∞ is a free variable

[Magnus Anderson]

Is that what you want it to mean?

Don't project. That's how everyone defines it. And yes, it is a number.

[Skepdick]

Is that what you want it to mean?

Don't project. That's how everyone defines it. And yes, it is a number.

Maybe you want to sit down and have a chat with wtf about that. You two seem to be getting close.

[Skepdick]

I am a heretic in any Orthodoxy, and I have zero tolerance for your intolerance.

You are a thought and discussion cancer.

You get all the bonus points for irony.

I show you a way of thinking free from normatives. And you hate it.

[promethean75]

Since there are no actual infinities in nature, as far as we know, we have that problem of using the word infinity in nonsensical ways, e.g., describing an unending or open set.

Along the lines of something your boy Witt said (not verbatim), the word use means only the extension of a rule; add one to the set, add another to the set, etc., but not a sense of the number of things. One doesn't describe anything with the predicate 'infinite' except a use or extension of the rule of adding.

[wtf]

Along the lines of something your boy Witt said (not verbatim), the word use means only the extension of a rule; add one to the set, add another to the set, etc., but not a sense of the number of things. One doesn't describe anything with the predicate 'infinite' except a use or extension of the rule of adding.

Actually not FWIW. To be sure, infinity is not a physical concept (as far as we currently know), but it's most definitely important in mathematics.

https://en.wikipedia.org/wiki/Transfinite_number

To be fair, the transfinite ordinals and cardinals are not the symbol ∞, which generally refers only to the extra points of the extended real number system.

But infinite sets, and symbols that refer to specific infinite numbers, are an essential part of math, especially set theory.

The predicate "infinite" refers to

(a) A set that is not finite; where a set is finite if its members can be placed into bijective correspondence with some natural number; or

(b) A set that can be placed into bijective correspondence with a proper subset of itself. (This condition is called Dedekind-infinite).

The two conditions are equivalent in the presence of the axiom of choice; but without choice, there are sets that are infinite yet Dedekind-finite.

See https://en.wikipedia.org/wiki/Dedekind-infinite_set

[Skepdick]

The predicate "infinite" refers to

(a) A set that is not finite...

This presupposes excluded middle where Finite and infiite are complementary notions.

NOT finite --(means)---> Infinite
NOT infinite ---(means)---> Finite

What about neither inifinite nor finite sets?

The two conditions are equivalent in the presence of the axiom of choice

And the axiom of choice implies excluded middle.

https://en.wikipedia.org/wiki/Diaconescu%27s_theorem

but without choice, there are sets that are infinite yet Dedekind-finite.

See https://en.wikipedia.org/wiki/Dedekind-infinite_set

The definition of Dedekind finite still uses choice/excluded middle...

A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists).

So back to my question.. What about neither (Dedekind) finite nor (Dedekind) infinite sets?

[Skepdick]

One doesn't describe anything with the predicate 'infinite' except a use or extension of the rule of adding.

This is immaterial. A (boolean) predicate determines truth-value.

Mortal(Socrates) -> True
Immortal(Socrates) -> False
Even(2) -> True
Even(3) -> False

The same goes for the predicate "infinite"

Infinite(X) -> {True, False}
Infinite({1,2,3}) -> False
Infinite({1,2,3,...}) -> Supposedly True

The crux of the matter is that if a predicate can determine that something is, in fact, infinite that means the determination of its "infiniteness" was made in finite time. Que?

Does Infinite(ℕ) perform a successful determination? You decide! That's a philosophical choice.

My philosophical choice is that the natural numbers don't form a set. They are a stream of data which can be generated on demand but never comes to an end, and so Infinite(ℕ) doesn't halt.

[wtf]

Quoting me: "(a) A set that is not finite..."

This presupposes excluded middle where Finite and infiite are complementary notions.

It's a definition. Nothing whatsoever to do with excluded middle. We define what it means for a set to be finite. If it doesn't happen to be finite, we say it's infinite.

Your monomania regarding the only thing you know causes you to fall into error.

If a set isn't finite we call it infinite. This has absolutely nothing to do with excluded middle. You can't see that because somewhere along the line you learned one thing or read one Wiki page and you just harp on that tedious topic whether it applies to the subject at hand or not.

What about neither inifinite nor finite sets?

There is no such thing by the definition I gave. If a set is not finite it's infinite. It's a definition. It's entirely independent of whether we have excluded middle or not. You can't see that because somewhere along the line your mind fell into a channel of one single thought that you repeat to yourself obsessively whether it applies or not.

And the axiom of choice implies excluded middle.

https://en.wikipedia.org/wiki/Diaconescu%27s_theorem

True, but what has that got to do with anything under discussion?

A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists).

You QUOTED SOMETHING I DIDN'T WRITE! Do you have any f*cking ethics at all? If that was from the Wiki article you needed to say so. If it wasn't ... why did you quote it?

So back to my question.. What about neither (Dedekind) finite nor (Dedekind) infinite sets?

It's apparently impossible for you to grasp a simple definition. If a set is not finite then it's infinite. By definition.

How did you get like this? Are you simply incapable of focussing on anything in the world outside of your monomania? Did you read a Wiki page one day and say "That's it, I never need to understand anything else in the world." Congratulations.

It's a DEFINITION. A set is finite if it bijects to a natural number (taking the natural numbers to be their von Neumann encoding as sets). If a set doesn't happen to be finite we call it infinite. Your monomania precludes you from grasping such a simple definition.

[Magnus Anderson]

This presupposes excluded middle where Finite and infiite are complementary notions.

It does not presuppose it. The middle is excluded by definition.

A set is said to be finite if its size is equal to an integer ( specifically, to an integer that is greater than or equal to zero, since a negative integer is not a valid size for a set. )

A set is said to be infinite if its size is equal to a number greater than every integer.

The two categories cover every logically possible / conceivable number of elements in a set.

As such, no sets that fall outside of the categories "finite sets" and "infinite sets" exist.

[Magnus Anderson]

Since there are no actual infinities in nature, as far as we know, we have that problem of using the word infinity in nonsensical ways, e.g., describing an unending or open set.

Well, if you believe that time and space are infinite in all directions and / or that they are infinitely divisible, then you also believe that infinite quantities exist in the world.

But the more important point is that it's irrelevant. Whether or not infinite quantities exist has nothing to do with whether or not the term "infinite quantity" is a meaningful one.

Unicorns and dragons do not exist yet the two terms are perfectly meaningful.

A term is meaningful insofar it has a meaning assigned to it. And the meaning of a term is simply the set of all conceivable things that the term can be used to represent.

The term "infinite quantity" simply means "a quantity larger than every integer".

Saying that infinity is not a number is a little futile given that anything that can validly, not necessarily truly, answer the question "How many people there are in the world?" is a number.

"Cat" is not a valid answer.
"Long" is not a valid answer.
"Big" is not a valid answer.
"Five" is a valid answer ( so it's a number. )
"A million" is a valid answer ( so it's a number. )
"An infinite number" is a valid answer ( so it's a number. )

[Skepdick]

Quoting me: "(a) A set that is not finite..."

This presupposes excluded middle where Finite and infiite are complementary notions.

It's a definition. Nothing whatsoever to do with excluded middle.

Do you understand what an inference rule is in logic? Yes? No?

Your definition implicitly entails precisely the following inference rule about the set X:

¬Finite (X) -> Infinite(X)
¬Infinite (X) -> Finite(X)

Surely you know what an involution is?

f(x) = y
f(y) = x

How is it that you don't recognize that as just another re-formulation of Excluded Middle?!?

The fommulatio of LEM is: P ∨ ¬ P

Let x:= P
let y:= ¬ P

¬(x) = y e.g ¬(P) => ¬ P
¬(y) = x e.g ¬(¬ P) => P

This the double negation elimination principle in classical logic.

It's so deeply wired into your psychology as a Mathematician that you can't even recognize it all over your definitions.

We define what it means for a set to be finite. If it doesn't happen to be finite, we say it's infinite.

Your monomania regarding the only thing you know causes you to fall into error.

If a set isn't finite we call it infinite. This has absolutely nothing to do with excluded middle.

You can't see that because somewhere along the line you learned one thing or read one Wiki page and you just harp on that tedious topic whether it applies to the subject at hand or not.

I can only explain it to you. I can't understand it for you.

I am sorry you find understanding to be so tedious.

The "only thing I know" is one more thing than you know about.

You know about finite and infinite sets.
I know about finite, infinite and neither finite nor infinite sets.

There is no such thing by the definition I gave. If a set is not finite it's infinite. It's a definition.

Yes, your definition fails to account for such sets. I understand your definition. Do you understand its limits?

Your definition is literally a false dichotomy!

It's entirely independent of whether we have excluded middle or not.

It's not.... it's precisely beause Finite and Infinite are seen as complimentary to each other (e.g negating the one gives you the other) is
why you can't account for neither finite nor infinite sets in your Mathematics.

You can't see that because somewhere along the line your mind fell into a channel of one single thought that you repeat to yourself obsessively whether it applies or not.

It applies. Can't you see it? I am trying to help you see it...

True, but what has that got to do with anything under discussion?

It has to do everything with the discussion.

if not Finite means Infinite; and not Infinite means Finite you are implicitly operating in a paradigm in which LEM/choice holds.

You QUOTED SOMETHING I DIDN'T WRITE! Do you have any f*cking ethics at all? If that was from the Wiki article you needed to say so. If it wasn't ... why did you quote it?

Spare me the moral outrage.

Is it a quote? Yes.
Is it a quote of you? No.

How can you tell? Because there's no explicit attribution to you.

See the difference be between this:

and this

It's apparently impossible for you to grasp a simple definition. If a set is not finite then it's infinite. By definition.

If it's so "impossible" for me to grasp it then how is it that I am grasping it?

There exist neither infinite nor finite sets.

In choosing to define Infinite and Finite as complementary notions (e.g by using Excluded Middle) your definition fails to account for the existence of such Mathematical objects.

How did you get like this? Are you simply incapable of focussing on anything in the world outside of your monomania? Did you read a Wiki page one day and say "That's it, I never need to understand anything else in the world." Congratulations.

I got like this by learning to understand the consequences of choices.

It's a DEFINITION. A set is finite if it bijects to a natural number (taking the natural numbers to be their von Neumann encoding as sets). If a set doesn't happen to be finite we call it infinite. Your monomania precludes you from grasping such a simple definition.

I understand your definition just fine.

I am asking you to explain what you call a set that happens to be neither finite nor happens to be infinite.

What is preventing you from answering a simple damn question?

[Skepdick]

It does not presuppose it. The middle is excluded by definition.

And I reject this exclusion. By definition.

As such, no sets that fall outside of the categories "finite sets" and "infinite sets" exist.

It follows trivially and by definition that the rejection of excluded middle amounts precisely to the following:

not(Excluded MIddle) -> not(Either(Finite, Infinite)) -> neither finite nor infinite

[Magnus Anderson]

And I reject this exclusion. By definition.

You can reject it all you want, it won't change a thing. The middle will still be excluded.

Reality does not give a damn about what you think.

As such, no sets that fall outside of the categories "finite sets" and "infinite sets" exist.

It follows trivially and by definition that the rejection of excluded middle amounts precisely to the following:

not(Excluded MIddle) -> not(Either(Finite, Infinite)) -> neither finite nor infinite

It follows trivially that rejecting the excluded middle amounts to insanity . . .

[Skepdick]

You can reject it all you want, it won't change a thing. The middle will still be excluded.

You can object all you want. I am rejecting its exclusion anyway.

Reality does not give a damn about what you think.

Reality doesn't give a damn about what you think either.

And neither do I.

It follows trivially that rejecting the excluded middle amounts to insanity . . .

OK. Then I am insane.

It seems to me being insane-yet-right is preferable to being sane-yet-wrong.

theorem.png

[Magnus Anderson]

You can reject it all you want, it won't change a thing. The middle will still be excluded.

You can object all you want. I am rejecting its exclusion anyway.

It has become fashionable, and it is particular so on this forum, to be fanatical about trying to be clever by mindlessly rejecting every widely accepted truth regardless of how basic it is.

A is not A.
1 isn't equal to 1.
2 + 2 is not 4.
Square-circles aren't oxymorons.
And so on.

This place is full of such brilliance.