One is free to propose different axioms that prove differently. — TonesInDeepFreeze

Since there is a whole lot of difference between the different types of numbers you outline, I think it a very good idea for a mathematician to look for a whole new set of axioms to better deal with the problem of having different types of numbers. This could avoid the problem of needing principles to relate the different types of numbers to each other, in an attempt to reconcile the sometimes irreconcilable difference between them. Attempting to reconcile the incompatibility between them tends to create a new type of infinity. So every time a new type of number is produced to deal with a specific problem that has arisen, a new type of infinity is produced. One could get rid of a whole lot of unnecessary complexity with a more comprehensive set of axioms..

The answer to your problem is quite simple. In mathematics things are done by axiom. If you want to count to infinity and beyond, simply produce an axiom which allows you to do that, and bingo the infinite is countable, and you're ready to go beyond. Look closely at the following: — Metaphysician Undercover

My belief is that we can't just produce axioms. We can only recognise truths about Existence such as 1 add 1 equals 2 or the angles in a triangle add up to 180 degrees or one cannot count to infinity.

But having a different concept and definition of infinitude doesn't thereby entail that there is a contradiction in set theory or mathematics. — TonesInDeepFreeze

You see, my position is that there is only one semantic/definition for the label "infinity". If we are focused on different semantics, we are not talking about the same thing. I think it would then help if we don't use the same label for that thing so as to make it clear that we are talking about different semantics.

Again, yes, there may be a contradiction between set theory and certain other formulations. But that does not entail that there is a contradiction within set theory. — TonesInDeepFreeze

I think a belief or theory has to be consistent with Existence as a whole, and not just consistent in isolation. To me, by definition, any theory or belief that encompasses the following belief "the set of all sets is contradictory" is a contradictory belief. It would be like any theory or belief encompassing the belief that "triangles are not triangular", which is contradictory belief to encompass.

set theory does not refer to an object named 'infinity' but rather to the property of being infinite, which is a crucial distinction. — TonesInDeepFreeze

Again, I think you are focused on a different semantic to me. To me, the semantic of infinity is one quantity or quality. It is a quantity or measure or quality that can never be reached. My position is best summed up with the following:

The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite. The only thing I view as Infinite, is Existence.

or the angles in a triangle add up to 180 degrees — Philosopher19

You should see non-Euclidean geometry where the angles in a triangle can be more or less than 180 degrees.

where there are an infinite number of elements (including fractions and irrationals) between 0 and 1 — punos

Compare the following:

1) There are an infinite number of elements between 0 and 1
2) There is no end to the number of elements between 0 and 1

If there is no end to something, how can another thing with no end be twice as large as it? Don't they both have no ends?

This is why there is a distinction between something that can go on forever and something that is infinite. Infinity allows for things like a number sequence to go on without end, but the thing that goes on without end is not infinite, it just goes on without end without actually reaching infinity just as one cannot count to infinity and reach infinity even if one was projected to count forever (so the number sequence is not infinite).

You should see non-Euclidean geometry where the angles in a triangle can be more or less than 180 degrees. — Michael

Imperfect triangles are imperfect by definition. I'm focused on absolutes.

My belief is that we can't just produce axioms. We can only recognise truths about Existence such as 1 add 1 equals 2 or the angles in a triangle add up to 180 degrees or one cannot count to infinity. — Philosopher19

Axioms are simply produced, created. The ones which prove to be useful are put to use, and they persist by becoming conventional. "Truths about Existence" is irrelevant to the mathematicians who create axioms.

1 add 1 equals 2 — Philosopher19

In binary, 1 + 1 equals 10. In mod1 arithmetics, 1+1 equals 0.

Suppose someone produces an axiom. Will it not be the case that that axiom will either be contradictory in relation to certain truths or consistent in relation to certain truths? Existence determines what is true and what is false. Whether any belief or axiom highlights truths or is contradictory to truth is determined by Existence/Truth. If not, there is no truth or semantics to work with to deduce further truths.

Imperfect triangles are imperfect by definition. I'm focused on absolutes. — Philosopher19

What do you mean by an "imperfect" triangle?

What do you mean by an "imperfect" triangle? — Michael

What you may call a non-euclidian triangle, I call an imperfect triangle. A perfect triangle has perfectly straight lines and its angles add up to 180 degrees. Another shape may resemble this without actually perfectly being this (like an imperfect triangle whose angles don't add up to 180 degrees but is near)

So all you're saying is that in Euclidean geometry the angles of a triangle add up to 180 degrees. And they do so because of the axioms of Euclidean geometry.

The angles in a true triangle add up to 180 degrees because that is the nature of Existence. It is not because someone said it or highlighted it.

The angles in a true triangle add up to 180 degrees because that is the nature of Existence. — Philosopher19

What is this supposed to mean?

What is this supposed to mean? — Michael

I think it's clear enough, therefore, I don't want to clarify further.

If there is no end to something, how can another thing with no end be twice as large as it? Don't they both have no ends? — Philosopher19

I believe the the concept of infinity is often misunderstood because it can be applied to different contexts, such as time and space, which are not necessarily equivalent. To explore the differences in sizes between different infinities, let's consider a few thought experiments that illustrate how infinity can vary in magnitude.

First, imagine you have achieved immortality and are presented with two options: to receive $1 every day forever or $1 every year. Intuitively, you would choose $1 every day because, over the same infinite duration, you would accumulate more money. This illustrates that while both options extend to infinity in time, the rate at which you receive money differs, leading to a larger "size" of wealth in one scenario over the other.

Now, let's consider a spatial analogy. Imagine two pipes, both of infinite length, but one has a diameter of 1 inch and the other has a diameter of 10 inches. Despite their lengths being equally infinite, the pipe with the larger diameter has a greater volume. This demonstrates that even with one dimension being infinite, other finite dimensions can contribute to a difference in "size" or capacity.

Interestingly, if we were to expand the diameter of the pipe to infinity as well, we would lose the essence of what makes a pipe a pipe. To maintain its identity, certain characteristics, like diameter, must remain finite. This constraint allows us to differentiate between pipes of different diameters, even if their lengths are infinite.

Lastly, consider an infinite number of pencils, each 6 inches long, laid end to end to form a line of infinite length. If we compare this to another line composed of an infinite number of 3-inch pencils, both lines would stretch to infinity. However, if you were to take one pencil from each line, there would be a clear difference in their lengths. This paradox highlights that while the total lengths of both lines are infinite, the "size" of their components is different, and this difference is observable when comparing individual elements.

So, the concept of infinity can indeed vary in magnitude depending on the context. Temporal infinity can differ based on the rate or frequency of an event, while spatial infinity can vary when other dimensions are considered. These examples show that not all infinities are created equal, and it is the nuances in their properties that allow us to distinguish between them.

You did a very good job covering the details of the formal math. About 8 hours ago. Again, helpfully.

I'll give my perspective. I think the way this math physically exists is only by physical brain state that is able to support it. So not everyone is going to be at the level of the math people.

Given there is a lot to know, my approach is to model it as mental algorithms to get a bird's eye view:

Brain; (Algorithm 1)
Brain; (Algorithm 2)
Brain;.(Algorithm 3)

And so on.

Obviously the math people pick up on a lot of these that the rest of us don't have, but for all of us, picking up on as many of these little recipes as we can can be a good strategy.

If this is so, then none of these concepts have any existence outside our brains. What we see and should expect is a lot of variation in approaches to problems unless they are standardized such as in formal math.

The angles in a true triangle add up to 180 degrees because that is the nature of Existence. It is not because someone said it or highlighted it. — Philosopher19

It's pretty clear you just make it up as you go along, and in retreat resort to the default tactic of, "That's just how I define it," insisting at the same time that how you define it is the way it is. You accomplish nothing in this but to build and reinforce your own redoubt of ignorance, which to remain within is the definition of stupidity.

As if in playing chess you changed the rules, yet at the same time insisted you were playing chess. And one can only wonder why anyone would bother to do that.

If I count 1, 2, 3, 4 ad infinitum, will I reach infinity? One cannot count to infinity, and even if something like a number sequence goes on forever, it will not reach infinity. — Philosopher19

You cannot start counting 1,2,3,4,... ad infinitum and reach somewhere, anywhere. Infinity has neither a start or an end.
Then, counting (natural) numbers you can never reach infinity because that infility would be also a number, and infinity is not a real or natural number.

To call {1,2,3,4,...} an infinite set is to imply that {1,2,3,4,...} consists of an infinite number of numbers. No doubt, even if 1, 2, 3, 4 goes on forever, an infinite number of numbers will never be reached. — Philosopher19

A set is a collection of objects (elements, members). I'm not sure if we can talk about an infinite set, although there are some theories about it (e.g. Zermelo–Fraenkel).
As I see it, an infinite set cannot be defined as one consisting of infinite numbers, because only the fact of being defined (limited) makes it (de)finite. An infinite set would be something limitless, hence undefinable.

All this raises questions about the infiniteness of the Universe, whether it started (created) from something or it always existed, etc. And, as I see it, since we don't have a proof that it is created from nothing, it must have always existed, even in the form of extremely high density and temperature, which at some point exploded (re: Big Bang), or in any other form. But I'm not the right person to talk about these things.

And, of course, I'm well aware that all I said is subject to debate ...

The angles in a true triangle add up to 180 degrees because that is the nature of Existence — Philosopher19

But Euclidian triangles don't exist in nature.

Another example that just came to mind related to Hilbert's infinite hotel thought experiment.

Consider a hotel with an infinite number of rooms, all of which are occupied. Due to the infinite guests, there are no vacant rooms. However, if each room in the hotel were to magically double into 2 rooms, the hotel would then have an additional infinity of rooms to accommodate an extra infinity of guests. Although the number of rooms seems the same in both cases, the capacity differs in some sense. In the first case, no more guests can be accommodated, while in the second case, an additional infinity of guests can be accommodated. This doubling (spacial sense) can continue (temporal sense) indefinitely in both time and space.

That's not Hilbert's paradox. There's no "magically doubling into 2 rooms" or anything like that. It's simply that whoever is in room 1 moves into room 2, and whoever is in room 2 moves into room 3, etc. In other words, each guest moves up a room. This leaves room 1 empty, ready for a new guest.

That's not Hilbert's paradox. — Michael

Yes, i am aware of that, but i didn't see the point in describing something one could just read anywhere. I was trying to show a different way of conceptualizing different sizes of infinities. That's all, but i'm more interested in if my example is a reasonable one or not.

Is it?

With yours, you confuse cardinality with simple quantity or amount. Transfinite cardinals are a generalization of quantity or amount. Thus "counting" into the transfinite is just plain different from ordinary counting, and you might use ordinals instead, but transfinite ordinal arithmetic comes with its own rules, and they're neither intuitive nor simple.

Yea, like i said im not up on all the terminology. I'm a little bit motivated now to look a little deeper into it, because i do find it interesting. I'm going to look up some of these concepts you mentioned like transfinite cardinals, and ordinal arithmetic. But i'd like to ask.

What was i describing in my last example about the infinite hotel. What is the correct terminology for what i described?

My view is that there might be no single concept of infinity. People talk about infinities using informal language, using mathematical language, and in the context of physics. If all coherent concepts of infinity turn out to be equivalent, this will be a surprise, and it must be demonstrated.

The statement that some infinities are bigger than others comes from set theory. The OP talks about infinities in the context of counting procedures. These are two different concepts of infinity.

In set theory, infinities are just infinite sets. In turn, infinite sets are those equinumerous to one of its proper subsets. That is, they have the same cardinality, which is defined by the existence of a bijective map between the two.

In set theory, an infinite set is bigger than another when there exists a surjective function from one to the other, but not vice-versa. What Cantor has proven is as simple as that: one cannot construct a surjective map from ℕ to ℝ.

Perhaps you would like to work with a different definition of cardinality, infinite set, or infinity (scraping sets altogether). That is fine, but keep in mind that you would be changing subjects, rather than disagreeing with set theory in general or Cantor in specific.

The statement that some infinities are bigger than others comes from set theory. The OP talks about infinities in the context of counting procedures. These are two different concepts of infinity. — DanCoimbra

Most of what i think about infinities comes from my own intuitions, so forgive me if i sound a bit ignorant of the well-established terms and procedures involved.

The "counting procedure" aspect is what i relate to the temporal sense of speaking or thinking about it. The other side seems to be more spatial in character, which instantiates an infinity all at once, outside time, so to say. It's just something i noticed recently and thought it might be useful to know when thinking about infinities. There are probably proper terms for these distinctions, and if there aren't then there should be.

Welcome to TPF! :smile:

Another:

Suppose that the universe has infinite space, and let's also say that there is an infinite number of particles in this space. For there to be space between the particles, would that not make space a bigger infinity than the infinite number of particles in the infinite space?

Notation I'll use:

'iff' for 'if and only if'

df. x is equinumerous with y iff there is a one-to-one correspondence between x and y

df. x is countable iff (x is finite or x is equinumerous with the set of natural numbers)

df. x is denumerable iff x is equinumerous with the set of natural numbers

'*' for cardinal multiplication

Your scenario can be boiled down to this:

There are denumerably many rooms. And if we multiply the number of rooms by 2, then there are still denumerably many rooms.

That reflects the set theoretic fact that if H is a denumerable cardinal and K is a countable cardinal, then H*K = H.

In this case, H is the number of rooms in the original hotel and K = 2.

First, imagine you have achieved immortality and are presented with two options: to receive $1 every day forever or $1 every year. Intuitively, you would choose $1 every day because, over the same infinite duration, you would accumulate more money. This illustrates that while both options extend to infinity in time, the rate at which you receive money differs, leading to a larger "size" of wealth in one scenario over the other. — punos

Thank you for that clear and easy to understand example.

When you say "extend to infinity in time", I assume you mean go on forever. It follows that both will forever add to their money. It also follows that one will always have more than the other. But it also follows that neither will ever have an infinite amount of dollars precisely because their wealth will not amass to infinity dollars. To say that it would is to say that one can count to infinity.

Now, let's consider a spatial analogy. Imagine two pipes, both of infinite length, but one has a diameter of 1 inch and the other has a diameter of 10 inches. Despite their lengths being equally infinite, the pipe with the larger diameter has a greater volume. This demonstrates that even with one dimension being infinite, other finite dimensions can contribute to a difference in "size" or capacity. — punos

I believe there is no contradiction in saying that something can go on forever. So I believe a pipe can go on forever. But to me, Infinity is the reason something can go on forever or be endlessly added to. It is not the measurement of the thing that goes on forever.

I think Infinite and Infinity should be exclusively used to refer to Existence, and a part of Existence is not equal to the whole of Existence. Trying to divide Infinity into parts seems contradictory to me. Another reason for why I think a pipe that measures infinite in length is an impossibility.