To me, Infinity and Existence denote the same. — Philosopher19

:roll:

According to the philosophy of intuitionism, a sequence that is said to be "without an end", is only taken to mean a sequence that is without a defined end. This is similar to computer programming, where an infinite loop that is declared in a computer program is only interpreted to imply that the program is to be stopped by the external user rather than internally by the program logic.

So in intuitionism (and computer programming), the difference between a finite sequence and an infinite sequence is taken to be epistemic rather than ontological. From the point of view of the producer of the sequence who gets to control it's eventual termination, the sequence could be said to be "finite", whereas from the consumer's point of view who has no knowledge and control of the sequence's termination, the same sequence could be said to be "infinite", or better, "potentially infinite". Or even better, the word "infinity" can be deprecated and replaced by finer-grained terminology that precisely conveys the information that one has at one's disposal in a given situation, without committing to the idea that the information one has is complete.

Amateur (and even some professional) philosophers demonstrate a profound gullibility, in their face-value interpretation of mathematical symbolism. To believe that infinity means "never ending" in an absolute sense just because an upper bound is omitted from a definition, is like believing that a blank cheque cannot bounce.

I’m a bit late to the thread. Just to put you at ease, mathematical infinities are not the same thing as philosophical infinities. They are precisely defined and used as means to perform mathematical equations. Which is pretty much what sime has said.

In Cantor's system counting is not 1,2,3,...it is about a one to one correspondence-
1 -> a
2 -> b
3 -> c etc.
If an infinity can be matched in this way it is Aleph Null. If not, it is bigger.

Draw a circle on the X, Y axis with radius pi. All points on the circumference except 4 of them are irrational numbers. No others are rational, even though there is and infinity of both rational and irrational. This is because the irrationals are denser.

Draw a circle on the X, Y axis with radius pi. All points on the circumference except 4 of them are irrational numbers. No others are rational, — EnPassant

Assuming you mean the ordered pairs of real numbers that identify points on the circumference have at least one member, x or y, irrational, what are those four points? x^2+y^2=pi^2.

"What are the 4 points"
That was a mistake. I was thinking about another function. No rational points.

"x^2+y^2=pi^2" - x and y will be irrational which is why all end points on they hypotenuse (pi) will be irrational.
Here is something on it https://mathoverflow.net/questions/71305/shortest-irrational-path

Two New Infinities Discovered

Here's an introduction to the exacting infinity and the ultra-exacting infinity.

New Scientist described exacting cardinals as being so large that they contain copies of themselves—sort of like a house with many full-scale copies of itself inside. Ultra-exacting copies additionally include mathematical rules on how to create them “as if the nested house was also wallpapered with blueprints of itself.”

How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached? — Philosopher19

There are "countably many" integers. That doesn't imply they can all be counted, but one can map a counting process to the set of integers. In the real world, that process would never end.

On the other hand, the real numbers can't be counted. There are infinitely many numbers between 1 and 2. In fact, there are infinitely many real numbers between any 2 real numbers. This is the rationale for stipulating that there are "more" real numbers. It's not "more" in the real-world sense of your intuitions; it's "more" in a mapping sense.

There are "countably many" integers. That doesn't imply they can all be counted, but one can map a counting process to the set of integers. — Relativist

Agree with all of the above. But you can't map one infinity to another with one being bigger than another because there isn't more than one. Infinity is one number just as 10 is one number. 10 and infinity do not come in different quantities.

There are infinitely many numbers between 1 and 2. In fact, there are infinitely many real numbers between any 2 real numbers. This is the rationale for stipulating that there are "more" real numbers. It's not "more" in the real-world sense of your intuitions; it's "more" in a mapping sense. — Relativist

if you map all the numbers from 1 to 2 to all the numbers from 1 to 4 or from any number to any number you would get infinity. Or rather you would get the possibility of an infinite number of numbers. But you'll never successfully map all the numbers from 1 to 2 to all the numbers from 1 to 4 because you cannot count them all. This is because of infinity. Infinitely speaking, there are no more numbers between 1 to 4 than there are between 1 to 2, but finitely speaking, there are more numbers between 1 to 4 than there are between 1 to 2.

Agree with all of the above. But you can't map one infinity to another with one being bigger than another because there isn't more than one. — Philosopher19

Infinity is not a thing that exists. It is a concept, and when it is applied to sets - it can lead to inconsistencies. There are infinitely many integers and infinitely many real numbers, but infinity is not a member of either set. Rather, "infinity" is a property of each of these sets. But is it the same property in both sets?

We can compare sets by defining a mapping between them. There is a 1:1 mapping between the set of even integers and the set of all integers. So although it may seem like there "more" integers than even-integers, that's not the relevant comparison. The comparison that is made is based on abstractly mapping the members of one set to the other. In this example, each integer can be mapped 1:1 to the set of even integers. 1->2, 2->4, 3->6...The mapping applies to all members of both sets; no members are left out.

However, there is no 1:1 mapping between the reals and the integers. Reals map into integers, covering all the integers, but you can't cover all the reals with integers. This is the basis for saying the "size" of the set of reals is greater than the "size" of the set of integers. The formal term for "size" is cardinality: the cardinality of the set of reals is greater that the cardinality of the set of integers. This is the basis for saying there are "more" reals than integers, but this isn't "more" in the everyday sense of the word.

However, there is no 1:1 mapping between the reals and the integers. Reals map into integers, covering all the integers, but you can't cover all the reals with integers. This is the basis for saying the "size" of the set of reals is greater than the "size" of the set of integers. — Relativist

This is where I disagree. I don't believe Cantor's diagonal argument shows anything. Infinity is one cardinality/size, it makes no sense for one infinity to be bigger than another in terms of size.

This is where I disagree. I don't believe Cantor's diagonal argument shows anything. Infinity is one cardinality/size, it makes no sense for one infinity to be bigger than another in terms of size. — Philosopher19

What's the basis for your claim that it makes no sense?

What's the basis for your claim that it makes no sense? — Relativist

It makes no sense for one quantity of 10 to be bigger than another quantity of 10. 10 is one quantity. Similarly, it makes no sense for one quantity of infinity to be bigger than another quantity of infinity. Infinity is one quantity.

An interesting question to think about that might help in regards to different sizes of infinity:

What is the length of the circumference of a circle with a radius of infinity?
The circumference would need to be 6.283185307... (Tau) times the size of the infinite radius.

If you can conceive that an infinite radius can form a circle, then it would logically make it necessary that the circumference be at least 6 times the size of the infinite radius. Perhaps this makes sense? If this is inconceivable to you, then disregard the suggestion.

It makes no sense for one quantity of 10 to be bigger than another quantity of 10. 10 is one quantity. Similarly, it makes no sense for one quantity of infinity to be bigger than another quantity of infinity. Infinity is one quantity. — Philosopher19

In the everyday use of the term, a "quantity" is always a fixed, real number (e.g. a number of liters, a number of tomatoes, a number of molecules in a mole...). Infinity is not a real number. Your mistake seems to be that you're treating it as one.

I posted an earlier draft of this comment by mistake. This one is less of a mess.

It makes no sense for one quantity of 10 to be bigger than another quantity of 10. 10 is one quantity. Similarly, it makes no sense for one quantity of infinity to be bigger than another quantity of infinity. Infinity is one quantity. — Philosopher19

In the everyday use of the term, a "quantity" is always a fixed, real number (e.g. a number of liters, a number of tomatoes, a number of molecules in a mole...). Infinity is not a real number. Your mistake seems to be that you're treating it as one. — Relativist

Forgive me for jumping in, but I think what is needed here is a closer look at what is going on here. I'm not a mathematician, so I hope that you will correct anything I say that is not properly forumulated.
In essence, the problem is that the the normal concepts of number don't apply once one has defined inifite sets. So the mathematical concepts here look very strange unless one looks closely at how they change in this new context.

In one way, this situation is unique. But the concept of number in mathematics has changed several times as mathematics has developed. The ancient Greeks, for example, did not consider that either 0 or 1 was a number; that seems strange to us, but we have got used to the new concept 0 and all the many developments that have happened since the Arabic mathematicians changed everything.

As a start, look more closely at the original question?

How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached? — Philosopher19

The definition of size here is the number of members. It is true that the number of members of an infinite set can never be specified, and the set is uncountable in that sense. (But we can confidently assert that the number of members - and hence the size - of an infinite set is larger than any finite set.)
However, the definition of "countable" in the context of infinite sets is that the counting can be started, not that it can be finished.
There's a misinterpretation of "infinity". Inifnity is not a target that can never be reached, but the recognition that counting can never be completed, that it will always be possible to take another step in the series. It is, in my (non-mathematical) book not a number at all.

However, mathematicians work around this, but defining a new kind of number.

In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers.

See Wikipedia - Transfinite Numbers
This does not posit that there is a largest finite number. It is an application of concepts that clearly exist in the case of inifinite sets like [1, 0.5, 0.25, .....] In the case of those sets, there is a number that is smaller than any of the members of the set - 0. It is, paradoxically, not a member of the set. It is called the limit. The analgous numbers in the case of the natural numbers is <omega> or <aleph-null>. But you should read further to more fully see what is going on. The Wikipedia article is a reasonable starting-point.

One might feel that Cantor's argument does not demonstrate its conclusion. But it does demonstrate that the relationship will persist at each step along the way. A counter-argument would have to show that it will break down at some point, and I don't see how such an argument could be made.

I hope I said enough to show how the apparently impossible conclusion can be established. I'm sure someone will correct anything that I have not formulated properly.

Your reasoning is correct, but I'll try to give a mathematician's perspective (I received a B.S. in math in 1976, so I'm not really a mathematician - but I'm familiar with some foundational points). The real number system, natural number system, and transfinite math, are "mathematical systems".

Here's a good definition (from this source) of the term:

A mathematical system consists of:

1) A set or universe, U.[/b]

2) Definitions: sentences that explain the meaning of concepts that relate to the universe. Any term used in describing the universe itself is said to be undefined. All definitions are given in terms of these undefined concepts of objects.

3. Axioms: assertions about the properties of the universe and rules for creating and justifying more assertions. These rules always include the system of logic that we have developed to this point.

4. Theorems: A true proposition derived from the axioms of a mathematical system based on the axioms and derived by the logic.

In my abstract algebra course, I had to learn about a variety of mathematical systems (e.g. groups, rings, fields), that have no relationship to the real world, and to prove theorems about them. As long as the system has the 4 components, it's valid math.


The real number system and integer number system are mathematical systems, and both of these relate directly to the world. They also relate to each other: the integers are a subset of the reals.

Transfinite math is just another mathematical system, and it's one with no direct relationship to the real world. It has an indirect relationship, in that it pertains to the sets of integers and the sets of reals - but that doesn't mean all the concepts of the reals&integers are applicable.

The mistake is to treat all of mathematics as a single system, with a single set of axioms and definitions.
Example: the ordinal numbers (the ordered set of integers) have a "successor function" - an operation that produces the next integer: "+1" There is no successor function with the reals, because there is no "next" real number.

More to the point, the real number axioms don't apply to transfinites. What matters is that there is a universe (the transfinite numbers) and that there are operations that can be performed with them - including a successor function for the transfinite ordinals - which allows treating them as greater than or less than.

It's still true that there is a conceptual relation between the transfinites and the reals and integers, and that was the basis for Cantor defining them. But it needs to be remembered that definitions (like "greater than" "less than" etc) are intra-system.

I'm glad you agree. Nor do I find fault with the definition of a mathematical system. However, there do seem to be some differences of perspective and approach between us; these are not necessarily questions of right and wrong, true or false.
I would not want to say that @Philosopher19 is wrong - just that the argument is based on a different approach to the idea of infinity. Specifically, there is a different understanding of what "countable" means even though there is a common understanding of what "size" means and that infinity means that there is no number that is the number of the members of an infinite set.
There is a constant tension here around the fact that counting cannot be completed and the temptation or desire to think of the infinite as some sort of destination or limit. Compare "the sky's the limit" or "the gold at the end of the rainbow" or even the concept of "transfinite numbers". I don't see that there is a basis here of talking of "correct" or "incorrect" or of mistakes - that requires a shared agreed system, which is not available.

The mistake is to treat all of mathematics as a single system, with a single set of axioms and definitions. — Relativist

I'm suggesting that, in the face of the concept of infinity, there is more than one way to apply the relevant concepts. If we can choose one way rather than another, we cannot apply correct and incorrect. We need a different kind of argument.

What matters is that there is a universe (the transfinite numbers) and that there are operations that can be performed with them - including a successor function for the transfinite ordinals - which allows treating them as greater than or less than.
It's still true that there is a conceptual relation between the transfinites and the reals and integers, and that was the basis for Cantor defining them. But it needs to be remembered that definitions (like "greater than" "less than" etc) are intra-system. — Relativist

But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion?

But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion? — Ludwig V

Actually, because the reals and integer systems are applicable to the real world (they were developed by analyzing aspects of the real world), the terms "greater than" and "less than" do apply meaningfully.

The transfinite system was not developed directly from real world analysis, but from analysis of implications of sets.

There is a constant tension here around the fact that counting cannot be completed and the temptation or desire to think of the infinite as some sort of destination or limit. — Ludwig V

Agreed- it results in people treating infinity like a natural, or real, number. Then when non-mathematicians hear of transfinite numbers, it reinforces that false view - because it turns infinities into "numbers" but only in a very specialized sense.

To me, Infinity and Existence denote the same. — Philosopher19

Existence is, from the beginning. It is eternal and infinite. That which exists is eternal. Finite things are events in eternity.

But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion? — Ludwig V

Actually, because the reals and integer systems are applicable to the real world (they were developed by analyzing aspects of the real world), the terms "greater than" and "less than" do apply meaningfully. — Relativist

Yes, I put that very badly. What I was getting at was that "largest number" or "smallest number" is not defined, or rather, the possibility is excluded by the definitions of "greater than" or "less than", or, more accurately, by the absence of any definition of "largest" or "smallest".
BTW, I'm actually not entirely happy that "number greater than (smaller than) every other number" is not a definition of "greatest" or "smallest". But I have to accept that in the context of mathematics, the rules allow it.

Infinity is one number just as 10 is one number. — Philosopher19

I'm afraid that, although I can see the sense of your conclusion, I do think your argument is mistaken at this point. My reason for accepting your conclusion is that infinity is not a number, so comparisons of size are meaningless.
The concept of "infinity" is a bit like the concept of the horizon. The horizon seems to be located in space, but as you approach it, it recedes; you can never get there. Each step in the number sequnce seems to get you closer to infinity, but it is always infinitely far away. More to the point, any attempt to apply ordinary arithmetic produces nonsense. If infinity was a number, you could add 1 to it and produce a larger one.
What one chooses to make of the transfinite numbers is another question, but, for the purposes of mathematics, I think we have to accept that they work in that context. But even they are not numbers of the same kind as the natural or real numbers.
As I said earlier, your definition of "countable" is different from the one used in mathematics, so your concept of infinity is different from the mathematical concept. It is not a question of right or wrong, but of different ways of thinking.

Classical mathematics doesn't need "infinity" as a sort of number with associated features. The limit concept works well. However, modern math defines the term(s) and goes into abstracta.