Thanks, I appreciate the correction.

What is your mathematical definition of 'infinites'?

Unlike the odd numbers, there is no set of all infinite sets nor of all infintie cardinals. So what are the infinities?

We could define "x is an infinity iff x is infinite'. But then 'is an infinity' is a predicate and doesn't stand for a quantity.

A cardinality is a quantity. But 'is infinite' is an adjective.

It's not a merely pedantic distinction. Ignoring the distinction causes real confusions about sets and set theory. Ignoring the distinction is typical of cranks (not you) who know nothing about set theory but try to refute it with incoherent arguments that conflate 'is infinite' as if 'infinity' is a noun.

Infinities - all of them - are cardinalities. — Banno

Actually all the infinities are ordinals. Even the cardinals are ordinals these days, though they didn't use to be. That was all explained in the post I wrote that you were kind enough to thank me for, while announcing that you weren't going to read it. It's true that most people have heard about the transfinite cardinals and not the ordinals, but FWIW, ordinals are logically prior to cardinals, in the modern formulation.

↪fishfry OK, I'm not reading all that... but thank you. — Banno

LOL. But thank you for saying thanks!

Actually all the infinities are ordinals. Even the cardinals are ordinals these days, though they didn't use to be. That was all explained in the post I wrote that you were kind enough to thank me for, while announcing that you weren't going to read it. It's true that most people have heard about the transfinite cardinals and not the ordinals, but FWIW, ordinals are logically prior to cardinals, in the modern formulation. — fishfry

Can you put a time for it when this change happened?

Sure, infinite oceans are impossible. — Banno

why waste your time trying to solve non existent problems

can god make a square circle? its not a limit of god. its just wordplay

infinite ocean is just wordplay. it doesn't eixst. therefore you dont need to solve any questions about it

Can you put a time for it when this change happened? — ssu

1923. Found this in John von Neumann and Hilbert's School of Foundations of Mathematics (pdf link)

"The definition of ordinals and cardinals was given by von Neumann in the paper Zur
Einführung der transniten Zahlen (1923)"

They also mention that von Neumann was still cleaning up his definition in 1928, since definitions by transfinite recursion were on shaky ground in that era.

The modern definition is the von Neumann cardinal assignment. Von Neumann defined a cardinal as the least ordinal having that cardinality.

Prior to that, cardinals were the equivalence class of all sets having that cardinality. The problem was that this was a proper class and not a set, so you couldn't manipulate cardinals using the rules of set theory. Von Neumann's definition defines each cardinal as a particular set, which is more convenient.

can god make a square circle? its not a limit of god. its just wordplay — MikeListeral

The unit circle in the taxicab metric is a square. There's a picture of a square circle on that page. Better to use "married bachelor," because in fact there are square circles!

why waste your time trying to solve non existent problems — MikeListeral

"Good sense about trivialities is better than nonsense about things that matter." -- Quote on a math professor's door that I saw once.

"Good sense about trivialities is better than nonsense about things that matter." -- Quote on a math professor's door that I saw once. — fishfry

That's pretty good. I like it.

is an infinitely expanding ocean fundamentally the same as an infinite ocean?
— Bradaction
Yes, except that it is expanding. — Banno

Perhaps then the question can be asked, are vertical infinity, horizontal infinity and infinity, all potentially different terms that could be given to different types of infinity?

are vertical infinity, horizontal infinity and infinity, all potentially different terms that could be given to different types of infinity? — Bradaction

'is infinte' can be qualified any way you can come up with a definition of your qualifier.

is countably infinite

is uncountaby infinite

is infinte in correspondence with the y axis

is infinite in correspondence with the x axis

Etc.

'is infinte' can be qualified any way you can come up with a definition of your qualifier.

is countably infinite

is uncountaby infinite

is infinte in correspondence with the y axis

is infinite in correspondence with the x axis — TonesInDeepFreeze

Should we then refer to these terms as different types of infinite?

Should we then refer to these terms as different types of infinite? — Bradaction

I wouldn't. I would say they are different predicates of the form: x is infinite & Rx.

The modern definition is the von Neumann cardinal assignment. Von Neumann defined a cardinal as the least ordinal having that cardinality. — fishfry

Isn't this circular? Doesn't "least" already imply cardinality, such that cardinality is already inherent within the ordinals, to allow the designation of a least ordinal? Then the claim that ordinals are logically prior to cardinals would actually be false, because more and less is already assumed within "ordinal".

I think it'll help if we make a distinction if we're to make any headway on this issue.

First let's begin with finite sets:

A = {w, o, r, k}

A has 4 elements

Take away the element w: A - {w} = B

and

I'm left with {o, r, k}

Element w was taken away. Effects:

1. The set B doesn't have the element w

2. The cardinality of B (3) is less than the cardinality of A (4)

The missing element in B produces a corresponding decrease in the cardinality of B (from 4 elements in A to 3 elements in B).

Let's now look at infinite sets:

N = {1, 2, 3,...}

Take away 1 as in, N - {1} = M = {2, 3, 4,...}

Effects:

3. M is missing the element 1

4. The cardinality of N = The cardinality of M = Infinity

The missing element in M (which is 1) fails to produce an effect on the cardinality of M.

See the difference?

Consider the fact that

A. Oceans aren't defined in terms of unions of droplets.

This means that atomically constructive definitions of oceans in terms of merging droplets together is irrelevant in terms of the logical characterisation of an ocean that assumes no physics. To mathematically define an ocean is to write it down instantaneously without constraining it's size.

B. Oceans are potentially infinite in terms of their number of droplets, but are not actually infinite.

This means that

1) An ocean is Dedekind-finite; there does not exist a constructable bijection between any number of droplets extracted from the ocean and a proper subset of those droplets.

2) An ocean is not specifiable a priori as a finite object in the sense that there is no a priori specifiable upper-bound on the number of droplets that can be extracted from it. In other words, an ocean, apriori, isn't equivalent to any a finite subset of droplets extracted from it. In mathematical parlance, oceans are therefore Kuratowski-infinite, like an infinite-loop in a computer program that isn't a priori equivalent to any finite number of loop iterations.

Together, 1 and 2 necessitate the rejection of the Axiom of Countable Choice, since that axiom forces all non-finite sets to be dedekind infinite.

Oceans are streams in a type-theoretical sense, which are lazily-evaluated lists

Ocean (0) = Ocean (no droplets so far extracted)
Ocean ( n) := [ droplet (n+1), Ocean (n+1) ] (n+1 droplets so far extracted)


Therefore we can say Ocean(0) > Ocean(1) > Ocean (2) .... without assigning a definite quantity to Ocean (0) and its predecessors, and without assuming that Ocean(i) is evaluated for all i, in the sense that only when we draw a droplet from ocean (i) does ocean (i) expand into [droplet(i+1), ocean (i +1) ].

And when the ocean eventually runs dry, our non-standard mathematical specification that is consciously aware of an a priori/ a posteriori distinction in mathematical meaning, isn't contradicted by reality, unlike in the case of classical set theory that in appealing to AC equivocates the a priori with the a posteriori.

Infinite is a quality, not a quantity.
— Possibility

Tell that to a mathematician. — Banno

OK, I will...

...Yep, I stand by my statement.

Isn't this circular? — Metaphysician Undercover

No, although it's slightly tricky. We are distinguishing between two sets having the same cardinality -- meaning that there is a bijection between them -- and assigning them a cardinal -- a specific mathematical object that can represent their cardinality.

Doesn't "least" already imply cardinality, — Metaphysician Undercover

No, "least" is in terms of ordinality, not cardinality.

such that cardinality is already inherent within the ordinals, — Metaphysician Undercover

Well it is, I agree with that. Take for example the ordinals $\omega$ and $\omega + 1$. They have the same cardinality, as they can be represented by two distinct orderings of the same set, as I endeavored to explain to you in the other thread. But they are different ordinals, with $\omega \lt \omega + 1$.

So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. In the old days, before von Neumann, we identified a cardinal number with the class of all sets having that cardinality. After von Neumann, we identified a cardinal with the least ordinal of all the ordinals having that cardinality. The benefit is that the latter definition makes a cardinal into a particular set; whereas the former definition is a class (extension of a predicate) but not a set.

to allow the designation of a least ordinal? — Metaphysician Undercover

Any nonempty collection of ordinals always has a least member, by the definition and construction of ordinals.

Then the claim that ordinals are logically prior to cardinals would actually be false, because more and less is already assumed within "ordinal". — Metaphysician Undercover

Not at all. Not "more or less," but "prior in the order," if you prefer more accurate verbiage.
You insist on conflating order with quantity, and that's an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. I can't do anything about your refusal to recognize the distinction between quantity and order.

What is your mathematical definition of 'infinites'? — TonesInDeepFreeze

Mine?

I just go to Wolfram: an unbounded quantity that is greater than every real number.

That way I can share the blame.

It's on my "to do" list.

It's on my "to do" list. — Banno

No prob, I regretted not adding a smiley to my earlier post. I didn't expect anyone to read all that, but the tl;dr is that ordinals are an important class of transfinite numbers even though fewer people have heard of them than have heard of cardinals. Smileys to make up for previous omission. :-) :-) :-) :-) :-)

Yes, there are points of infinity on the extended real line. So if by 'infinities' we mean such points and others in different number systems and such, fine.

But you said they are cardinalities. It is not required that such points have infinite cardinality, though in some treatments they might. Cardinality is a different subject.

That Wolfram article is poorly conceived.

It defines 'infinity' in the sense of points such as on the real line.

But then it mentions infinity with regard to infinite sets. Notice that 'infinite' is the word there, not 'infinity'.

'infinity' is a noun. So it is a name for a certain object, such as the point of positive infinity on the extended real line. Note that that point does not itself have to be an infinite set, even if in some treatments it may be.

'infinite' is an adjective not a noun. It is not name of a certain object and so it is not the name of a certain cardinality. Rather, it is property of certain sets and a property of certain cardinalities.

The Wolfram article sets up confusion by glibly conflating 'infinity' with 'infinite'.

As I've said previously in this thread and elsewhere, and now again:

'infinity' - a noun - does not refer to a cardinality. It couldn't even do that, since there are many infinite cardinalities. But 'infinity' may refer to things like a point on the extended real line.

What refers to cardinality is 'infinite' - an adjective. The predicate 'is infinite' applies to sets, as a set either is or is not infinite.

Wolfram uses "real", which I suppose is better than "cardinal" or "Ordinal" a
s a definition.

But it's authoritative that it's a quantity, not a quality, contra . So I'm worried by

a quality of sorts. — jgill

.

I'm not wanting to contradict our resident mathematician.

That Wolfram article is poorly conceived. — TonesInDeepFreeze

I think the expectation is that folk will look at the related articles for more detail.

Or alternately, I'll have to disavow it's authority, which I am loath to do.

Wolfram uses "real", which I suppose is better than "cardinal" or "Ordinal" a
s a definition. — Banno

The $\pm \infty$ of the extended real numbers are not the same as the transfinite ordinals and cardinals. The extended reals are the standard reals with two meaningless symbols $\pm \infty$ adjoined, and given certain formal properties such as $a + \infty = \infty$ and so forth, entirely for the purpose of being able to say things like, "as x goes to infinity" rather than, "as x gets arbitrarily large." The extended reals are a notational convenience in calculus and integration theory. They should not be confused with the transfinite ordinals and cardinals. I didn't look at the Wolfram article but if they contributed to this confusion, then bad Wolfram!

ps -- Ok I looked at the article. First they start out by talking about infinity as one of the points adjoined to the real line to make the extended reals. Then they casually conflate this to Cantor's work.

Bad Wolfram. Bad article.

it's a quantity, not a quality — Banno

What does 'it' refer to there?

I think the expectation is that folk will look at the related articles for more detail. — Banno

That's a terrible excuse. One shouldn't initiate further study by first publishing a dictionary entry that conflates important concepts.

wo meaningless symbols — fishfry

I agree with the basics in your post.

One technical point though:

Yes, in many (probably most or even just about all) writings, the points of infinity are just arbitrary points, and they are not specified to be any particular mathematical objects. But in some treatments, the points are specified to be certain objects, so that the set of reals with extensions is a definite set.

Infinity.

Infinity. — Banno

It is not required that the extension points have infinite cardinality.

That's a terrible excuse. One shouldn't initiate further study by first publishing a dictionary entry that conflates important concepts. — TonesInDeepFreeze

So what would your replacement be?

But in some treatments, the points are specified to be certain objects, so that the set of reals with extensions is a definite set. — TonesInDeepFreeze

I did not think this was an appropriate context in which to mention the two-point compactification of the real line. Do you? You must have driven your teachers crazy. That's ok, I did too.