Georg Cantor thought so ...

The question is not well formed. It is not apparent what "infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely… (etc.) infinities" means.

But here are exact statements that might answer what the poster is wondering about:

Any set of all the real numbers in any non-empty interval is infinite.

Any set of all the real numbers in one non-empty interval is equinumerous with the set of all the real numbers in any other non-empty interval.

There is no greatest cardinality.

For any infinite set of cardinalities, there are greater infinite sets of cardinalities.

does that mean that there are infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely… (etc.) infinities? — an-salad

Hopefully this doesn't contradict what @TonesInDeepFreeze has said. It seems that if the word "infinity" was being used as a noun, then yes, there would be an infinite number of infinities. However, the word "infinity" is not being used as a noun, but rather is being used as an adjective, in which case there is only one infinity. IE, "infinity" as an adjective means something along the lines "any known set of real numbers can be added to".

if there are infinite whole numbers, and there are infinite decimals between 0 and 1, and there are infinite decimals between 0.1 and 0.12, and there are infinite decimals between 0.1111111 and 0.1111112, and (etc.) does that mean that there are infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely… (etc.) infinities? — an-salad

Georg Cantor thought so ... — 180 Proof

As @180 Proof said, set theory goes like that. And since you gave in the example of just rational numbers (0,1111111 and 0,1111112,..) then this is equivalent to the infinity of natural numbers, a countable infinity. With real numbers we get into the more interesting questions.

And @TonesInDeepFreeze correctly asked you just what you mean by "infinitely infinite...". We are puzzled what you mean by this. But before you answer that, please read the following:

You see, it comes down to if can you have a way to count, at least theoretically, those infinities themselves, then they can be put into 1-to-1 correspondence with the Natural Numbers. Then it's easy. Just how different the math is, you can see for instance from the example of Hilbert's Hotel.

Here's an easy primer on this short video:

Yes, there are an "infinite number" of infinite sets:

$\{0.1,\text{ }0.01,\text{ }0.001,\text{ }...\}\\\{0.2,\text{ }0.02,\text{ }0.002,\text{ }...\}\\\{0.3,\text{ }0.03,\text{ }0.003,\text{ }...\}\\...\\\{0.11,\text{ }0.011,\text{ }0.0011,\text{ }...\}\\\{0.12,\text{ }0.012,\text{ }0.0012,\text{ }...\}\\\{0.13,\text{ }0.013,\text{ }0.0013,\text{ }...\}\\...$

Wee, another thread on infinity.
Yes, that follows from the axioms of "standard mathematics". You can build any sort of mathematics (if you wanna call it that) depending on what axioms you choose, the matter is whether it is useful to do so and whether it matches at least something in reality.
There is a retired Australian professor of mathematics called Norman Wildberger, whose project is to build mathematics without mention of infinity — within the doctrine of finitism. I would not really recommend looking into it however if you are not deeply knowledgeable in mathematics.

You can build any sort of mathematics (if you wanna call it that) depending on what axioms you choose, the matter is whether it is useful to do and whether it matches at least something in reality. — Lionino

That's actually a philosophical view in mathematics. And thus quite well fits a Philosophy Forum.

But of course you can argue that the most permissive math is simply the one where we start with an axiom of 0=1.

Anything goes. Wee! :razz:

In Philosophy, they tend to analyse concepts and propositions for truth or falsity. That's what they do. End of the story.

But maybe the mathematicians and scientists do things differently. They don't ask what the concepts mean as long as they are in the textbook. They just accept them, and work on.

"INFINITY definition =1. time or space that has no end: 2. a place that is so far away that it cannot be reached:" - the Cambridge English Dictionary.

It implies that if you know what it is, then you don't know what it is. If you don't know what it is, then you know that you don't know. It is a paradoxical concept, which has to be branded as a contradiction in Philosophy.

That's actually a philosophical view in mathematics. — ssu

And it is a view that I hold, as a "methological physicalist" (as 180 proof puts it), I don't subscribe to abstract objects, so I could not be a platonist about mathematics :razz:

I will likely make a thread about the Grundlagenkrise in the coming weeks.

with an axiom of 0=1.

:gasp:

A: 0 = 1
B: 0 = S(0) (follows from definition)
C: The set of natural numbers only has the element 0 (follows from B)
C contradicts B. Proof of the contradiction? :smirk:

In Philosophy, they tend to analyse concepts and propositions for truth or falsity. That's what they do. End of the story.

But maybe the mathematicians and scientists do things differently — Corvus

I think you got it a bit wrong. Those who are obsessed about truth or falsity are mathematicians. Even if they sometimes have different axiomatic systems, then it's about right or wrong in that formal system.

It's the Philosophers who are interested about a lot more. Things like morals or aesthetics, which obviously aren't about truth or falsity.

I will likely make a thread about the Grundlagenkrise in the coming weeks. — Lionino

Great! Like to see that one...

But as a non-mathematician, try to keep it as simple and understandable, because the paradoxes are interesting. After all, it's everything to do with infinity.

I think you got it a bit wrong. Those who are obsessed about truth or falsity are mathematicians. Even if they sometimes have different axiomatic systems, then it's about right or wrong in that formal system. — ssu

I think you got it wrong too. Philosophers don't care about the truths and falsity as the answers in the answer sheets. Philosophers are more concerned with the truth and falsity in the concepts, propositions, and logic.

It's the Philosophers who are interested about a lot more. Things like morals or aesthetics, which obviously aren't about truth or falsity. — ssu

Yes, Philosophy used to be the parents of all sciences and mathematics. It is the mother of all subjects, and we cannot deny the fact.

I will likely make a thread about the Grundlagenkrise in the coming weeks. — Lionino

Blimey I was going to make a thread about "Science as a superstition".

Philosophers don't care about the truths and falsity as the answers in the answer sheets. Philosophers are more concerned with the truth and falsity in the concepts, propositions, and logic. — Corvus

? :yikes:

I don't get your point here.

? :yikes:

I don't get your point here. — ssu

Math and Science pursues the answers in the answer book. You are either right or wrong. Philosophy is more into your arguments and logic for the answers, hence there is no such thing as the answers in the answer book i.e. truth and falsity they pursue are different in nature.

Math and Science pursues the answers in the answer book. — Corvus

What answer book?

I think mathematics is especially interested in logic. I would dare to say that math is part of logic.

The starting foundations of Science accepts that we cannot find some ultimate truth, hence things are theories, not laws. We can in the find out something new that alters our present views. And mathematicians do understand that especially when you look at the foundations of mathematics, there are philosophical arguments and philosophical schools. Hence you have the philosophy of Mathematics.

Just look at wrote above. Now I don't know if he is a mathematician, but at least he totally understands that philosophy is part of mathematics.

What answer book?

I think mathematics is especially interested in logic. I would dare to say that math is part of logic. — ssu

Sure, not denying that at all. They are all parts of each other we could say that. They are all inter-related too. But the methodologies they employ and the ideas of their goals might be different depending on the folks who are doing them.

Just look at ↪Lionino wrote above. Now I don't know if he is a mathematician, but at least he totally understands that philosophy is part of mathematics. — ssu

Never said math is not part of philosophy. That is what you are saying for some reason.
I said math and philosophy have different way of doing things.

Math and Science pursues the answers in the answer book.
— Corvus
What answer book? — ssu

Have you not read a single math book? If you read any math book, it will have Exercises and Examples after or in the middle of a chapter. The answers for the Exercises will be either at the back of the book, or as a separate Answer Book that you must acquire, if you needed it.

Now I don't know if he is a mathematician, but at least he totally understands that philosophy is part of mathematics. — ssu

I am not a professional mathematician, but my area does use lots and lots of mathematics inherently, my interest in the foundations of mathematics are coincidental.

I think jgill is one though.

A question for mathematicians: looking at what I've done above, can this be written as a matrix like this?

$\begin{bmatrix}0.1 & 0.01 & 0.001 & \cdots \\ 0.2 & 0.02 & 0.002 & \cdots \\ 0.3 & 0.03 & 0.003 & \cdots\\\vdots & \vdots & \vdots & \vdots \\ 0.11 & 0.011 & 0.0011 & \cdots \\ 0.12 & 0.012 & 0.0012 & \cdots \\ 0.13 & 0.013 & 0.0013 & \cdots\\\vdots & \vdots & \vdots & \ddots\end{bmatrix}$

We can then say that $a_{m,n} = {m\over10^{n+\lfloor\log_{10}(m)\rfloor}}$?

We can then say that am,n=m10n+⌊log10(m)⌋��,�=�0�+⌊log10(�)⌋? — Michael

The series m/(10^(m+log(10,m))) converges to a power of 1/10 as m goes to infinity and n is any given number, and to 0 as n goes to infinity and m is any given number, and the diagonal also converges to 0. The two conditions that converge to 0 seem fine, as we are multiplying by a power of 1/10. But for the condition that converges to powers of 1/10 I am not sure.
So as you go down the column 1, it should converge to 1/10 and in column 3 to 1/1000. Is that what you were looking for? I am not sure if that is what is represented by your matrix.
Disclaimer: Not a mathematician.

But urgently, how do you write matrices and footnotes and equations here?

Have you not read a single math book? If you read any math book, it will have Exercises and Examples after or in the middle of a chapter. The answers for the Exercises will be either at the back of the book, or as a separate Answer Book that you must acquire, if you needed it. — Corvus

Umm... that's a school math book. Have you even studied a math course in the University? They are a bit different.

And if you study philosophy, you will similarly (hopefully) be given a exam where you have to answer too.

I assume that true math is more about giving proofs.

But urgently, how do you write matrices and footnotes and equations here? — Lionino

https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts/p1

Can there be infinite infinities?

Can there be an infinite set of (infinite set of numbers)?

The word "infinite" is not a noun but an adjective qualifying the noun "set".

Therefore, there can be infinite infinities because the word "infinity" is an adjective.

Umm... that's a school math book. Have you even studied a math course in the University? They are a bit different. — ssu

I have a few university Calculus and Algebra and Trigonometry books lying around here, and they are full of questions and answers. Studying math means you read the definitions in the books and work on the questions for the answers purely using your reasonings.

And if you study philosophy, you will similarly (hopefully) be given a exam where you have to answer too. — ssu

No. That is not the case. If you study philosophy for the degree, you must read, and write dissertations which you must defend it at a 'viva voce'.

Norman Wildberger, whose project is to build mathematics without mention of infinity — within the doctrine of finitism — Lionino

Where can one see the project?

Wildberger's video on set theory is atrocious, appalling, obnoxious intellectual dishonesty.

jquill — Lionino

jgill.

You make it sound like he's a sleep medicine.

https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts/p1 — Michael

Very nice, that should be pinned imo.

Therefore, there can be infinite infinities because the word "infinity" is an adjective. — RussellA

The fact that you can stack a property onto a substance to make an object does not mean that that object is instantiated in real life, especially because many objects are contradictory and cannot exist (rectangular circle or blue orange).

Where can one see the project? — TonesInDeepFreeze

No clue, I could not find it, I only know that he works on it lol

I said math and philosophy have different way of doing things — Corvus

They certainly do, which is why I’m wondering what a thread on mathematics is doing on a philosophy forum.

I’m wondering what a thread on mathematics is doing on a philosophy forum. — Joshs

The philosophy of mathematics is a rich area.

(1) Unfortunately, cranks, who are ignorant and confused about the mathematics post incorrect criticisms of the mathematics, from either a crudely conceived philosophical or a crudely imagined mathematical perspective. That calls for correcting their misinformation about the mathematics itself.

It is great to challenge classical mathematics, but a meanginful challenge needs to not misrepresent that mathematics. Otherwise the effect is inimical to knowledge and understanding of the subject.

(2) And sometimes people post questions about mathematical subjects that have bearing on philosophy, such as about infinities, incompleteness and computability. The debate on realism v nominalism has as one of its major battlegrounds the ontological status of mathematical objects, especially infinitistic ones. And some may think that questions in epistemology are informed by such things as the incompleteness theorem and the unsolvability of the halting problem.

Brouwer v Hilbert itself is one of the very great debates in the history of the philosophy of mathematics, carried on by two mathematicians.

/

Meanwhile, one could also ask what are threads on such things as the U.S. presidential election, Gaza, and candy bars doing in a philosophy website. (Don't get me wrong, I am in no way saying those should not be in this website. Very much I say live and let live.)