that there are long lists of consecutive whole numbers counting from zero, but each such list is always finite none-the-less. There is not an actually infinite number of primes in the list.

I took the liberty of editing what you wrote, because I couldn’t understand it as written. However, I had to guess at the end. If you intended to assert that there doesn’t exist a prime number that is infinitely large, then I agree. However, I assert that a set of numbers can be an infinite set even though each element of it is finite as a number.

Alternatively, are you saying that there isn’t an actually infinite list? If you are saying that, then are you relying upon the meaning of the word “list” to refer to something like an infinitely long sheet of paper that actually has every item written on it in pencil or ink?

We write such equations as 1/3 = 0.33333 …

Nowadays, there is nothing particularly mysterious about such an equation. It is possible to reformulate the equation to avoid using the “dot dot dot” symbol. If you have an objection to the “dot dot dot” symbol, then I can sympathize with your concerns.

Would you say that the equation is simply incorrect on the grounds that we are forbidden from writing down “0.33333 …”? Do you not believe that there is any true idea that somebody might have in mind when claiming (as a first attempt to formulate the idea) that 1/3 is equal to 0.33333 …?

 

[Hatikvah]

Well, OK, but who is trying to define infinity in this way?

ICXC NIKA

There is a popular idea among irreligious people that the universe is infinitely vast – no spatial beginning and no spatial end. This may be targeted towards that idea if we’re talking space-time (or just space and its contents, perhaps).

Would the Christian say “there is nothing outside the universe”? I’m not sure, all we can be sure of is that God is omnipresent and greater than the universe, His creation.

 

[JuanFlorencio]

Quoting the relevant part of what PseuTonym posted:

I took the liberty of editing what you wrote, because I couldn’t understand it as written. However, I had to guess at the end. If you intended to assert that there doesn’t exist a prime number that is infinitely large, then I agree. However, I assert that a set of numbers can be an infinite set even though each element of it is finite as a number.

Alternatively, are you saying that there isn’t an actually infinite list? If you are saying that, then are you relying upon the meaning of the word “list” to refer to something like an infinitely long sheet of paper that actually has every item written on it in pencil or ink?

We write such equations as 1/3 = 0.33333 …

Nowadays, there is nothing particularly mysterious about such an equation. It is possible to reformulate the equation to avoid using the “dot dot dot” symbol. If you have an objection to the “dot dot dot” symbol, then I can sympathize with your concerns.

Would you say that the equation is simply incorrect on the grounds that we are forbidden from writing down “0.33333 …”? Do you not believe that there is any true idea that somebody might have in mind when claiming (as a first attempt to formulate the idea) that 1/3 is equal to 0.33333 …?

Does anybody claim that the “equation” 1/3 = 0.33333 … means nothing else but that when you add more digits to the right side your approximation is better and better, though never exact?

 

[PseuTonym]

Does anybody claim that the “equation” 1/3 = 0.33333 … means nothing else but that when you add more digits to the right side your approximation is better and better, though never exact?

Do you want me to name names? Generally speaking, I don’t know what people claim. I can read what they write, but then I have to ask questions before I can do anything other than repeat their claims word-for-word. However, I am neither good at nor talented at memorizing, so it would be more trouble than it is worth.

Edited to add:
There are two constants, one on each side, and there is an equals sign between them. What motivated you to use quotation marks around the word equation?

 

[JuanFlorencio]

Do you want me to name names?

Edited to add:
There are two constants, one on each side, and there is an equals sign between them. What motivated you to use quotation marks around the word “equation”?

If behind the names there are some rational arguments, why not? But the arguments would be enough to me.

I wrote “equation” because I think 1/3 = 0.33333… is not an equation.

 

[PseuTonym]

I wrote “equation” because I think 1/3 = 0.33333… is not an equation.

How about 1/4=0.25?

Do you consider that to be an equation?

 

[JuanFlorencio]

How about 1/4=0.25?

Do you consider that to be an equation?

Sure, PseuTonym! That’s is an equation.

 

[Pat_Albertson]

Let S = 1 + 1/2 + 1/4 + 1/8 + . . .

Then 2S = 2 + 1 + 1/2 + 1/4 + 1/8 + . . .

So, 2S - S = 2 + (1 - 1) + (1/2 - 1/2) … where all the fractional terms cancel out.

2S - S = 2

S = 2

Your infinite sum totals 2.

rossum

Yes, that is essentially what I wanted the OP to do.

What is the infinite sum of 1 + 1/2 + 1/4 + 1/8 + . . . and so on forever and ever? Can fractions be both distinct and finite? Why did I start off with the number 1 as opposed to the diagram in post #3 which started off with a semicircle of 1/2 as the diameter? What is this type of series? Have you taken 1st year college calculus (I suppose they offer this in some high schools too though.)?

I’m not trying to be challenging here, just probing what you mean, and where you are mathematically is all.

The fractions are all distinct, finite and infinite (or tending toward infinity if you’re a purist bursting with rigour) in number with the length indistinguishable from 2.

I started off with 1 because 1/2 ^ 0 = 1 as in 1/2 ^ 0 + 1/2 ^ 1 + 1/2 ^ 2 + 1/2 ^ 3 + . . . = 1 + 1/2 + 1/4 + 1/8 + . . .

This type of series is known as a power series. If we can call 1/2 by the variable name of r, then all such series converge to 1/(1 - r).

These things are typically introduced in a first year’s college calculus class. I wanted the OP to stop worrying about infinity and do what they tell physics students of quantum mechanics baffled by the meaning of it all to do: just shut up and compute.*

*No actual rudeness implied here, that really is one of the standard answers to people who try to fit quantum mechanics into standard English understandable by the layman but without the use of mathematics.

 

[Pat_Albertson]

Yes, that is essentially what I wanted the OP to do.
The fractions are all distinct, finite and infinite (or tending toward infinity if you’re a purist bursting with rigour) in number with the length indistinguishable from 2.

I started off with 1 because 1/2 ^ 0 = 1 as in 1/2 ^ 0 + 1/2 ^ 1 + 1/2 ^ 2 + 1/2 ^ 3 + . . . = 1 + 1/2 + 1/4 + 1/8 + . . .

This type of series is known as a power series. If we can call 1/2 by the variable name of r, then all such series converge to 1/(1 - r).

These things are typically introduced in a first year’s college calculus class. I wanted the OP to stop worrying about infinity and do what they tell physics students of quantum mechanics baffled by the meaning of it all to do: just shut up and compute.*

*No actual rudeness implied here, that really is one of the standard answers to people who try to fit quantum mechanics into standard English understandable by the layman but without the use of mathematics.

Correction: power series should have been geometric series.