Two picture the first two orders of infinity, we can start with the integers:

1, 2, 3, 4, 5 ...

This is called a “countable” infinity, because all the items in it can be perused by a discrete counting operation.

The next order of infinity is the real number line, including all points on it.

I can’t write this out, because it is an uncountable infinity. But think of all the points along a ruler. For every two integers, there are an infinite number of points in between: an uncountable number of points.

That still sounds like a difference in density of numbers within the limits of infinity, not a difference in size.

ok, taking the Hotel example. I am uncomfortable with the analogy of adding guests, which expands the number of rooms. That presents two problematic scenarios.
a) even if we add adults (requiring separate rooms), adding rooms does not necessarily expand the size of the hotel. If the hotel is infinitely large then it can never be filled up in the first place and should have empty rooms available anyway.
b) if we add children of guests already occupying rooms, then there is no need for adding rooms at all. there are just more people in the hotel (maybe with roll-out beds)
To say that counting the numbers which fall in between whole numbers increases the overall size of infinity is indeed counter intuitive to me. This is why I posed the question if an infinity of feet yields a larger infinity as an infinity of inches, which are increments of feet and fall in between the measurement of a foot, thus do not add to the size, but to the density of numbers.
I do understand the concept that one set yields a larger amount of numbers and as such one might say there is a greater number of inches in an infinity of inches than there are of feet, but it does not necessarily increase the actual size of the measurement of infinity, but rather the density of numbers within that measurement. But that is self evident. This can be applied using any set of increments, the smaller the increment, the greater the amount of numbers we have, without affecting the result.
pi is single number and while the notation of it is infinitely long, it cannot be considered a set of numbers, because the number 3 is repeated ad infinitum and can only be counted once in the set. Moreover, while the string of pi is infinitely long, its actual value is less than 4. The same can be said to the decimal notation of 1 1/3=1.333333333….ad infinitum, or any open ended decimal number.
What is the point of all this? Sounds to me just a theoretical mental exercise, without real application.

To me, infinity (in mathematics) means the sum total (either by counting or by the sum product.) of all numbers and all sets and subsets of numbers, regardless if they are “countable” or “uncountable”. Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.

To me, infinity (in mathematics) means the sum total (either by counting or by the sum product.) of all numbers and all sets and subsets of numbers, regardless if they are “countable” or “uncountable”. Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.

I agree, because mathematics is a calculus. If an algorithm of counting comes out with conflicting results, then either the algorithm of counting is wrong or we have miscounted.

Let us consider the odd numbers of (O) as oranges, the even numbers of (E) as apples and the natural numbers of (N) as orange and apples.

For finite sets, using one-to-one correspondence to determine whether there are more oranges, apples or oranges and apples, it is obvious since there are no apples in (O) and no oranges in (E) while (N) has both oranges and apples, therefore (N) must be bigger than either (O) or (E).

However, for infinite sets, the elements of (N), (O) and (E) are inexhaustible, by substituting apples for oranges and vice versa when we compare the size of (N), (O) and (E) (using one-to-one correspondence), the result is (N)=(O)=(E), which is how Cantor defined infinite sets.

Why is there a contradiction in the two results? Cantor wrote that as we “ascent into the realm of the infinite, we must accept this paradoxical result”. Are we asked to accept this absurd result which only satisfy Hume’s Principle, but contradicts Euclid’s principle: The whole is greater than the part? Has he any evidence to justify this claim or is it just an arbitrary rule so that he can define an infinite set?

So, where is the catch here?

1. We must compare apples with apples or oranges with oranges, not apples with oranges and vice versa. This rule must be applied rigorously to determine the size of sets with distinct identical elements.

2. Infinity is not a number, like 99999 and therefore cannot be treated as such.

3. An infinite set is a recursive rule to generate elements, it is not a list like the elements of a finite set. One should not compare a recursive rule to a list. They are different things.

4. Infinity is not defined precisely and cannot be defined precisely, otherwise it is not infinite. Therefore, one cannot proceed to (mathematically) treat infinity as though it is a known mathematical entity.

5. Infinity + infinity is meaningless (does not make any sense).

Cantor’s definition of an infinite set as (N)=(O)=(E) is wrong. The whole edifice of infinite mathematics is based on the wrong assumption of only using Hume’s Principle to compare the size of sets with distinct identical elements, (O), (E) and (N).

In sum, because a mathematical extension is necessarily a finite sequence of symbols, an infinite mathematical extension is a contradiction-in-terms.

Since a mathematical set is a finite extension, we cannot meaningfully quantify over an infinite mathematical domain, simply because there is no such thing as an infinite mathematical domain (i.e., totality, set), and, derivatively, no such things as infinite conjunctions or disjunctions.

Perhaps an infinity of infinities: where each infinity set is equal to all other sets and equal to the whole…..?

I don’t think so, no. Since there are different orders of infinities (an infinite number of different orders, in fact), “each infinity” is not going to equal “all other sets”, nor is going to equal “the whole”. Again, I forget the details, but IIRC it was impossible to construct a meaningful symbol that would encompass “the whole” or “all the infinities”, in the same way that people have constructed symbols for some of the lower orders of infinities.

In elementary set theory, Cantor’s theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself.

But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite.

And so, it goes on and on….....an infinity of infinities.

To me, infinity (in mathematics) means the sum total (either by counting or by the sum product.) of all numbers and all sets and subsets of numbers, regardless if they are “countable” or “uncountable”. Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.

Correct, mathematics is a calculus. If a scheme of counting comes out with conflicting results, then either the scheme of counting is wrong or we have miscounted.

Let us take (O) as oranges, (E) as apples and (N) as orange and apples.

For finite sets, using one-to-one correspondence to count whether there are more oranges, apples or oranges and apples, it is obvious since there are no apples in (O) and no oranges in (E) and(N) has both oranges and apples, therefore (N) must be bigger than either (O) or (E).

However, if the elements of (N), (O) and (E) are inexhaustible,

therefore, there is an inexhaustible supply of oranges or apples. Why not substitute apples for oranges and vice versa when we compare the size of (N), (O) and (E) (using one-to-one correspondence)? The result now, is (N)=(O)=(E) which Cantor defined as infinite sets.

Why is there a contradiction in the two results? Cantor wrote that as we ascent into the realm of the infinite, we must accept this paradoxical result. Are we asked to accept this absurd result which contradicts Euclid’s principle: The whole is greater than the part?

So, where is the catch here?

1. We must compare apples with apples or oranges with oranges, not apples with oranges and vice versa. This rule must be applied rigorously.

Thank you for that clear and concise analysis. (kinda like the principles of double entry bookkeeping)
For efficiency, one could also just exchange and switch the set letters and definitions of (O) and (E) instead of trying to match all those apples and oranges one-to-one.

Perhaps an infinity of infinities: where each infinity set is equal to all other sets and equal to the whole…..?

I don’t think so, no. Since there are different orders of infinities (an infinite number of different orders, in fact), “each infinity” is not going to equal “all other sets”, nor is going to equal “the whole”. Again, I forget the details, but IIRC it was impossible to construct a meaningful symbol that would encompass “the whole” or “all the infinities”, in the same way that people have constructed symbols for some of the lower orders of infinities.

In elementary set theory, Cantor’s theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself.

But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite.

And so, it goes on and on….....an infinity of infinities.

What you have quoted is true, but doesn’t refer to my point. What it’s saying is that the number of infinities is uncountably infinite. We already knew that. The point is that it is impossible to write a symbol within the mathematics that refers to them all. (Again, IIRC from logic class).

To me, infinity (in mathematics) means the sum total (either by counting or by the sum product.) of all numbers and all sets and subsets of numbers, regardless if they are “countable” or “uncountable”. Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.

Not sure what this means. But if you mean that the infinity of the reals is the same size as the infinity of the integers, then this has been proven wrong and is necessarily false.

What is the point of all this? Sounds to me just a theoretical mental exercise, without real application.

This is often said about theoretical mathematics. It was said before about imaginary numbers (those that are the square root of negative one), and yet real applications were later found for them.

It is not a good idea to claim that theoretical mental exercises in mathematics cannot have real applications.

(and for all others who want to amuse themselves with infinity. If the link does not work, search it manually.)

From the article:

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.

(...)

Some find this state of affairs profoundly counterintuitive. The properties of infinite “collections of things” are quite different from those of finite “collections of things”.

Thank you for that clear and concise analysis. (kinda like the principles of double entry bookkeeping)
For efficiency, one could also just exchange and switch the set letters and definitions of (O) and (E) instead of trying to match all those apples and oranges one-to-one.

As an accountant who is used to analyzing accounts, I think you have the credentials to be a good philosopher/mathematician. Your suggestion to switch is neat. FYI, Luca Pacioli, mathematician and Franciscan friar, codified the double entry accounting system and he is widely regarded as the “Father of Accounting”.

Philosophers/mathematicians, however, like to use fancy sounding counting techniques like one-to-one correspondence to explain their theories. However, if (O) and (E) do not have distinct elements to match distinct identical elements of (N), then one-to-one correspondence must be used to determine size.

Thanks, GdB. I am familiar with the fascinating Hilbert’s Grand Hotel paradox. In reality, there is no such hotel, it only exists in the mind of Hilbert (who was a staunch supporter of Cantor) as an illustration of the weird property of the infinite. It is like imagining a golden mountain. Just because it is conceivable (like a golden mountain) does not make it real.

What is the point of all this? Sounds to me just a theoretical mental exercise, without real application.

This is often said about theoretical mathematics. It was said before about imaginary numbers (those that are the square root of negative one), and yet real applications were later found for them.

It is not a good idea to claim that theoretical mental exercises in mathematics cannot have real applications.

Are there any real applications for the mathematics of the infinite found since Cantor proposed and worked it out after more than a hundred years? Perhaps, in the field of quantum vagueness or Turing’s Halting problem?

I say this not to denigrate pure mathematicians. However, one should remember that mathematics is now considered as a creative human invention (cf. Wittgenstein), like music, literature, art and mathematicians can have “mathematistic license”. OTOH, pure mathematical proofs are getting so long and complex that eventually only computers can complete them.

All one can ask of the formal computer verification of proofs is that they perform better than human beings, in the sense that they find mistakes in proofs that humans have missed and that humans recognize once they are pointed out. In the field of software and chip design verification this has already happened,and it is to be expected that it will become more common in mathematics itself.

A number of mathematicians are very concerned about where this revolution is leading us. If the goal of mathematics is understanding, then one cannot deny that computer-assisted proofs do not supply it in full measure.

Hilbert’s goal of achieving perfect certainty by the laying of firm foundations died with Godel’s work, but the problem of complexity would have killed his dreams with equal finality fifty years later.