I read your work.
I can't give an opinion but you have put some work into it to your credit.

I think my issue is in mapping an infinity to a known finite. A one to one mapping will use up the finite and end. The unmapped trailing infinity becomes a useless appendage.

And there is the issue of logic working for the finite but failing in the infinite.

Interesting to see other people's opinions.

I might be thinking mapping an infinity to a larger infinity also leaves this useless appendage....not sure, just my instinct not real math. Still...parameters are arbitrary.

And if you want a better understanding of the issues here, see Chapter Four of Open Logic.

Unlike Philosophy Forum, it's guaranteed free of psychoceramics.

I can't give an opinion but you have put some work into it to your credit. — Mark Nyquist

Thanks Mark

I think it's clear that one cannot count to infinity So one cannot say that x is an infinite sequence of numbers just because it goes on forever. If I count forever I will not reach infinity. I cannot say assume I completed my count with this set of numbers and that set of numbers and then argue that that set is a bigger infinite set than the other.

Just think: how can one infinite quantity be bigger than another when the quantity of infinity is one quantity?

I think it's clear that one cannot count to infinity So one cannot say that x is an infinite sequence of numbers just because it goes on forever. — Philosopher19

How does one not laugh at this?

Two questions were asked, no answers were given:

How would a difference in size be established between two sets when there is no counting of the number of items in the sets involved?

If there is counting involved, how has one reached an infinite number of items?

I also asked an additional question:

If infinity is a quantity, how is it more than one different quantity?

If I ask how many items in that set and the answer is infinite and I ask how many items in that other set, it is surely contradictory for someone to say to me and even bigger infinity. There is no beyond one infinity for there to be the possibility of a bigger infinity.

How would a difference in size be established between two sets when there is no counting of the number of items in the sets involved? — Philosopher19

By bijection. See Open Logic Ch.4.

If there is counting involved, how has one reached an infinite number of items? — Philosopher19

"Counting", and ill-defined notion, is not involved in bijection, although "enumeration", a well-defined notion, is.

If infinity is a quantity, how is it more than one different quantity? — Philosopher19

See Cantor's diagonal argument.

By bijection. See Open Logic Ch.4. — Banno

A one to one to correspondence implies a count of one side compared to the other. But infinity is not reached or exhausted and cannot be counted to

"Counting", and ill-defined notion, is not involved in bijection, although "enumeration", a well-defined notion, is. — Banno

Is it not? Do you not count how many maps onto how many?

See Cantor's diagonal argument. — Banno

I have already seen. Tell me what about it suggests that infinity is more than one possible quantity despite it being the case that infinity is one semantic as opposed to two. Note that 5 is one semantic as opposed to two.

Yes, if you were to measure both distances at a specific point in time, but outside the context of a finite time measurement, the distance is probably equally infinite for both.

I think that to make sense of infinities, one has to have a system for extracting their finite properties, as I mentioned in my prior post, or by looking, for example, at the difference between one element in a sequence and the next, which has a specific finite value. This specific value for example can be considered a fundamental component of a periodically regular sequence, by which any periodically regular sequence can be constructed, including infinite ones.

Maybe it'sthe terminology. "Infinity" is just a very abstract term used usually informally and non-rigorously to refer to quantities either very large or larger than any specific quantity, or otherwise unlimited. As for technical definitions, I leave that to you to look up. And you can search transfinite cardinals.

There are many, many brief Youtube videos on this topic, and a lot of threads here you can search. And Banno's reference looks comprehensive. So much is available, and much of it so well-done, that there is no excuse for prolonged ignorance here. So, take flight and learn!

I think that's good advice for me. Seems like the direction of the discussion and the actual math departed ways.

The issue here is not one of logic, but of pedagogy. The logic is clear, there are multiple infinities. The issue is why some folk cannot see that to be the case, even when presented with the proof.

Consider:

A one to one to correspondence implies a count of one side compared to the other. But infinity is not reached or exhausted and cannot be counted to — Philosopher19

Is it that Philosopher19 has a picture of infinity such that, since one cannot count to infinity, one cannot have a grasp of infinity?

One way infinity is introduced to children by showing them that for any number, we can construct a bigger number - by adding one, or some other finite number. Then comes "Infinity plus one!". The child will have understood infinity not as something one counts to, but as the ability to carry on in the same way...

In a way, Cantor showed the child's "infinity plus one" to be a reality... :wink:

So Philosopher19 it seems has a notion of infinity that is dependent on actually counting to infinity, rather than "carry on in the same way...", and hence takes it as granted that a one-to-one correspondence must involve counting. Two approaches occur to me, when I put on my long-discarded teacher's hat: to show a variety of infinite one-to-one correspondences, making the point that we do not need to count them all, or even at all, to see that they go forever; and to look at infinity in other contexts - art, perhaps - in order to show that one can understand infinity apart from counting.

Anyway, we are not being paid to teach Philosopher19, so that goes by the by.

Usually, in mathematics we do not use 'infinity' as a noun. There is not an object that we call 'infinity'.*

Rather, we use the adjective 'is infinite'.

To define 'is infinite' we may:

First, define 'finite'. That can be done in various ways. One way is to define:

x is an ordinal if and only if (x is membership-transitive and x is well ordered by membership)

n is a natural number if and only if (n is an ordinal and n is well ordered by the inverse of membership)

x is finite if and only if there is a natural number n and a one-to-one function from n onto x

x is infinite if and only if x is not finite.

/

In mathematics, 'equinumerous' is defined:

x is equinumerous with y if and only if there is a one-to-one function from x onto y.

That corresponds to the utterly basic intuition that sets have the same number of elements if and only if they can be put in one-to-one correspondence. For example, proverbially, there are the same number of sheep in the flock as there are stones in the pile if and only if for each stone there is a corresponding sheep and no stone corresponds to more than one sheep.

Set theory uses this definition for both finite and infinite sets.

Given this definition, we have prove that the set of natural numbers is not equinumerous with the set of real numbers.

Now, if one has different intuitions about equinumerosity, then one may imagine set theory not to use the word 'equinumerous' but instead 'zequinumerous' or whatever, for mathematics does not depend on the words it happens to use but rather on the formal relations, no matter what natural language words we use to nickname those relations.

/

* But there also are the points of 'infinity' and 'negative infinity' on the extended real line. But these are not indications of CARDINALITY. They regard two elements in addition to the set of real numbers and an ordering that such that infinity is greater than every real and greater than negative infinity, and negative infinity is less than every real and less than infinity. The points themselves may be of any cardinality.

* We also use phrasing and/or the lemniscate, for example, "as n goes to infinity". But this is a facon de parler that can be explicated in various ways such as "n rangers over the natural numbers" or "the domain of the function is the set of natural numbers".

If you start with a set of integers 1 to a million and another set of integers one to infinity and pair one to one up to a million then the set of infinity unpaired is infinity minus one million which is meaningless and undefined.

That is still an issue not answered. Can logic apply to an undefined set.

I'm rusty at this but does someone know?

No doubt, even if 1, 2, 3, 4 goes on forever, an infinite number of numbers will never be reached. — Philosopher19

You can't apply natural language words to mathematics. Things like "goes on" and "be reached" mean nothing in mathematics, those phrases can only informally refer to real mathematical concepts such as addition or limits, otherwise you end up with gibberish like here.

1, 2, 3, 4... goes on forever. The verb go implies movement. Therefore the natural set of numbers moves through time and space!

...infinity minus one million which is meaningless and undefined. — Mark Nyquist

Infinity minus any finite number is still infinity. Doubtless others might make this informal answer rigourous.

I don't see what is "undefined" here, let alone "meaningless".

There is no definition of cardinal subtraction with infinite cardinals in a way such as with the integers.

So infinity minus one million is defined?
No it is not defined.

There is no object named 'infinity'. Rather, there is the property of being infinite.

There is not a "minus" operation involving infinite cardinals in the same way as with integers.

Rather:

For any set x, there is its cardinality called card(x).

Now let '\' stand for the complement operation:

x\y = {p | p in x and p not in y}

If x is infinite and y is finite then

card(x) = card(x\y)

That is meaningful and correct.

Saying x-y is not even meaningful.

If you take away a million from any set there are a million less elements. That has priority over abstract infinity. Maybe that has been my issue not addressed.

Again, it is not meaningful to say 'infinity' as if there is an object named by it.

Rather, there are various sets that have the property of being infinite.

And there is no "minus" operation as being bandied here.

Okay these are really just mental abstractions and it looks like your framework really is arbitrary.
I'm sure it's standardized academically but there are still problems.

We don't use 'less' in that sense with infinite sets.

Rather, if x is an infinite set, and y is a set with n number of elements, then there are n number of elements that are in x but not in {p | p in x and p not in y}.

Casually speaking you might say "there are n less elements" but to be mathematically accurate, we need to not use 'less' in that imprecise way and instead say " there are n number of elements that are in x but not in {p | p in x and p not in y}."

'less than' has a mathematically exact definition, and it is not used in the way being bandied here along with 'minus'.

You are perhaps right, but I don't see as it helps.

Again, the point is pedagogic, not logical. Here's the question:

If you start with a set of integers 1 to a million and another set of integers one to infinity and pair one to one up to a million then the set of infinity unpaired is infinity minus one million which is meaningless and undefined. — Mark Nyquist

One might understand this as: What is the cardinality of the integers that come after one million? It's still ℵ₀.

The framework is arbitrary just as definitions in general are arbitrary.

One can have whatever formalization of mathematics one wants to have and any definitions one wants to have. But if we wish to know exactly what is the case with the usual formulations and definitions in mathematics then we need to discuss them as they actually are formulated and defined in mathematics.

That's helpful.

It helps because talking about "minus" in that way is incorrect and leads to confusions. If we are talking with people unfamiliar with the arithmetic of infinite cardinals, and we go along with their mistaken notions born of incorrect use of the terminology, then we are doomed to their confusions and even allow them to score points to which they are not entitled.

And it first starts with using 'infinity' as a noun in a context such as this. That just sets up all kinds of misunderstandings.

And, yes for any natural number n, the cardinality of the set of integers greater than n is aleph_0.

Hmm. Maybe I misunderstood 's issue. I had taken him to be suggesting bijection would not do the task set it...

Is it that Philosopher19 has a picture of infinity such that, since one cannot count to infinity, one cannot have a grasp of infinity? — Banno

No I think we have a grasp of infinity or an awareness of the semantic. Some are more focused on this awareness than others. Some are more sincere to this awareness than others. Part of that awareness entails one cannot count to infinity. You can add one to any quantity, except of course infinity. Such is the nature of infinity. Yet, it seems to me that some seem to believe "beyond infinity" is meaningful.
What more can I say?

I believe I've said enough in this discussion and that beyond this is time not well spent.

Peace

If one claimed that the definition of 'equinumerous' must lead to a definition of cardinal subtraction, moreover a requirement that a definition of cardinal subtraction adheres in certain ways to integer subtraction, then one would be mistaken.

But to be positive about this, I would suggest: What definition of cardinal subtraction does anyone here have in mind? That is to provide the formula P in the following (where P has no free variables other than x, y and z; and for each <x y> there is a unique z such that P):

If x and y are cardinals, then x-y = z <-> P.